Finite Group Databasebeta

← #15922847

Subgroups — #15922847

Order 3328.

854 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718706C211yesmaximal, characteristic, complemented
2C2×#11718625C211yesmaximal, characteristic, complemented
3#13117785C211yesmaximal, characteristic
4#13116379C211yesmaximal, characteristic, complemented
5#13116312C211yesmaximal, characteristic, complemented
6#13116272C211yesmaximal, characteristic, complemented
7#13116029C211yesmaximal, characteristic, complemented
8#13115474C211yesmaximal, characteristic, complemented
9#13115425C211yesmaximal, characteristic, complemented
10#13115270C211yesmaximal, characteristic, complemented
11#13115149C211yesmaximal, characteristic, complemented
12#13115134C211yesmaximal, characteristic
13#13115033C211yesmaximal, characteristic, complemented
14#13115029C211yesmaximal, characteristic, complemented
15C13×#2417C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C22×C13×C4⋊C4C2211yescharacteristic
20C2×#88180C2211yescharacteristic
21C2×#88180C2211yescharacteristic
22C22×#6099C2211yescharacteristic
23C13×#398—21no—
24C13×#396—21no—
25C13×#391C2211yescharacteristic
26C13×#389C2211yescharacteristic
27C13×#386—21no—
28C13×#380—21no—
29C13×#380C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34#11718882—21no—
35#11718873C2211yescharacteristic
36#11718873C2211yescharacteristic
37C2×C22⋊#1055C2211yescharacteristic
38C2×C22⋊#1055C2211yescharacteristic
39C2×C22⋊#1055C2211yescharacteristic
40#11718706—24no—
41C4⋊C4×Dic13—21no—
42#11718625—24no—
43C2×#88048C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45C2×#88048C2211yescharacteristic
46C2×#88048C2211yescharacteristic
47C2×#88047C2211yescharacteristic
48C2×#88047C2211yescharacteristic
49#11718612—21no—
50#11718612—21no—
51#11718596C2211yescharacteristic
52C4×C4⋊Dic13C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C22×D4×C13D411yescharacteristic
63C23×C4×C13—21no—
64C23×C4×C13C2311yescharacteristic
65C23×C4×C13C2311yescharacteristic
66C13×C22⋊Q8—22no—
67C13×C22⋊Q8—22no—
68C13×C22⋊Q8—22no—
69C13×C22⋊Q8—41no—
70C4×D4×C13—22no—
71C4×D4×C13—22no—
72C2×C13×C4⋊C4—22no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4—41no—
78C2×C13×C4⋊C4D411yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C13×C4⋊C4C2311yescharacteristic
82C2×C13×C22⋊C4—21no—
83C2×C13×C22⋊C4—21no—
84C2×C13×C22⋊C4—21no—
85C2×C13×C22⋊C4D411yescharacteristic
86C2×C13×C22⋊C4D411yescharacteristic
87C2×C42×C13C2311yescharacteristic
88C2×C22⋊Dic13—21no—
89C2×C22⋊Dic13—21no—
90C2×C22⋊Dic13—21no—
91C2×C22⋊Dic13—21no—
92#88180—22no—
93#88180—22no—
94#88180—41no—
95#88180—41no—
96C2×C4⋊Dic13—21no—
97C2×C4⋊Dic13—21no—
98C2×C4⋊Dic13C2311yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×C4⋊Dic13C2311yescharacteristic
101C2×C4⋊Dic13C2311yescharacteristic
102C2×C4×Dic13—21no—
103C2×C4×Dic13—21no—
104C2×C4×Dic13—21no—
105C2×#6099—22no—
106C2×#6099—22no—
107C2×#6099—22no—
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110C22×#1055C2311yescharacteristic
111C22×#1055C2311yescharacteristic
112C13×#95—21no—
113C13×#95—21no—
114C13×#95—21no—
115C13×#95—21no—
116#88075—22no—
117#88075—22no—
118#88075—22no—
119#88075—41no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123C22⋊#1055—22no—
124C22⋊#1055—22no—
125C22⋊#1055—22no—
126#88048—22no—
127#88048—22no—
128#88048—22no—
129#88048—41no—
130#88047—22no—
131#88047D414yescomplemented
132#26155—131nomaximal, Sylow
133C24×C13C2×D411yescharacteristic
134C24×C13C2×D411yescharacteristic
135C2×Q8×C13—21no—
136C2×Q8×C13—21no—
137C2×Q8×C13—21no—
138C2×Q8×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×D4×C13—22no—
142C2×D4×C13—22no—
143C2×D4×C13—41no—
144C22×C4×C13—21no—
145C22×C4×C13—21no—
146C22×C4×C13—21no—
147C22×C4×C13—21no—
148C22×C4×C13—21no—
149C22×C4×C13—21no—
150C22×C4×C13—21no—
151C22×C4×C13—21no—
152C22×C4×C13—21no—
153C22×C4×C13—21no—
154C22×C4×C13—21no—
155C22×C4×C13—21no—
156C22×C4×C13—21no—
157C22×C4×C13—21no—
158C22×C4×C13—42no—
159C22×C4×C13—42no—
160C22×C4×C13—81no—
161C22×C4×C13C2×D411yescharacteristic
162C22×C4×C13C2×D411yescharacteristic
163C22×C4×C13C2×D411yescharacteristic
164C22×C4×C13C2×D411yescharacteristic
165C22×C4×C13#4112yes—
166C22×C4×C13#4112yes—
167C22×C4×C13C2411yescharacteristic
168C22×Dic13—21no—
169C22×Dic13—21no—
170C22×Dic13—21no—
171C22×Dic13—21no—
172C13×C4⋊C4—22no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—42no—
179C13×C4⋊C4—42no—
180C13×C4⋊C4—42no—
181C13×C4⋊C4—81no—
182C13×C4⋊C4C2×D412yes—
183C13×C22⋊C4—22no—
184C13×C22⋊C4—22no—
185C13×C22⋊C4—42no—
186C13×C22⋊C4—42no—
187C13×C22⋊C4—81no—
188C42×C13—21no—
189C42×C13—21no—
190C22⋊Dic13—42no—
191C22⋊Dic13—42no—
192C22⋊Dic13—81no—
193C22⋊Dic13—81no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—21no—
201C4⋊Dic13—42no—
202C4⋊Dic13—81no—
203C4⋊Dic13C2×D412yes—
204C4×Dic13—42no—
205C4×Dic13—42no—
206C4×Dic13—42no—
207#6099—28no—
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×#1055C2×D412yes—
219C2×#1055#4112yes—
220C2×#485—131no—
221C2×#428—131no—
222#2417D1311yescharacteristic
223#2354—131no—
224#2096—131no—
225#2084—131no—
226#2077—131no—
227#2035—131no—
228#1928—131no—
229#1921—131no—
230#1894—131no—
231#1873—131no—
232#1870—131no—
233#1854—131no—
234#1853—131no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13—21no—
238C23×C13—21no—
239C23×C13—21no—
240C23×C13—21no—
241C23×C13—21no—
242C23×C13—21no—
243C23×C13—21no—
244C23×C13—21no—
245C23×C13—21no—
246C23×C13—21no—
247C23×C13—21no—
248C23×C13#12312yes—
249C23×C13C22×D411yescharacteristic
250Q8×C13—41no—
251Q8×C13—41no—
252Q8×C13—41no—
253Q8×C13—41no—
254D4×C13—44no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—22no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—22no—
267C2×C4×C13—21no—
268C2×C4×C13—21no—
269C2×C4×C13—21no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—41no—
274C2×C4×C13—41no—
275C2×C4×C13—41no—
276C2×C4×C13—41no—
277C2×C4×C13—41no—
278C2×C4×C13—41no—
279C2×C4×C13—41no—
280C2×C4×C13—41no—
281C2×C4×C13—41no—
282C2×C4×C13—41no—
283C2×C4×C13—41no—
284C2×C4×C13—41no—
285C2×C4×C13—84no—
286C2×C4×C13#12312yes—
287C2×C4×C13#13712yes—
288C2×C4×C13C22×D411yescharacteristic
289C2×C4×C13C2×#4111yescharacteristic
290C2×C4×C13C2×#4111yescharacteristic
291C2×C4×C13C2×#4111yescharacteristic
292C2×C4×C13H(5,𝔽2)11yescharacteristic
293C2×C4×C132−1+411yescharacteristic
294C2×Dic13—22no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—41no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300C2×Dic13—41no—
301C2×Dic13—41no—
302C2×Dic13—41no—
303C2×Dic13—41no—
304C2×Dic13—41no—
305C2×Dic13—41no—
306#1055—22no—
307#1055—22no—
308#1055—41no—
309#1055—41no—
310C22×M16—131no—
311C2×C22⋊Q8C2×D1311yescharacteristic
312C2×C22⋊Q8C2×D1311yescharacteristic
313C2×C4×D4C2×D1311yescharacteristic
314C2×#117—131no—
315C2×#117—131no—
316C22×C4⋊C4C2×D1311yescharacteristic
317#485—264no—
318#428—264no—
319C2×#107—131no—
320C2×#107—131no—
321C2×#106—131no—
322C2×#106—131no—
323C2×#106—131no—
324C2×#106—131no—
325C2×#103—131no—
326C2×#103—131no—
327C2×#103—131no—
328C2×#103—131no—
329C2×C22⋊C8—131no—
330C2×C22⋊C8—131no—
331C2×C22⋊C8—131no—
332#398—21no—
333#396—21no—
334#391C2×D1311yescharacteristic
335#390—261no—
336#389C2×D1311yescharacteristic
337#388—261no—
338#388—261no—
339#386—21no—
340#384—131no—
341#380—21no—
342#380—131no—
343#380—131no—
344#380C2×D1311yescharacteristic
345C4×C4⋊C4—131no—
346C4×C4⋊C4—261no—
347#340—131no—
348#340—131no—
349#340—131no—
350#340—131no—
351#340—131no—
352#340—131no—
353#336—131no—
354#336—131no—
355C22×C13—22no—
356C22×C13—22no—
357C22×C13—22no—
358C22×C13—41no—
359C22×C13—41no—
360C22×C13—41no—
361C22×C13—41no—
362C22×C13—41no—
363C22×C13—41no—
364C22×C13—41no—
365C22×C13—41no—
366C22×C13—41no—
367C22×C13—41no—
368C22×C13—41no—
369C22×C13C2×#12311yescharacteristic
370C22×C13D4211yescharacteristic
371C22×C13#54711yescharacteristic
372C22×C13C2×#13711yescharacteristic
373C22×C13#57511yescharacteristic
374C22×C13#57711yescharacteristic
375C22×C13#57711yescharacteristic
376C4×C13—22no—
377C4×C13—22no—
378C4×C13—41no—
379C4×C13—41no—
380C4×C13—42no—
381C4×C13—81no—
382C4×C13—81no—
383C4×C13—81no—
384C4×C13#52511yescharacteristic
385C4×C13#53911yescharacteristic
386C4×C13#54811yescharacteristic
387C4×C13#55311yescharacteristic
388Dic13—42no—
389Dic13—81no—
390Dic13—81no—
391Dic13—81no—
392C22×Q8C22×D1311yescharacteristic
393C22×D4C22⋊D1311yescharacteristic
394C23×C4—21no—
395C23×C4C22×D1311yescharacteristic
396C23×C4C22×D1311yescharacteristic
397C2×M16—262no—
398C2×M16—262no—
399C2×M16—262no—
400C22×C8—131no—
401C22×C8—131no—
402C22×C8—131no—
403C22×C8—131no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8—41no—
408C4×D4—22no—
409C4×D4—22no—
410#117—262no—
411#117—262no—
412#117—521no—
413#117—521no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—41no—
420C2×C4⋊C4—131no—
421C2×C4⋊C4—131no—
422C2×C4⋊C4—131no—
423C2×C4⋊C4—131no—
424C2×C4⋊C4—261no—
425C2×C4⋊C4—261no—
426C2×C4⋊C4C22⋊D1311yescharacteristic
427C2×C4⋊C4C22×D1311yescharacteristic
428C2×C4⋊C4C22×D1311yescharacteristic
429C2×C4⋊C4C22×D1311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—261no—
434C2×C22⋊C4—261no—
435C2×C22⋊C4—261no—
436C2×C22⋊C4—261no—
437C2×C22⋊C4C22⋊D1311yescharacteristic
438C2×C22⋊C4C22⋊D1311yescharacteristic
439C2×C42—261no—
440C2×C42—261no—
441C2×C42—261no—
442C2×C42C22×D1311yescharacteristic
443#107—134no—
444#107—262no—
445#106—262no—
446#106—262no—
447#106—262no—
448#106—521no—
449#103—262no—
450#103—262no—
451#103—262no—
452#103—521no—
453C22⋊C8—262no—
454C22⋊C8—262no—
455C22⋊C8—262no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—261no—
461#95—261no—
462#95—261no—
463C2×C13—42no—
464C2×C13—81no—
465C2×C13#244311yescharacteristic
466C2×C13#309511yescharacteristic
467C2×C13#309911yescharacteristic
468C2×C13#326711yescharacteristic
469C2×C13#327811yescharacteristic
470C2×C13#334811yescharacteristic
471C2×C13#335611yescharacteristic
472C24C2×C22⋊D1311yescharacteristic
473C24C2×C22⋊D1311yescharacteristic
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—41no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—261no—
501C22×C4—261no—
502C22×C4—261no—
503C22×C4—261no—
504C22×C4#612912yes—
505C22×C4#612912yes—
506C22×C4C2×C22⋊D1311yescharacteristic
507C22×C4C2×C22⋊D1311yescharacteristic
508C22×C4C2×C22⋊D1311yescharacteristic
509C22×C4C2×C22⋊D1311yescharacteristic
510C22×C4C23×D1311yescharacteristic
511M16—268no—
512C2×C8—132no—
513C2×C8—132no—
514C2×C8—261no—
515C2×C8—261no—
516C2×C8—261no—
517C2×C8—261no—
518C2×C8—261no—
519C2×C8—261no—
520C2×C8—261no—
521C2×C8—261no—
522C2×C8—261no—
523C2×C8—261no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—81no—
534C4⋊C4—132no—
535C4⋊C4—261no—
536C4⋊C4—261no—
537C4⋊C4—261no—
538C4⋊C4—261no—
539C4⋊C4—261no—
540C4⋊C4—261no—
541C4⋊C4—261no—
542C4⋊C4—522no—
543C4⋊C4—1041no—
544C4⋊C4D4×D1312yes—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—522no—
551C22⋊C4—522no—
552C22⋊C4—1041no—
553C22⋊C4—1041no—
554C42—21no—
555C42—21no—
556C42—522no—
557C42—522no—
558C42—522no—
559C13#2615511yescharacteristic, complemented, Sylow
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#8818912yes—
574C23C22×C22⋊D1311yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579D4—44no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—262no—
612C2×C4—521no—
613C2×C4—521no—
614C2×C4—521no—
615C2×C4—521no—
616C2×C4—521no—
617C2×C4—521no—
618C2×C4—521no—
619C2×C4—521no—
620C2×C4—521no—
621C2×C4—521no—
622C2×C4—521no—
623C2×C4#8818912yes—
624C2×C4#8819612yes—
625C2×C4C2×D4×D1311yescharacteristic
626C2×C4C2×#612911yescharacteristic
627C2×C4C2×#612911yescharacteristic
628C2×C4#8825111yescharacteristic
629C2×C4#41×D1311yescharacteristic
630C2×C4#8825711yescharacteristic
631C8—262no—
632C8—262no—
633C8—521no—
634C8—521no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#8818911yescharacteristic
650C22D4×C22⋊D1311yescharacteristic
651C22C2×#8819611yescharacteristic
652C22#1171952711yescharacteristic
653C22#1171952911yescharacteristic
654C22#1171954111yescharacteristic
655C22#1171954211yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—522no—
665C4—1041no—
666C4—1041no—
667C4—1041no—
668C4#1171930311yescharacteristic
669C4#1171930611yescharacteristic
670C4#1171930711yescharacteristic
671C4#1171931111yescharacteristic
672C2—42no—
673C2—81no—
674C2#1311856711yescharacteristic
675C2#1312478011yescharacteristic
676C2#1312482111yescharacteristic
677C2#1312635711yescharacteristic
678C2#1312649811yescharacteristic
679C2#1312734511yescharacteristic
680C2#1312745111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2417C2
socleC23×C13C22×D4
prime radicalC22×D4×C13D4
hypercenter#2417D13
hypocenterC13#26155
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2417

Complements

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

NH ≅ G/Ntype
1. C2×#11718706C2semidirect
2. C2×#11718625C2semidirect
4. #13116379C2semidirect
5. #13116312C2semidirect
6. #13116272C2semidirect
7. #13116029C2semidirect
8. #13115474C2semidirect
9. #13115425C2semidirect
10. #13115270C2semidirect
11. #13115149C2semidirect
13. #13115033C2semidirect
14. #13115029C2semidirect
131. #88047D4semidirect
559. C13#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#117187062
C2×#117186252
#131177852
#131163792
#131163122
#131162722
#131160292
#131154742
#131154252
#131152702
#131151492
#131151342
#131150332
#131150292
C13×#24172
#2615513