Finite Group Databasebeta

← #10702232

Subgroups — #10702232

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
1C2×C3×#391C211yesmaximal, characteristic
2C2×#5225C211yesmaximal, characteristic
3C2×#5225C211yesmaximal, characteristic
4#82157C211yesmaximal, characteristic
5C2×#4530C211yesmaximal, characteristic
6C2×#4523C211yesmaximal, characteristic
7C2×#4523C211yesmaximal, characteristic
8#11064—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C2×#926C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C3×#391—22no—
17C3×#391—22no—
18C2×C3×#95—22no—
19#5225—22no—
20#5225—22no—
21#5225—22no—
22#5225C414yescomplemented
23C2×#835—21no—
24C2×#835—21no—
25C2×#835—22no—
26#5210—22no—
27#4977—21no—
28#4975—21no—
29#4972—22no—
30#4530—22no—
31#4523—22no—
32#4523—22no—
33C2×#419—31no—
34C2×#419—31no—
35#2317—31no—
36C2×#391D311yescharacteristic
37C2×#343—31no—
38C2×#339—31no—
39C2×#339—31no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8—21no—
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8—21no—
50C22×C3×Q8—21no—
51C22×C3×Q8C2×C412yes—
52C22×C3×Q8D412yes—
53C22×C3×Q8Q812yes—
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C42—21no—
65C2×C3×C42—21no—
66C2×C3×C42—21no—
67C2×C3×C42—21no—
68C2×C3×C42—21no—
69C2×C3×C42D412yes—
70C2×#217—21no—
71C2×#217—21no—
72#926—21no—
73#926—21no—
74#926—21no—
75#926—21no—
76#926—21no—
77#926—21no—
78#926C2×C412yes—
79#926D412yes—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3D411yescharacteristic
82C2×C4⋊Dic3D412yes—
83C2×Dic3⋊C4—24no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—21no—
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—28no—
101#835—24no—
102#835—24no—
103#835—28no—
104C3⋊#95—24no—
105C23×Q8C2×D311yescharacteristic
106C22×M16—31no—
107C22×M16—31no—
108C2×#117—31no—
109C2×#117—31no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C42C2×D311yescharacteristic
112#419—34no—
113#419—62no—
114#419—62no—
115#419—62no—
116C2×#103—62no—
117C2×#103—61no—
118C2×#103—61no—
119#391—22no—
120#391—22no—
121#389—62no—
122#385—62no—
123#384—61no—
124#382—61no—
125C2×#95—22no—
126#343—62no—
127#339—62no—
128#339—62no—
129C24×C3C2×D411yescharacteristic
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—41no—
146C2×C3×Q8—41no—
147C2×C3×Q8—41no—
148C2×C3×Q8—41no—
149C2×C3×Q8—41no—
150C2×C3×Q8—41no—
151C2×C3×Q8—41no—
152C2×C3×Q8—41no—
153C2×C3×Q8—41no—
154C2×C3×Q8—41no—
155C2×C3×Q8—41no—
156C2×C3×Q8—41no—
157C2×C3×Q8C4⋊C414yescomplemented
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—41no—
171C22×C3×C4—41no—
172C22×C3×C4—41no—
173C22×C3×C4—44no—
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×Q811yescharacteristic
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4#4111yescharacteristic
184C22×C3×C4#4111yescharacteristic
185C22×C3×C4#4111yescharacteristic
186C22×C3×C4#4111yescharacteristic
187C22×C3×C4#4111yescharacteristic
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—41no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—41no—
198C3×C42—41no—
199#217—22no—
200#217—22no—
201#217—41no—
202#217—41no—
203C4⋊Dic3—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3C22⋊C414yes—
209Dic3⋊C4—84no—
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—21no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8Dic612yes—
240C22×Q8C4×D312yes—
241C22×Q8#6712yes—
242C23×C4—21no—
243C23×C4—21no—
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—32no—
261#117—32no—
262#117—61no—
263#117—61no—
264#117—61no—
265#117—61no—
266#117—61no—
267#117—61no—
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—32no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—64no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42#6712yes—
288#103—68no—
289#103—64no—
290#103—64no—
291#95—28no—
292#95—64no—
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
302C3×Q8—24no—
303C3×Q8—24no—
304C3×Q8—42no—
305C3×Q8—42no—
306C3×Q8—42no—
307C3×Q8—42no—
308C3×Q8—42no—
309C3×Q8—42no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—44no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—44no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—82no—
326C2×C3×C4C2×C22⋊C411yescharacteristic
327C2×C3×C4C2×C4⋊C411yescharacteristic
328C2×C3×C4#11711yescharacteristic
329C2×C3×C4C4×D411yescharacteristic
330C2×C3×C4C4×D411yescharacteristic
331C2×C3×C4C4×D411yescharacteristic
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4#12111yescharacteristic
335C2×C3×C4#12111yescharacteristic
336C2×C3×C4#12111yescharacteristic
337C2×C3×C4#12111yescharacteristic
338C2×C3×C4#12111yescharacteristic
339C2×C3×C4C22⋊Q811yescharacteristic
340C2×C3×C4C22⋊Q811yescharacteristic
341C2×C3×C4C22⋊Q811yescharacteristic
342C2×C3×C4C22⋊Q811yescharacteristic
343C2×C3×C4C22⋊Q811yescharacteristic
344C2×C3×C4#12311yescharacteristic
345C2×C3×C4#12311yescharacteristic
346C2×C3×C4#12311yescharacteristic
347C2×C3×C4#12311yescharacteristic
348C2×C3×C4#12311yescharacteristic
349C2×C3×C4#12411yescharacteristic
350C2×C3×C4#12411yescharacteristic
351C2×C3×C4#12611yescharacteristic
352C2×C3×C4#12611yescharacteristic
353C2×C3×C4#12711yescharacteristic
354C2×Dic3—24no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C3⋊C8—42no—
362C3⋊C8—42no—
363C24C2×#6711yescharacteristic
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8Dic3⋊C414yescomplemented
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—44no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4C2×Dic611yescharacteristic
412C22×C4C2×C4×D311yescharacteristic
413C22×C4#23511yescharacteristic
414C22×C4#23511yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4#23711yescharacteristic
421C22×C4#23711yescharacteristic
422C22×C4#23711yescharacteristic
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—34no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—62no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—244no—
450C22⋊C4—62no—
451C22⋊C4—62no—
452C22⋊C4—121no—
453C22⋊C4—121no—
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—44no—
481C3×C4—82no—
482C3×C4#38011yescharacteristic
483C3×C4#38511yescharacteristic
484C3×C4#38811yescharacteristic
485C3×C4#39011yescharacteristic
486C3×C4#39411yescharacteristic
487C3×C4#39611yescharacteristic
488C3×C4#39711yescharacteristic
489C3×C4#39911yescharacteristic
490Dic3—44no—
491Dic3—81no—
492Dic3—81no—
493C23—22no—
494C23—22no—
495C23#92411yescharacteristic
496C23C4×#6711yescharacteristic
497C23#93611yescharacteristic
498C23#93711yescharacteristic
499C23#93811yescharacteristic
500C23#93911yescharacteristic
501C23#95311yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—64no—
526C2×C4—82no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4#87711yescharacteristic
534C2×C4#87811yescharacteristic
535C2×C4#87811yescharacteristic
536C2×C4#87811yescharacteristic
537C2×C4#87811yescharacteristic
538C2×C4#87911yescharacteristic
539C2×C4#87911yescharacteristic
540C2×C4D3×C22⋊C411yescharacteristic
541C2×C4#67⋊C411yescharacteristic
542C2×C4#67⋊C411yescharacteristic
543C2×C4#88311yescharacteristic
544C2×C4#88311yescharacteristic
545C2×C4#88411yescharacteristic
546C2×C4#88411yescharacteristic
547C2×C4#88511yescharacteristic
548C2×C4#88511yescharacteristic
549C2×C4C2×Dic3⋊C411yescharacteristic
550C2×C4#92411yescharacteristic
551C2×C4C4×#6711yescharacteristic
552C2×C4#92911yescharacteristic
553C2×C4#93511yescharacteristic
554C2×C4#93511yescharacteristic
555C2×C4#93711yescharacteristic
556C2×C4#93811yescharacteristic
557C2×C4#93911yescharacteristic
558C2×C4#93911yescharacteristic
559C2×C4#93911yescharacteristic
560C2×C4#94011yescharacteristic
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—22no—
573C22—22no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—22no—
578C22—21no—
579C22—21no—
580C22#494511yescharacteristic
581C22#494711yescharacteristic
582C22#494811yescharacteristic
583C22#495611yescharacteristic
584C22#506112yes—
585C22#509512yes—
586C22#521011yescharacteristic
587C22#522211yescharacteristic
588C22#522311yescharacteristic
589C4—44no—
590C4—44no—
591C4—82no—
592C4—124no—
593C4—241no—
594C4—241no—
595C4#494211yescharacteristic
596C4#494411yescharacteristic
597C4#494611yescharacteristic
598C4#494811yescharacteristic
599C4#494911yescharacteristic
600C4#495211yescharacteristic
601C4#495711yescharacteristic
602C4#495911yescharacteristic
603C3#1106411yescharacteristic, complemented, Sylow
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2#6967312yes—
609C2#6969312yes—
610C2#7154711yescharacteristic
611C2#7173111yescharacteristic
612C2#7179511yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#391C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#391D3
hypocenterC3#11064
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
22. #5225C4semidirect
157. C2×C3×Q8C4⋊C4semidirect
391. C2×Q8Dic3⋊C4semidirect
603. C3#11064semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C2×#52252
#821572
C2×#45302
C2×#45232
#110643