Finite Group Databasebeta

← #15889452

Subgroups — #15889452

Order 3328.

732 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#88139C211yesmaximal, characteristic
2#13115775C211yesmaximal, characteristic, complemented
3#13115774C211yesmaximal, characteristic, complemented
4#13115452C211yesmaximal, characteristic, complemented
5#13115451C211yesmaximal, characteristic, complemented
6#13115450C211yesmaximal, characteristic, complemented
7#13115447C211yesmaximal, characteristic, complemented
8#13115133C211yesmaximal, characteristic
9#13115130C211yesmaximal, characteristic
10#13115019C211yesmaximal, characteristic, complemented
11#13115017C211yesmaximal, characteristic, complemented
12#13115017C211yesmaximal, characteristic, complemented
13#13114952C211yesmaximal, characteristic, complemented
14#13114951C211yesmaximal, characteristic, complemented
15C13×#1851C211yesmaximal, characteristic, complemented
16C22×C13×M16C2211yescharacteristic
17C2×C4×C22⋊D13C2211yescharacteristic
18C22×#6094C2211yescharacteristic
19C2×#88139C411yescharacteristic
20C2×C4×Dic26C2211yescharacteristic
21C2×C13×C4⋊C8C2211yescharacteristic
22C2×C13×C4⋊C8C2211yescharacteristic
23C2×C13×C22⋊C8C2211yescharacteristic
24C4×C13×C22⋊C4C2211yescharacteristic
25C4×C22⋊Dic13C2211yescharacteristic
26C2×C22⋊#1055C2211yescharacteristic
27C2×C22⋊#1055C2211yescharacteristic
28C2×#88060C2211yescharacteristic
29C2×#88060C2211yescharacteristic
30C2×#88060C2211yescharacteristic
31C2×#88055C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×#88054C2211yescharacteristic
34C2×#88054C2211yescharacteristic
35#11718618C411yescharacteristic
36#11718612C411yescharacteristic
37#11718605C411yescharacteristic
38C4×#6103C2211yescharacteristic
39C4×#6103C2211yescharacteristic
40C42×Dic13C2211yescharacteristic
41C2×C4⋊#1055C2211yescharacteristic
42C2×#88043C2211yescharacteristic
43#11718331C411yescharacteristic
44#11718330C411yescharacteristic
45#11718311C411yescharacteristic
46#11718307C411yescharacteristic
47C13×#336C2211yescharacteristic
48C13×#336C2211yescharacteristic
49#11718142C2211yescharacteristic
50#11718142C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C22×C22⋊D13C2×C411yescharacteristic
61C22×C4×D13C2311yescharacteristic
62C22×Dic26C2×C411yescharacteristic
63C2×C13×M16—41no—
64C2×C13×M16—41no—
65C2×C13×M16—41no—
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C2×C13×C22⋊C4C2×C411yescharacteristic
71C2×C13×C22⋊C4C2×C411yescharacteristic
72C2×C42×C13C2311yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×C22⋊Dic13C2×C411yescharacteristic
75C2×C22⋊Dic13C2×C411yescharacteristic
76C2×C22⋊Dic13C2×C411yescharacteristic
77C4×C22⋊D13—41no—
78C4×C22⋊D13—41no—
79C2×#6103C2×C411yescharacteristic
80C2×#6103C2×C411yescharacteristic
81C2×#6103C2×C411yescharacteristic
82C2×#6103C2×C411yescharacteristic
83C2×#6103C2×C411yescharacteristic
84C2×C4⋊Dic13C2×C411yescharacteristic
85C2×#6101C2×C411yescharacteristic
86C2×#6101C2×C411yescharacteristic
87C2×#6101C2×C411yescharacteristic
88C2×C4×Dic13C2×C411yescharacteristic
89C2×C4×Dic13C2×C411yescharacteristic
90C2×C4×Dic13C2×C411yescharacteristic
91C2×C4×Dic13C2×C411yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C2×#6094—41no—
100C2×#6094—41no—
101C2×#6094—41no—
102#88139—24no—
103C4×Dic26—41no—
104C4×Dic26—41no—
105C13×C4⋊C8—41no—
106C13×C4⋊C8—41no—
107C13×C22⋊C8—41no—
108C13×#95C2×C411yescharacteristic
109C13×#95C2×C411yescharacteristic
110#88071C2×C411yescharacteristic
111#88071C2×C411yescharacteristic
112#88071C2×C411yescharacteristic
113#88071C2×C411yescharacteristic
114#88071C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116C22⋊#1055—41no—
117C22⋊#1055—41no—
118#88060—41no—
119#88060—41no—
120#88060—41no—
121#88055—41no—
122#88054—41no—
123#88054—41no—
124#88054—41no—
125C4⋊#1055—41no—
126#88043—41no—
127#23353—131nomaximal, Sylow
128C24×C13C22×C411yescharacteristic
129C23×D13C22×C411yescharacteristic
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13C22×C411yescharacteristic
137C22×C4×C13C22×C411yescharacteristic
138C22×C4×C13C22×C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×C4×C13C22×C411yescharacteristic
141C22×C4×C13C2411yescharacteristic
142C2×C22⋊D13—22no—
143C2×C22⋊D13—41no—
144C2×C22⋊D13—41no—
145C22×Dic13C22×C411yescharacteristic
146C22×Dic13C22×C411yescharacteristic
147C22×Dic13C22×C411yescharacteristic
148C22×Dic13C22×C411yescharacteristic
149C22×Dic13C22×C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C2×C4×D13—21no—
153C2×C4×D13—21no—
154C2×C4×D13—21no—
155C2×C4×D13—21no—
156C2×C4×D13—21no—
157C2×C4×D13—21no—
158C2×Dic26—22no—
159C2×Dic26—41no—
160C2×Dic26—41no—
161C13×M16—44no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13—21no—
171C2×C8×C13—21no—
172C2×C8×C13—21no—
173C2×C8×C13—21no—
174C13×C22⋊C4—22no—
175C13×C22⋊C4—41no—
176C42×C13—21no—
177C42×C13—21no—
178C42×C13—21no—
179C42×C13—21no—
180C22⋊Dic13—22no—
181C22⋊Dic13—41no—
182C22⋊Dic13—41no—
183#6103—22no—
184#6103—22no—
185#6103—41no—
186#6103—41no—
187#6103—41no—
188C4⋊Dic13—41no—
189#6101—41no—
190#6101—41no—
191#6101—41no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—21no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—41no—
199C4×Dic13—41no—
200C4×Dic13—41no—
201C4×Dic13M1614yescomplemented
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—21no—
213C2×#1055—21no—
214#6094—44no—
215C4×#124—131no—
216#1985—131no—
217#1985—131no—
218#1924—131no—
219#1924—131no—
220#1924—131no—
221#1924—131no—
222#1870—131no—
223#1870—131no—
224#1851—131no—
225#1851—131no—
226#1851—131no—
227#1851D1311yescharacteristic, complemented
228#1831—131no—
229#1831—131no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13C23×C411yescharacteristic
238C22×D13—21no—
239C22×D13—21no—
240C22×D13—21no—
241C22×D13—21no—
242C22×D13—21no—
243C22×D13—21no—
244C22×D13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13C2×M1612yes—
264C2×C4×C13H(5,𝔽2)11yescharacteristic
265C2×C4×C13H(5,𝔽2)11yescharacteristic
266C2×C4×C13H(5,𝔽2)11yescharacteristic
267C2×C4×C13H(5,𝔽2)11yescharacteristic
268C2×C4×C132−1+411yescharacteristic
269C2×C4×C132−1+411yescharacteristic
270C22⋊D13—44no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—21no—
286C2×Dic13—21no—
287C2×Dic13—21no—
288C2×Dic13—21no—
289C2×Dic13—21no—
290C2×Dic13C2×M1612yes—
291C2×Dic13C2×M1612yes—
292C4×D13—41no—
293C4×D13—41no—
294C4×D13—41no—
295C4×D13—41no—
296Dic26—44no—
297C8×C13—41no—
298C8×C13—41no—
299C8×C13—41no—
300C8×C13—41no—
301#1055—41no—
302#1055—41no—
303#1055—41no—
304#1055—41no—
305C22×M16—131no—
306C22×M16C2×D1311yescharacteristic
307C2×#124—131no—
308C2×C4×Q8—131no—
309C2×C4×D4—131no—
310C2×C4⋊C8—131no—
311C2×C4⋊C8—131no—
312C2×C4⋊C8—131no—
313C2×C4⋊C8—131no—
314C2×C4⋊C8C2×D1311yescharacteristic
315C2×C4⋊C8C2×D1311yescharacteristic
316C2×C22⋊C8—131no—
317C2×C22⋊C8—131no—
318C2×C22⋊C8—131no—
319C2×C22⋊C8—131no—
320C2×C22⋊C8—131no—
321C2×C22⋊C8C2×D1311yescharacteristic
322C2×#97—131no—
323C2×#97—131no—
324#391—131no—
325#390—131no—
326#388—131no—
327#388—131no—
328#388—131no—
329#388—131no—
330#382—131no—
331C4×C22⋊C4—131no—
332C4×C22⋊C4—131no—
333C4×C22⋊C4—131no—
334C4×C22⋊C4C2×D1311yescharacteristic
335C43—131no—
336#336—131no—
337#336—131no—
338#336—131no—
339#336—131no—
340#336—131no—
341#336—131no—
342#336—131no—
343#336—131no—
344#336—131no—
345#336—131no—
346#336C2×D1311yescharacteristic
347#336C2×D1311yescharacteristic
348C22×C13—41no—
349C22×C13—41no—
350C22×C13—41no—
351C22×C13—41no—
352C22×C13—41no—
353C22×C13—41no—
354C22×C13—41no—
355C22×C13#51811yescharacteristic
356C22×C13#52011yescharacteristic
357C22×C13#52011yescharacteristic
358C22×C13C22×M1611yescharacteristic
359C22×C13#56811yescharacteristic
360C22×C13#56811yescharacteristic
361C22×C13#56811yescharacteristic
362C2×D13—41no—
363C2×D13—41no—
364C2×D13—41no—
365C2×D13—41no—
366C2×D13—41no—
367C2×D13—41no—
368C2×D13—41no—
369C4×C13—22no—
370C4×C13—41no—
371C4×C13—41no—
372C4×C13—41no—
373C4×C13—41no—
374C4×C13H(3,𝔽4)11yescharacteristic
375C4×C13#56211yescharacteristic
376C4×C13#56211yescharacteristic
377C4×C13#56311yescharacteristic
378Dic13—22no—
379Dic13—22no—
380Dic13—41no—
381Dic13—41no—
382Dic13—41no—
383Dic13—41no—
384Dic13—41no—
385C22×Q8—131no—
386C22×D4—131no—
387C23×C4—131no—
388C23×C4C22×D1311yescharacteristic
389C2×M16—41no—
390C2×M16—41no—
391C2×M16—41no—
392C2×M16—521no—
393C2×M16—521no—
394C2×M16—521no—
395C22×C8—131no—
396C22×C8—131no—
397C22×C8—131no—
398C22×C8—131no—
399C22×C8C22×D1311yescharacteristic
400C22×C8C22×D1311yescharacteristic
401C22×C8C22×D1311yescharacteristic
402C22×C8C22×D1311yescharacteristic
403#124—264no—
404C4×Q8—521no—
405C4×Q8—521no—
406C4×D4—521no—
407C4×D4—521no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—131no—
411C2×C4⋊C4—131no—
412C2×C22⋊C4—131no—
413C2×C22⋊C4—131no—
414C2×C22⋊C4—131no—
415C2×C22⋊C4—131no—
416C2×C22⋊C4—131no—
417C2×C22⋊C4—131no—
418C2×C22⋊C4—131no—
419C2×C22⋊C4—131no—
420C2×C22⋊C4C4×D1311yescharacteristic
421C2×C22⋊C4C4×D1311yescharacteristic
422C2×C42—131no—
423C2×C42—131no—
424C2×C42—131no—
425C2×C42—131no—
426C2×C42—131no—
427C2×C42—131no—
428C2×C42—131no—
429C2×C42C22×D1311yescharacteristic
430C2×C42C22×D1311yescharacteristic
431C4⋊C8—41no—
432C4⋊C8—41no—
433C4⋊C8—521no—
434C4⋊C8—521no—
435C4⋊C8—521no—
436C4⋊C8—521no—
437C22⋊C8—41no—
438C22⋊C8—521no—
439C22⋊C8—521no—
440C22⋊C8—521no—
441C22⋊C8—521no—
442C22⋊C8—521no—
443#97—521no—
444#97—521no—
445#95—131no—
446#95—131no—
447#95—131no—
448#95—131no—
449#95—131no—
450#95—131no—
451#95C4×D1311yescharacteristic
452#95C4×D1311yescharacteristic
453C2×C13—81no—
454C2×C13#237911yescharacteristic
455C2×C13#297511yescharacteristic
456C2×C13#298011yescharacteristic
457C2×C13#298911yescharacteristic
458C2×C13#298911yescharacteristic
459C2×C13#299511yescharacteristic
460C2×C13#299511yescharacteristic
461D13—81no—
462C24—131no—
463C24C2×C4×D1311yescharacteristic
464C2×Q8—262no—
465C2×Q8—521no—
466C2×Q8—521no—
467C2×D4—262no—
468C2×D4—521no—
469C2×D4—521no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—131no—
477C22×C4—131no—
478C22×C4—131no—
479C22×C4—131no—
480C22×C4—131no—
481C22×C4—131no—
482C22×C4—131no—
483C22×C4—261no—
484C22×C4—261no—
485C22×C4—261no—
486C22×C4—261no—
487C22×C4—261no—
488C22×C4—261no—
489C22×C4C2×C4×D1311yescharacteristic
490C22×C4C2×C4×D1311yescharacteristic
491C22×C4C2×C4×D1311yescharacteristic
492C22×C4C2×C4×D1311yescharacteristic
493C22×C4C2×C4×D1311yescharacteristic
494C22×C4C23×D1311yescharacteristic
495M16—44no—
496M16—524no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—261no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8—261no—
513C2×C8—261no—
514C2×C8—261no—
515C2×C8—261no—
516C2×C8—261no—
517C2×C8—261no—
518C2×C8—261no—
519C2×C8—261no—
520C2×C8—261no—
521C4⋊C4—521no—
522C4⋊C4—521no—
523C4⋊C4—521no—
524C4⋊C4—521no—
525C22⋊C4—22no—
526C22⋊C4—41no—
527C22⋊C4—262no—
528C22⋊C4—262no—
529C22⋊C4—262no—
530C22⋊C4—521no—
531C22⋊C4—521no—
532C22⋊C4—521no—
533C22⋊C4—521no—
534C22⋊C4—521no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—134no—
540C42—261no—
541C42—261no—
542C42—261no—
543C42—261no—
544C42—261no—
545C42—261no—
546C42—521no—
547C42—521no—
548C42—521no—
549C13#2335311yescharacteristic, complemented, Sylow
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—261no—
558C23—261no—
559C23—261no—
560C23—261no—
561C23—261no—
562C23—261no—
563C23—261no—
564C23C22×C4×D1311yescharacteristic
565Q8—524no—
566D4—524no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—132no—
586C2×C4—132no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—261no—
596C2×C4—261no—
597C2×C4—261no—
598C2×C4—261no—
599C2×C4—261no—
600C2×C4—261no—
601C2×C4—261no—
602C2×C4—261no—
603C2×C4—261no—
604C2×C4—261no—
605C2×C4—261no—
606C2×C4—521no—
607C2×C4—521no—
608C2×C4—521no—
609C2×C4—521no—
610C2×C4M16×D1312yes—
611C2×C4#8825111yescharacteristic
612C2×C4#8825111yescharacteristic
613C2×C4#8825111yescharacteristic
614C2×C4#8825411yescharacteristic
615C2×C4#8825611yescharacteristic
616C2×C4#8825711yescharacteristic
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C8—521no—
622C8—521no—
623C8—521no—
624C8—521no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—521no—
633C22—521no—
634C22—521no—
635C22—521no—
636C22—521no—
637C22—521no—
638C22—521no—
639C22#1171920911yescharacteristic
640C22#1171925011yescharacteristic
641C22#1171925111yescharacteristic
642C22C2×M16×D1311yescharacteristic
643C22#1171945011yescharacteristic
644C22#1171945511yescharacteristic
645C22#1171945511yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—262no—
652C4—262no—
653C4—521no—
654C4—521no—
655C4—521no—
656C4—521no—
657C4—521no—
658C4#1171929911yescharacteristic
659C4#1171935111yescharacteristic
660C4#1171935211yescharacteristic
661C4#1171937211yescharacteristic
662C2—81no—
663C2—1041no—
664C2#1311800911yescharacteristic
665C2#1312387411yescharacteristic
666C2#1312390411yescharacteristic
667C2#1312394811yescharacteristic
668C2#1312395011yescharacteristic
669C2#1312400011yescharacteristic
670C2#1312400411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC13×#1851C2
prime radicalC22×C22⋊D13C2×C4
cosocleC22×C4×C13C24
hypercenter#1851D13
hypocenterC13#23353
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ #1851

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #13115775C2semidirect
3. #13115774C2semidirect
4. #13115452C2semidirect
5. #13115451C2semidirect
6. #13115450C2semidirect
7. #13115447C2semidirect
10. #13115019C2semidirect
11. #13115017C2semidirect
12. #13115017C2semidirect
13. #13114952C2semidirect
14. #13114951C2semidirect
15. C13×#1851C2semidirect
201. C4×Dic13M16semidirect
227. #1851D13semidirect
549. C13#23353semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#881392
#131157752
#131157742
#131154522
#131154512
#131154502
#131154472
#131151332
#131151302
#131150192
#131150172
#131149522
#131149512
C13×#18512
#2335313