Finite Group Databasebeta

← #15881160

Subgroups — #15881160

Order 3328.

978 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718789C211yesmaximal, characteristic
2C2×#11718522C211yesmaximal, characteristic, complemented
3#13117766C211yesmaximal, characteristic
4#13117761C211yesmaximal, characteristic
5#13115605C211yesmaximal, characteristic, complemented
6#13115601C211yesmaximal, characteristic, complemented
7#13115035C211yesmaximal, characteristic, complemented
8#13115032C211yesmaximal, characteristic, complemented
9#13115029C211yesmaximal, characteristic, complemented
10#13115027C211yesmaximal, characteristic, complemented
11#13114999C211yesmaximal, characteristic, complemented
12#13114870C211yesmaximal, characteristic, complemented
13#13114869C211yesmaximal, characteristic, complemented
14#106×Dic13C211yesmaximal, characteristic, complemented
15C2×C13×#428C211yesmaximal, characteristic, complemented
16C22×C13×M16C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C2×C4×C22⋊D13C2211yescharacteristic
20C22×#6094C2211yescharacteristic
21C2×#88146C2211yescharacteristic
22C2×C4⋊C4×D13C2211yescharacteristic
23C13×#428—24no—
24C2×C13×#107C2211yescharacteristic
25C2×C13×#107C2211yescharacteristic
26C2×C13×#106C2211yescharacteristic
27C2×C13×#106C2211yescharacteristic
28#11718874—21no—
29#11718873C2211yescharacteristic
30#11718873C2211yescharacteristic
31C4×C22⋊Dic13—21no—
32C2×C22⋊#1055C2211yescharacteristic
33#11718789C418yescomplemented
34C2×#88060C2211yescharacteristic
35C2×#88058C2211yescharacteristic
36C2×#88057C2211yescharacteristic
37C2×#88054C2211yescharacteristic
38C2×C8×Dic13C2211yescharacteristic
39#11718651—21no—
40#11718648C2211yescharacteristic
41#11718648C2211yescharacteristic
42#11718639—21no—
43#11718638C2211yescharacteristic
44#11718638C2211yescharacteristic
45C4⋊C4×Dic13C2211yescharacteristic
46C4⋊C4×Dic13C2211yescharacteristic
47C2×#88048C2211yescharacteristic
48C2×#88048C2211yescharacteristic
49C2×#88048C2211yescharacteristic
50C2×#88047C2211yescharacteristic
51#11718617—21no—
52#11718606—21no—
53C4×#6103—21no—
54#11718594—21no—
55#11718522—24no—
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718121C2211yescharacteristic
62C23×C4×C13—21no—
63C23×C4×C13C2311yescharacteristic
64C22×C22⋊D13D411yescharacteristic
65C22×C4×D13—21no—
66C22×C4×D13C2311yescharacteristic
67C2×C13×M16—22no—
68C2×C13×M16—22no—
69C2×C13×M16C2×C414yescomplemented
70C22×C8×C13C2311yescharacteristic
71C22×C8×C13C2311yescharacteristic
72C13×#117—22no—
73C13×#117—41no—
74C2×C13×C4⋊C4—22no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4—41no—
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C13×C22⋊C4—21no—
82C2×C42×C13—21no—
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13—21no—
85C2×C22⋊Dic13D411yescharacteristic
86C4×C22⋊D13C2×C414yes—
87C4×C22⋊D13Q814yes—
88C2×#6103—21no—
89C2×#6103—21no—
90C2×#6103—21no—
91C2×#6103D411yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×#6101—21no—
97C2×#6101D411yescharacteristic
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C2×#6094—22no—
105C2×#6094—22no—
106C2×#6094C2×C414yescomplemented
107#88146—22no—
108#88146—41no—
109C4⋊C4×D13—22no—
110C4⋊C4×D13—41no—
111C13×#107—22no—
112C13×#107—22no—
113C13×#106—22no—
114C13×#106D414yescomplemented
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125C22⋊#1055C2×C414yescomplemented
126#88060C2×C414yescomplemented
127#88058—22no—
128#88057—22no—
129#88054C2×C414yescomplemented
130C8×Dic13C2×C414yescomplemented
131#88048—22no—
132#88048—41no—
133#88048—41no—
134#88047—22no—
135#22666—131nomaximal, Sylow
136C24×C13C2×D411yescharacteristic
137C23×D13C2×D411yescharacteristic
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—21no—
145C22×C4×C13—21no—
146C22×C4×C13—42no—
147C22×C4×C13—42no—
148C22×C4×C13—81no—
149C22×C4×C13C22×C412yes—
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×Q812yes—
152C22×C4×C13C2411yescharacteristic
153C2×C22⋊D13—41no—
154C2×C22⋊D13—41no—
155C2×C22⋊D13C4⋊C414yes—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
159C22×Dic13—21no—
160C22×Dic13C2×D411yescharacteristic
161C22×Dic13C2×D411yescharacteristic
162C22×Dic13C2×D411yescharacteristic
163C2×C4×D13—21no—
164C2×C4×D13—21no—
165C2×C4×D13—21no—
166C2×C4×D13—21no—
167C2×C4×D13—42no—
168C2×C4×D13—42no—
169C2×C4×D13—81no—
170C2×C4×D13C22×C412yes—
171C2×C4×D13C2×Q812yes—
172C13×M16—28no—
173C2×C8×C13—21no—
174C2×C8×C13—21no—
175C2×C8×C13C22×C412yes—
176C2×C8×C13C22×C412yes—
177C2×C8×C13C2×D412yes—
178C2×C8×C13#4112yes—
179C13×C4⋊C4—21no—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—21no—
184C13×C4⋊C4—21no—
185C13×C4⋊C4—42no—
186C13×C4⋊C4C2×D412yes—
187C13×C4⋊C4C2×D412yes—
188C13×C22⋊C4—42no—
189C42×C13—42no—
190C22⋊Dic13—42no—
191C22⋊Dic13—81no—
192C22⋊Dic13C4⋊C414yes—
193#6103—42no—
194#6103—42no—
195#6103—81no—
196#6103C4⋊C414yes—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—21no—
201C4⋊Dic13—21no—
202C4⋊Dic13—21no—
203C4⋊Dic13—21no—
204C4⋊Dic13—21no—
205#6101—42no—
206#6101C4⋊C414yes—
207C4×Dic13—42no—
208C4×Dic13—81no—
209C4×Dic13—81no—
210C4×Dic13C22×C412yes—
211C4×Dic13C2×Q812yes—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#1055C22×C412yes—
217C2×#1055C22×C412yes—
218#6094—28no—
219C2×#436—131no—
220C2×#428—131no—
221C2×#428D1311yescharacteristic, complemented
222#2352—131no—
223#2352—131no—
224#1949—131no—
225#1949—131no—
226#1854—131no—
227#1854—131no—
228#1853—131no—
229#1853—131no—
230#1846—131no—
231#1809—131no—
232#1809—131no—
233C4×#106—131no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13—21no—
238C23×C13—21no—
239C23×C13—21no—
240C23×C13C2×C4⋊C412yes—
241C23×C13C22×D411yescharacteristic
242C22×D13—21no—
243C22×D13—21no—
244C22×D13—21no—
245C22×D13—21no—
246C22×D13—21no—
247C22×D13—21no—
248C22×D13C2×C4⋊C412yes—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—22no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—84no—
268C2×C4×C13C2×C4⋊C412yes—
269C2×C4×C13C23×C411yescharacteristic
270C2×C4×C13C22×D411yescharacteristic
271C2×C4×C13C22×Q811yescharacteristic
272C2×C4×C13C2×#4111yescharacteristic
273C2×C4×C13H(5,𝔽2)11yescharacteristic
274C2×C4×C132−1+411yescharacteristic
275C22⋊D13—44no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13C2×C4⋊C412yes—
295C2×Dic13C2×C4⋊C412yes—
296C2×Dic13C2×C4⋊C412yes—
297C4×D13—22no—
298C4×D13—22no—
299C4×D13—22no—
300C4×D13—22no—
301C4×D13—84no—
302C8×C13—22no—
303C8×C13C4×D414yes—
304#1055—22no—
305#1055—22no—
306C22×M16—131no—
307C22×M16C2×D1311yescharacteristic
308C2×C4×D4—131no—
309C2×#117—131no—
310C2×#117C2×D1311yescharacteristic
311C22×C4⋊C4—131no—
312C22×C4⋊C4C2×D1311yescharacteristic
313#436—138no—
314#428—24no—
315#428—264no—
316C2×#107—131no—
317C2×#107—131no—
318C2×#107C2×D1311yescharacteristic
319C2×#107C2×D1311yescharacteristic
320C2×#106—131no—
321C2×#106—131no—
322C2×#106—131no—
323C2×#106—131no—
324C2×#106C2×D1311yescharacteristic
325C2×#106C2×D1311yescharacteristic
326C2×C4⋊C8—131no—
327C2×C22⋊C8—131no—
328C2×C22⋊C8—131no—
329C2×C4×C8—131no—
330#388—261no—
331#388—261no—
332#387—261no—
333#387—261no—
334#386—261no—
335#386—261no—
336#382—131no—
337#382—131no—
338#380—131no—
339#380—131no—
340#380—131no—
341#380—131no—
342C4×C4⋊C4—131no—
343C4×C4⋊C4—131no—
344C4×C22⋊C4—261no—
345C4×C22⋊C4—261no—
346#340—131no—
347#340—131no—
348#340—131no—
349#340—131no—
350#340—131no—
351#340—131no—
352C22×C13—22no—
353C22×C13—22no—
354C22×C13—22no—
355C22×C13—41no—
356C22×C13—41no—
357C22×C13—41no—
358C22×C13—41no—
359C22×C13C22×C4⋊C411yescharacteristic
360C22×C13D4211yescharacteristic
361C22×C13#54711yescharacteristic
362C22×C13#57511yescharacteristic
363C22×C13#57511yescharacteristic
364C22×C13#57711yescharacteristic
365C22×C13#57711yescharacteristic
366C2×D13—22no—
367C2×D13—22no—
368C2×D13—22no—
369C2×D13—41no—
370C2×D13—41no—
371C2×D13—41no—
372C2×D13—41no—
373C4×C13—22no—
374C4×C13—42no—
375C4×C13—42no—
376C4×C13—81no—
377C4×C13—81no—
378C4×C13C2×C4×D411yescharacteristic
379C4×C13#52011yescharacteristic
380C4×C13D4×Q811yescharacteristic
381C4×C13#55411yescharacteristic
382Dic13—22no—
383Dic13—22no—
384Dic13—22no—
385Dic13—81no—
386Dic13—81no—
387Dic13—81no—
388Dic13—81no—
389C22×D4—131no—
390C23×C4—21no—
391C23×C4—131no—
392C23×C4—261no—
393C23×C4C22×D1311yescharacteristic
394C2×M16—22no—
395C2×M16—22no—
396C2×M16—134no—
397C2×M16—262no—
398C2×M16—262no—
399C2×M16C4×D1314yescomplemented
400C22×C8—131no—
401C22×C8—131no—
402C22×C8C22×D1311yescharacteristic
403C22×C8C22×D1311yescharacteristic
404C4×D4—134no—
405C4×D4—134no—
406#117—22no—
407#117—41no—
408#117—262no—
409#117—521no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—41no—
413C2×C4⋊C4—131no—
414C2×C4⋊C4—131no—
415C2×C4⋊C4—131no—
416C2×C4⋊C4—131no—
417C2×C4⋊C4—131no—
418C2×C4⋊C4—262no—
419C2×C4⋊C4—261no—
420C2×C4⋊C4—521no—
421C2×C4⋊C4C22×D1311yescharacteristic
422C2×C4⋊C4C22×D1311yescharacteristic
423C2×C4⋊C4C22×D1311yescharacteristic
424C2×C4⋊C4C22×D1311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—131no—
427C2×C22⋊C4—131no—
428C2×C22⋊C4—261no—
429C2×C22⋊C4—261no—
430C2×C22⋊C4—261no—
431C2×C22⋊C4—261no—
432C2×C22⋊C4—261no—
433C2×C42—21no—
434C2×C42—131no—
435C2×C42—261no—
436C2×C42—261no—
437C2×C42—261no—
438#107—22no—
439#107—22no—
440#107—262no—
441#107—262no—
442#106—22no—
443#106—262no—
444#106—262no—
445#106—521no—
446#106—521no—
447#106C22⋊D1314yescomplemented
448C4⋊C8—134no—
449C22⋊C8—134no—
450C22⋊C8—134no—
451C4×C8—134no—
452#95—261no—
453#95—261no—
454#95—261no—
455#95—261no—
456#95—261no—
457#95—261no—
458#95—261no—
459#95—261no—
460#95—261no—
461#95—261no—
462C2×C13—42no—
463C2×C13D4×C4⋊C411yescharacteristic
464C2×C13#291111yescharacteristic
465C2×C13#291211yescharacteristic
466C2×C13#327511yescharacteristic
467C2×C13#327511yescharacteristic
468C2×C13#335111yescharacteristic
469C2×C13#335111yescharacteristic
470D13—42no—
471C24—131no—
472C24D4×D1311yescharacteristic
473C2×D4—134no—
474C2×D4—521no—
475C2×D4—521no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—131no—
488C22×C4—131no—
489C22×C4—131no—
490C22×C4—132no—
491C22×C4—132no—
492C22×C4—261no—
493C22×C4—261no—
494C22×C4—261no—
495C22×C4—261no—
496C22×C4—261no—
497C22×C4—261no—
498C22×C4—261no—
499C22×C4—261no—
500C22×C4—522no—
501C22×C4—522no—
502C22×C4—1041no—
503C22×C4C2×C4×D1312yes—
504C22×C4D4×D1311yescharacteristic
505C22×C4Q8×D1312yes—
506C22×C4C23×D1311yescharacteristic
507M16—28no—
508M16—268no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—132no—
512C2×C8—132no—
513C2×C8—261no—
514C2×C8—261no—
515C2×C8—261no—
516C2×C8—261no—
517C2×C8C2×C4×D1312yes—
518C2×C8C2×C4×D1312yes—
519C2×C8#612712yes—
520C2×C8C2×C22⋊D1312yes—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—42no—
528C4⋊C4—134no—
529C4⋊C4—261no—
530C4⋊C4—261no—
531C4⋊C4—261no—
532C4⋊C4—261no—
533C4⋊C4—261no—
534C4⋊C4—261no—
535C4⋊C4—261no—
536C4⋊C4—261no—
537C4⋊C4—522no—
538C4⋊C4C2×C22⋊D1312yes—
539C4⋊C4C2×C22⋊D1312yes—
540C22⋊C4—42no—
541C22⋊C4—134no—
542C22⋊C4—134no—
543C22⋊C4—522no—
544C22⋊C4—522no—
545C22⋊C4—522no—
546C22⋊C4—1041no—
547C22⋊C4—1041no—
548C42—42no—
549C42—132no—
550C42—132no—
551C42—522no—
552C42—1041no—
553C42—1041no—
554C13#2266611yescharacteristic, complemented, Sylow
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—132no—
562C23—261no—
563C23—261no—
564C23—261no—
565C23—261no—
566C23—261no—
567C23—261no—
568C23C4⋊C4×D1312yes—
569C23C2×D4×D1311yescharacteristic
570D4—524no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—132no—
591C2×C4—132no—
592C2×C4—132no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—261no—
596C2×C4—261no—
597C2×C4—261no—
598C2×C4—261no—
599C2×C4—262no—
600C2×C4—262no—
601C2×C4—262no—
602C2×C4—262no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—521no—
606C2×C4—521no—
607C2×C4—521no—
608C2×C4—521no—
609C2×C4—521no—
610C2×C4—521no—
611C2×C4—521no—
612C2×C4—521no—
613C2×C4—521no—
614C2×C4—521no—
615C2×C4—1044no—
616C2×C4C4⋊C4×D1312yes—
617C2×C4C22×C4×D1311yescharacteristic
618C2×C4C2×#612711yescharacteristic
619C2×C4#8825111yescharacteristic
620C2×C4C2×Q8×D1311yescharacteristic
621C2×C4#8825411yescharacteristic
622C2×C4C22×C22⋊D1311yescharacteristic
623C8—22no—
624C8—262no—
625C8—262no—
626C8C4×C22⋊D1314yes—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—262no—
635C22—262no—
636C22—262no—
637C22—521no—
638C22—521no—
639C22—521no—
640C22—521no—
641C22C2×C4⋊C4×D1311yescharacteristic
642C22#1171921211yescharacteristic
643C22#1171947811yescharacteristic
644C22#1171948011yescharacteristic
645C22#1171948211yescharacteristic
646C22#1171948311yescharacteristic
647C22D4×C22⋊D1311yescharacteristic
648C4—22no—
649C4—42no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—262no—
654C4—262no—
655C4—262no—
656C4—1041no—
657C4—1041no—
658C4—1041no—
659C4—1041no—
660C4#1171920911yescharacteristic
661C4#1171921611yescharacteristic
662C4C2×C4×C22⋊D1311yescharacteristic
663C4Q8×C22⋊D1311yescharacteristic
664C2—42no—
665C2—522no—
666C2C4⋊C4×C22⋊D1311yescharacteristic
667C2#1312342311yescharacteristic
668C2#1312343011yescharacteristic
669C2#1312646111yescharacteristic
670C2#1312646711yescharacteristic
671C2#1312739211yescharacteristic
672C2#1312739711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#428C2
socleC23×C13C22×D4
prime radicalC22×C22⋊D13D4
cosocleC22×C4×C13C24
hypercenterC2×#428D13
hypocenterC13#22666
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#11718522C2semidirect
5. #13115605C2semidirect
6. #13115601C2semidirect
7. #13115035C2semidirect
8. #13115032C2semidirect
9. #13115029C2semidirect
10. #13115027C2semidirect
11. #13114999C2semidirect
12. #13114870C2semidirect
13. #13114869C2semidirect
14. #106×Dic13C2semidirect
15. C2×C13×#428C2semidirect
33. #11718789C4semidirect
69. C2×C13×M16C2×C4semidirect
106. C2×#6094C2×C4semidirect
114. C13×#106D4semidirect
125. C22⋊#1055C2×C4semidirect
126. #88060C2×C4semidirect
129. #88054C2×C4semidirect
130. C8×Dic13C2×C4semidirect
221. C2×#428D13semidirect
399. C2×M16C4×D13semidirect
447. #106C22⋊D13semidirect
554. C13#22666semidirect

Maximal subgroups

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

subgroupindex
C2×#117187892
C2×#117185222
#131177662
#131177612
#131156052
#131156012
#131150352
#131150322
#131150292
#131150272
#131149992
#131148702
#131148692
#106×Dic132
C2×C13×#4282
#2266613