Finite Group Databasebeta

← #10702227

Subgroups — #10702227

Order 768.

958 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2317C211yesmaximal, characteristic
2C2×#5004C211yesmaximal, characteristic
3C2×#5004C211yesmaximal, characteristic
4C2×#4973C211yesmaximal, characteristic
5C2×#4532C211yesmaximal, characteristic
6C2×#4532C211yesmaximal, characteristic
7C2×#4530C211yesmaximal, characteristic
8#11064—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C4×Dic3C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C3×#389—22no—
17C3×#385—22no—
18C3×#384—21no—
19C3×#382—21no—
20C2×C3⋊#95—22no—
21#5004—22no—
22#5004—22no—
23#5004—22no—
24#5004C414yescomplemented
25#4973—22no—
26#4973—22no—
27C2×#810—21no—
28C2×#810—21no—
29C2×#810—22no—
30#4532—22no—
31#4532—22no—
32#4530—22no—
33C2×#419—31no—
34C2×#419—31no—
35#2317D311yescharacteristic
36C2×#391—31no—
37C2×#343—31no—
38C2×#339—31no—
39C2×#339—31no—
40C23×C3×C4D412yes—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—21no—
51C22×Dic6—21no—
52C22×Dic6—21no—
53C22×Dic6—21no—
54C22×Dic6—21no—
55C22×Dic6C2×C412yes—
56C22×Dic6D412yes—
57C22×Dic6Q812yes—
58C3×#117—21no—
59C3×#117—21no—
60C3×#117—21no—
61C3×#117—21no—
62C3×#117—21no—
63C3×#117—21no—
64C3×#117C2×C412yes—
65C3×#117D412yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D412yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×C3×C42—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3—21no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3D412yes—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208C2×C412yes—
97C2×#208D412yes—
98C22×C3⋊C8—21no—
99C22×C3⋊C8—21no—
100C3×#95—24no—
101C3⋊#95—28no—
102#810—24no—
103#810—24no—
104#810—28no—
105C23×Q8—31no—
106C22×M16—31no—
107C22×M16—31no—
108C2×#117C2×D311yescharacteristic
109C2×#117C2×D311yescharacteristic
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C42—31no—
112#419—34no—
113#419—62no—
114#419—62no—
115#419—62no—
116C2×#103—61no—
117C2×#103—61no—
118C2×#103—62no—
119#391—62no—
120#391—62no—
121#389—22no—
122#385—22no—
123#384—21no—
124#382—21no—
125C2×#95—62no—
126#343—62no—
127#339—62no—
128#339—62no—
129C24×C3C2×D411yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—44no—
137C22×C3×C4C22×C411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×Q811yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×Dic3—24no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—42no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—41no—
179C2×Dic6—41no—
180C2×Dic6—41no—
181C2×Dic6—41no—
182C2×Dic6—41no—
183C2×Dic6—41no—
184C2×Dic6—41no—
185C2×Dic6—41no—
186C2×Dic6—41no—
187C2×Dic6—41no—
188C2×Dic6—41no—
189C2×Dic6—41no—
190C2×Dic6C4⋊C414yescomplemented
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4—84no—
201C3×C4⋊C4C22⋊C414yes—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—41no—
209C3×C42—41no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212C4×Dic3—41no—
213C4×Dic3—41no—
214#208—22no—
215#208—22no—
216#208—22no—
217#208—22no—
218#208—41no—
219#208—41no—
220#208—41no—
221#208—41no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—41no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8—41no—
227C2×C3⋊C8—41no—
228C22×Q8—32no—
229C22×Q8—32no—
230C22×Q8—32no—
231C22×Q8—61no—
232C22×Q8—61no—
233C22×Q8—61no—
234C22×Q8—61no—
235C22×Q8—61no—
236C22×Q8—61no—
237C22×Q8—61no—
238C22×Q8—61no—
239C22×Q8—61no—
240C22×Q8—61no—
241C22×Q8—61no—
242C23×C4—61no—
243C23×C4—61no—
244C23×C4#6712yes—
245C23×C4C22×D311yescharacteristic
246C2×M16—32no—
247C2×M16—32no—
248C2×M16—61no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C22×C8—61no—
259C22×C8—61no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117C4×D312yes—
267#117D1212yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4#6711yescharacteristic
276C2×C4⋊C4#6711yescharacteristic
277C2×C4⋊C4#6712yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—32no—
283C2×C42—61no—
284C2×C42—61no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42—61no—
288#103—64no—
289#103—64no—
290#103—68no—
291#95—24no—
292#95—68no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3C4×D411yescharacteristic
296C23×C3#12011yescharacteristic
297C23×C3#12011yescharacteristic
298C23×C3#12111yescharacteristic
299C23×C3#12111yescharacteristic
300C23×C3C22⋊Q811yescharacteristic
301C23×C3#12411yescharacteristic
302C2×C3×C4—24no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—44no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—44no—
311C2×C3×C4—42no—
312C2×C3×C4—82no—
313C2×C3×C4C2×C22⋊C411yescharacteristic
314C2×C3×C4C2×C4⋊C411yescharacteristic
315C2×C3×C4#11711yescharacteristic
316C2×C3×C4C4×D411yescharacteristic
317C2×C3×C4C4×D411yescharacteristic
318C2×C3×C4C4×D411yescharacteristic
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4#12111yescharacteristic
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4C22⋊Q811yescharacteristic
327C2×C3×C4C22⋊Q811yescharacteristic
328C2×C3×C4C22⋊Q811yescharacteristic
329C2×C3×C4C22⋊Q811yescharacteristic
330C2×C3×C4C22⋊Q811yescharacteristic
331C2×C3×C4#12311yescharacteristic
332C2×C3×C4#12311yescharacteristic
333C2×C3×C4#12311yescharacteristic
334C2×C3×C4#12311yescharacteristic
335C2×C3×C4#12311yescharacteristic
336C2×C3×C4#12411yescharacteristic
337C2×C3×C4#12411yescharacteristic
338C2×C3×C4#12611yescharacteristic
339C2×C3×C4#12611yescharacteristic
340C2×C3×C4#12711yescharacteristic
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—44no—
344C2×Dic3—42no—
345C2×Dic3—44no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353Dic6—24no—
354Dic6—24no—
355Dic6—42no—
356Dic6—42no—
357Dic6—42no—
358Dic6—42no—
359Dic6—42no—
360Dic6—42no—
361C3⋊C8—42no—
362C3⋊C8—42no—
363C24C2×#6711yescharacteristic
364C2×Q8—34no—
365C2×Q8—62no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—62no—
373C2×Q8—62no—
374C2×Q8—62no—
375C2×Q8—62no—
376C2×Q8—62no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8—121no—
381C2×Q8—121no—
382C2×Q8—121no—
383C2×Q8—121no—
384C2×Q8—121no—
385C2×Q8—121no—
386C2×Q8—121no—
387C2×Q8—121no—
388C2×Q8—121no—
389C2×Q8—121no—
390C2×Q8—121no—
391C2×Q8—121no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—44no—
399C22×C4—64no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—122no—
405C22×C4—121no—
406C22×C4—121no—
407C22×C4—121no—
408C22×C4—121no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4C2×C4×D311yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4#23511yescharacteristic
414C22×C4#23511yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4#23711yescharacteristic
419C22×C4D3×Q811yescharacteristic
420C22×C4#23911yescharacteristic
421C22×C4#23911yescharacteristic
422C22×C4#23911yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425M16—62no—
426M16—62no—
427M16—62no—
428M16—62no—
429M16—121no—
430M16—121no—
431M16—121no—
432M16—121no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—121no—
436C2×C8—121no—
437C2×C8—121no—
438C2×C8—121no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—84no—
449C4⋊C4(C2×D3)⋊C414yes—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—62no—
459C42—62no—
460C42—121no—
461C42—121no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3#38511yescharacteristic
471C22×C3#38611yescharacteristic
472C22×C3#38811yescharacteristic
473C22×C3#39211yescharacteristic
474C22×C3#39311yescharacteristic
475C22×C3#39411yescharacteristic
476C22×C3#39611yescharacteristic
477C22×C3#45512yes—
478C22×C3#45612yes—
479C3×C4—44no—
480C3×C4—81no—
481C3×C4—82no—
482C3×C4—81no—
483C3×C4#38011yescharacteristic
484C3×C4#38511yescharacteristic
485C3×C4#38811yescharacteristic
486C3×C4#39011yescharacteristic
487C3×C4#39411yescharacteristic
488C3×C4#39611yescharacteristic
489C3×C4#39711yescharacteristic
490C3×C4#39911yescharacteristic
491Dic3—44no—
492Dic3—44no—
493C23—22no—
494C23—22no—
495C23C4×#6711yescharacteristic
496C23#93011yescharacteristic
497C23#93711yescharacteristic
498C23#93911yescharacteristic
499C23#94611yescharacteristic
500C23#94711yescharacteristic
501C23#95311yescharacteristic
502Q8—64no—
503Q8—64no—
504Q8—122no—
505Q8—122no—
506Q8—122no—
507Q8—122no—
508Q8—122no—
509Q8—122no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—82no—
523C2×C4—122no—
524C2×C4—124no—
525C2×C4—124no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4D3×C4⋊C411yescharacteristic
534C2×C4#89211yescharacteristic
535C2×C4#89311yescharacteristic
536C2×C4#89311yescharacteristic
537C2×C4#89411yescharacteristic
538C2×C4#89411yescharacteristic
539C2×C4#89411yescharacteristic
540C2×C4#89411yescharacteristic
541C2×C4#89511yescharacteristic
542C2×C4#89511yescharacteristic
543C2×C4#89511yescharacteristic
544C2×C4#89611yescharacteristic
545C2×C4#89611yescharacteristic
546C2×C4#89711yescharacteristic
547C2×C4#89811yescharacteristic
548C2×C4#89811yescharacteristic
549C2×C4C2×(C2×D3)⋊C411yescharacteristic
550C2×C4C4×#6711yescharacteristic
551C2×C4#92911yescharacteristic
552C2×C4#93011yescharacteristic
553C2×C4#93711yescharacteristic
554C2×C4#93811yescharacteristic
555C2×C4#93911yescharacteristic
556C2×C4#94011yescharacteristic
557C2×C4#94611yescharacteristic
558C2×C4#94611yescharacteristic
559C2×C4#94711yescharacteristic
560C2×C4#94711yescharacteristic
561C8—122no—
562C8—122no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3#143911yescharacteristic
568C2×C3#188212yes—
569C2×C3#200411yescharacteristic
570C2×C3#200511yescharacteristic
571C2×C3#202812yes—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22#498911yescharacteristic
581C22#499011yescharacteristic
582C22#499111yescharacteristic
583C22#499411yescharacteristic
584C22#521311yescharacteristic
585C22#522311yescharacteristic
586C22#523111yescharacteristic
587C22#524212yes—
588C22#524612yes—
589C4—44no—
590C4—82no—
591C4—81no—
592C4—81no—
593C4—124no—
594C4—124no—
595C4#498711yescharacteristic
596C4#498811yescharacteristic
597C4#498911yescharacteristic
598C4#499011yescharacteristic
599C4#499111yescharacteristic
600C4#499211yescharacteristic
601C4#499311yescharacteristic
602C4#499411yescharacteristic
603C3#1106411yescharacteristic, complemented, Sylow
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2#7157011yescharacteristic
609C2#7166612yes—
610C2#7185711yescharacteristic
611C2#7186811yescharacteristic
612C2#7190512yes—
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2317C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2317D3
hypocenterC3#11064
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
24. #5004C4semidirect
190. C2×Dic6C4⋊C4semidirect
603. C3#11064semidirect

Maximal subgroups

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

subgroupindex
C3×#23172
C2×#50042
C2×#49732
C2×#45322
C2×#45302
#110643