Finite Group Databasebeta

← #15912994

Subgroups — #15912994

Order 3328.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718716C211yesmaximal, characteristic
2C2×#11718715C211yesmaximal, characteristic
3#13120044C211yesmaximal, characteristic, complemented
4#13116244C211yesmaximal, characteristic, complemented
5#13116239C211yesmaximal, characteristic
6#13116158C211yesmaximal, characteristic, complemented
7#13116153C211yesmaximal, characteristic
8#13115788C211yesmaximal, characteristic, complemented
9#13115560C211yesmaximal, characteristic, complemented
10#13115177C211yesmaximal, characteristic, complemented
11#13115170C211yesmaximal, characteristic
12#13114841C211yesmaximal, characteristic, complemented
13#13114836C211yesmaximal, characteristic
14#13114669C211yesmaximal, characteristic, complemented
15C13×#2367C211yesmaximal, characteristic
16C2×C13×#124C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C22×#6107C2211yescharacteristic
20C22×#6105C2211yescharacteristic
21C2×#88123C2211yescharacteristic
22C2×C4×Dic26C2211yescharacteristic
23C13×#391C2211yescharacteristic
24C13×#390C2211yescharacteristic
25C13×#388—21no—
26C13×#388—21no—
27C13×#385—21no—
28C4×C13×C4⋊C4C2211yescharacteristic
29C4×C13×C4⋊C4C2211yescharacteristic
30C4×C13×C22⋊C4—21no—
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88072C2211yescharacteristic
35C2×#88072C2211yescharacteristic
36C2×#88072C2211yescharacteristic
37#11718716—24no—
38#11718715—24no—
39#11718646—21no—
40#11718646—21no—
41C2×#88050C2211yescharacteristic
42C2×#88050C2211yescharacteristic
43#11718596C2211yescharacteristic
44#11718596C2211yescharacteristic
45#11718590C2211yescharacteristic
46#11718590C2211yescharacteristic
47C2×C4⋊#1055C2211yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×#88043C2211yescharacteristic
51#11718317—21no—
52#11718317—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C22×D4×C13C2311yescharacteristic
63C23×C4×C13—21no—
64C22×Dic26—21no—
65C22×Dic26C2311yescharacteristic
66C13×#124—22no—
67C13×#124D414yes—
68C4×Q8×C13—22no—
69C4×Q8×C13—41no—
70C4×D4×C13—22no—
71C4×D4×C13—41no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4D411yescharacteristic
74C2×C13×C4⋊C4D411yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4—21no—
80C2×C13×C22⋊C4—21no—
81C2×C13×C22⋊C4—21no—
82C2×C42×C13—21no—
83C2×C42×C13—21no—
84C2×C42×C13C2311yescharacteristic
85C2×C42×C13C2311yescharacteristic
86C2×C42×C13C2311yescharacteristic
87C2×#6107—22no—
88C2×#6107—22no—
89C2×#6107—41no—
90C2×#6105—22no—
91C2×#6105—22no—
92C2×#6105—41no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13—21no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13—21no—
97C2×C4⋊Dic13—21no—
98C2×C4⋊Dic13C2311yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×C4⋊Dic13C2311yescharacteristic
101C2×#6101—21no—
102C2×C4×Dic13—21no—
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C22×#1055C2311yescharacteristic
107#88123—22no—
108#88123—22no—
109C4×Dic26—22no—
110C4×Dic26—22no—
111C13×#95—21no—
112C13×#95—21no—
113C13×#95—21no—
114C13×#95D411yescharacteristic
115C13×#95D411yescharacteristic
116#88075—22no—
117#88075—22no—
118#88075—22no—
119#88072—22no—
120#88072—22no—
121#88072—22no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126#88050—22no—
127#88050—22no—
128C4⋊#1055—22no—
129C4⋊#1055—41no—
130C4⋊#1055—41no—
131#88043—22no—
132#25336—131nomaximal, Sylow
133C24×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×Q8×C13—21no—
137C2×Q8×C13—21no—
138C2×Q8×C13—21no—
139C2×Q8×C13C2×D412yes—
140C2×D4×C13—21no—
141C2×D4×C13—21no—
142C2×D4×C13—21no—
143C2×D4×C13—21no—
144C2×D4×C13—21no—
145C2×D4×C13C2×D412yes—
146C22×C4×C13—21no—
147C22×C4×C13—21no—
148C22×C4×C13—21no—
149C22×C4×C13—42no—
150C22×C4×C13—42no—
151C22×C4×C13—81no—
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13C2×D411yescharacteristic
154C22×C4×C13C2×D411yescharacteristic
155C22×C4×C13C2×D411yescharacteristic
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2×D411yescharacteristic
158C22×C4×C13C2411yescharacteristic
159C22×Dic13—21no—
160C22×Dic13—21no—
161C22×Dic13—21no—
162C22×Dic13—21no—
163C2×Dic26—21no—
164C2×Dic26—21no—
165C2×Dic26—21no—
166C2×Dic26—21no—
167C2×Dic26—21no—
168C2×Dic26—42no—
169C2×Dic26—81no—
170C2×Dic26—81no—
171C2×Dic26#4112yes—
172C13×C4⋊C4—22no—
173C13×C4⋊C4—22no—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—42no—
178C13×C4⋊C4#4112yes—
179C13×C22⋊C4—24no—
180C13×C22⋊C4—24no—
181C13×C22⋊C4—42no—
182C13×C22⋊C4—42no—
183C13×C22⋊C4—81no—
184C42×C13—21no—
185C42×C13—21no—
186C42×C13—21no—
187C42×C13—21no—
188C42×C13—42no—
189C42×C13—81no—
190C42×C13C2×D412yes—
191C42×C13#4112yes—
192#6107—44no—
193#6105—44no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—42no—
201C4⋊Dic13—81no—
202C4⋊Dic13—81no—
203C4⋊Dic13—81no—
204C4⋊Dic13—81no—
205#6101—42no—
206C4×Dic13—42no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#489—131no—
220C2×#464—131no—
221#2567—131no—
222#2367D1311yescharacteristic
223#2072—131no—
224#2071—131no—
225#2056—131no—
226#2055—131no—
227#1988—131no—
228#1940—131no—
229#1878—131no—
230#1877—131no—
231#1803—131no—
232#1802—131no—
233#1763—131no—
234C23×C13—22no—
235C23×C13—41no—
236C23×C13—41no—
237C23×C13—41no—
238C23×C13—41no—
239C23×C13—41no—
240C23×C13—41no—
241C23×C13C22×D411yescharacteristic
242Q8×C13—22no—
243Q8×C13—22no—
244Q8×C13—41no—
245Q8×C13—41no—
246D4×C13—22no—
247D4×C13—22no—
248D4×C13—41no—
249D4×C13—41no—
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—22no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—22no—
262C2×C4×C13—21no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—84no—
274C2×C4×C13#12112yes—
275C2×C4×C13#12112yes—
276C2×C4×C13#13712yes—
277C2×C4×C13#13712yes—
278C2×C4×C13C22×D411yescharacteristic
279C2×C4×C13C2×#4111yescharacteristic
280C2×C4×C13C2×#4111yescharacteristic
281C2×C4×C13C2×#4111yescharacteristic
282C2×C4×C13H(5,𝔽2)11yescharacteristic
283C2×C4×C132−1+411yescharacteristic
284C2×Dic13—22no—
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×Dic13—41no—
295C2×Dic13—41no—
296Dic26—22no—
297Dic26—22no—
298Dic26—41no—
299Dic26—41no—
300Dic26—84no—
301#1055—41no—
302#1055—41no—
303#1055—41no—
304#1055—41no—
305C22×Q16—131no—
306C22×SD16—131no—
307C2×C4⋊Q8—131no—
308C2×#124C2×D1311yescharacteristic
309C2×C4×Q8—131no—
310C2×C4×Q8C2×D1311yescharacteristic
311C2×C4×D4C2×D1311yescharacteristic
312#489—264no—
313#464—264no—
314C2×C4⋊C8—131no—
315C2×C4⋊C8—131no—
316C2×C4⋊C8—131no—
317C2×#103—131no—
318C2×#103—131no—
319C2×#103—131no—
320C2×#103—131no—
321C2×#103—131no—
322C2×#102—131no—
323C2×#102—131no—
324C2×#102—131no—
325C2×#97—131no—
326#395—261no—
327#395—261no—
328#391—131no—
329#391—131no—
330#391C2×D1311yescharacteristic
331#390C2×D1311yescharacteristic
332#389—261no—
333#389—261no—
334#388—21no—
335#388—21no—
336#385—21no—
337#384—131no—
338#384—131no—
339C4×C4⋊C4C2×D1311yescharacteristic
340C4×C4⋊C4C2×D1311yescharacteristic
341C4×C22⋊C4—21no—
342#340—131no—
343#340—131no—
344#340—131no—
345#340—131no—
346#340—131no—
347#340—131no—
348#336—131no—
349#336—131no—
350C22×C13—42no—
351C22×C13—42no—
352C22×C13—42no—
353C22×C13—81no—
354C22×C13—81no—
355C22×C13—81no—
356C22×C13—81no—
357C22×C13C2×#12111yescharacteristic
358C22×C13#54611yescharacteristic
359C22×C13#54811yescharacteristic
360C22×C13C2×#13711yescharacteristic
361C22×C13C2×#13711yescharacteristic
362C22×C13#57611yescharacteristic
363C22×C13#57711yescharacteristic
364C4×C13—22no—
365C4×C13—22no—
366C4×C13—22no—
367C4×C13—22no—
368C4×C13—42no—
369C4×C13—41no—
370C4×C13—42no—
371C4×C13—41no—
372C4×C13—81no—
373C4×C13#52911yescharacteristic
374C4×C13#54211yescharacteristic
375C4×C13#54611yescharacteristic
376C4×C13#54711yescharacteristic
377Dic13—42no—
378Dic13—81no—
379Dic13—81no—
380Dic13—81no—
381C22×Q8—131no—
382C22×Q8—261no—
383C22×Q8C22×D1311yescharacteristic
384C22×D4C22×D1311yescharacteristic
385C23×C4—21no—
386C2×Q16—262no—
387C2×Q16—262no—
388C2×Q16—521no—
389C2×SD16—262no—
390C2×SD16—262no—
391C2×SD16—521no—
392C22×C8—131no—
393C22×C8—131no—
394C22×C8—131no—
395C22×C8—131no—
396C4⋊Q8—262no—
397C4⋊Q8—262no—
398#124—22no—
399#124D5214yes—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×Q8—262no—
403C4×Q8—262no—
404C4×D4—22no—
405C4×D4—41no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—261no—
411C2×C4⋊C4—261no—
412C2×C4⋊C4—261no—
413C2×C4⋊C4—261no—
414C2×C4⋊C4—261no—
415C2×C4⋊C4—261no—
416C2×C4⋊C4C22⋊D1311yescharacteristic
417C2×C4⋊C4C22⋊D1311yescharacteristic
418C2×C4⋊C4C22×D1311yescharacteristic
419C2×C4⋊C4C22×D1311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C42—21no—
426C2×C42—21no—
427C2×C42—261no—
428C2×C42C22×D1311yescharacteristic
429C2×C42C22×D1311yescharacteristic
430C2×C42C22×D1311yescharacteristic
431C4⋊C8—262no—
432C4⋊C8—521no—
433C4⋊C8—521no—
434#103—262no—
435#103—262no—
436#103—262no—
437#103—262no—
438#103—262no—
439#102—262no—
440#102—262no—
441#102—262no—
442#97—262no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—261no—
447#95—261no—
448#95—261no—
449#95—261no—
450#95C22⋊D1311yescharacteristic
451#95C22⋊D1311yescharacteristic
452C2×C13—82no—
453C2×C13#242711yescharacteristic
454C2×C13#303911yescharacteristic
455C2×C13#304511yescharacteristic
456C2×C13#331411yescharacteristic
457C2×C13#331611yescharacteristic
458C2×C13#336911yescharacteristic
459C2×C13#337511yescharacteristic
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—132no—
467C2×Q8—261no—
468C2×Q8—261no—
469C2×Q8—261no—
470C2×Q8—261no—
471C2×Q8—261no—
472C2×Q8—522no—
473C2×Q8—1041no—
474C2×Q8—1041no—
475C2×Q8C2×D5212yes—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4C2×D5212yes—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—261no—
489C22×C4—261no—
490C22×C4—261no—
491C22×C4—261no—
492C22×C4C2×C22⋊D1311yescharacteristic
493C22×C4C2×C22⋊D1311yescharacteristic
494C22×C4C2×C22⋊D1311yescharacteristic
495C22×C4C2×C22⋊D1311yescharacteristic
496C22×C4C2×C22⋊D1311yescharacteristic
497C22×C4C2×C22⋊D1311yescharacteristic
498C22×C4C23×D1311yescharacteristic
499Q16—524no—
500SD16—524no—
501C2×C8—261no—
502C2×C8—261no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8—261no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—261no—
520C4⋊C4—261no—
521C4⋊C4—261no—
522C4⋊C4—261no—
523C4⋊C4—261no—
524C4⋊C4—261no—
525C4⋊C4—522no—
526C4⋊C4—522no—
527C4⋊C4—1041no—
528C4⋊C4—1041no—
529C4⋊C4—1041no—
530C4⋊C4—1041no—
531C4⋊C4#612912yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—81no—
543C42—522no—
544C42C2×D5212yes—
545C42#612712yes—
546C13#2533611yescharacteristic, complemented, Sylow
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×C22⋊D1311yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—262no—
560Q8—262no—
561Q8—521no—
562Q8—521no—
563Q8—1044no—
564D4—22no—
565D4—22no—
566D4—41no—
567D4—41no—
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—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—262no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—521no—
596C2×C4—521no—
597C2×C4—521no—
598C2×C4—521no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4#8818412yes—
605C2×C4#8818412yes—
606C2×C4#8820512yes—
607C2×C4#8820512yes—
608C2×C4C22×D5211yescharacteristic
609C2×C4C2×#612711yescharacteristic
610C2×C4C2×#612911yescharacteristic
611C2×C4#8825111yescharacteristic
612C2×C4#41×D1311yescharacteristic
613C2×C4#8825711yescharacteristic
614C8—521no—
615C8—521no—
616C8—521no—
617C8—521no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22C2×#8818411yescharacteristic
626C22#1171951111yescharacteristic
627C22C2×#8820511yescharacteristic
628C22C2×#8820511yescharacteristic
629C22#1171953611yescharacteristic
630C22#1171954011yescharacteristic
631C22#1171954211yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—81no—
641C4—522no—
642C4—1041no—
643C4—1041no—
644C4—1041no—
645C4#1171925911yescharacteristic
646C4#1171926011yescharacteristic
647C4#1171926211yescharacteristic
648C4#1171926611yescharacteristic
649C2—82no—
650C2#1311840711yescharacteristic
651C2#1312432311yescharacteristic
652C2#1312437511yescharacteristic
653C2#1312691711yescharacteristic
654C2#1312694711yescharacteristic
655C2#1312757211yescharacteristic
656C2#1312764611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2367C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
hypercenter#2367D13
hypocenterC13#25336
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
3. #13120044C2semidirect
4. #13116244C2semidirect
6. #13116158C2semidirect
8. #13115788C2semidirect
9. #13115560C2semidirect
10. #13115177C2semidirect
12. #13114841C2semidirect
14. #13114669C2semidirect
546. C13#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#117187162
C2×#117187152
#131200442
#131162442
#131162392
#131161582
#131161532
#131157882
#131155602
#131151772
#131151702
#131148412
#131148362
#131146692
C13×#23672
#2533613