Finite Group Databasebeta

← #10702682

Subgroups — #10702682

Order 768.

942 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2#84062C211yesmaximal, characteristic
3C22×#836C211yesmaximal, characteristic
4C2×#4964C211yesmaximal, characteristic
5#71558C211yesmaximal, characteristic
6C2×#4523C211yesmaximal, characteristic
7C2×#4523C211yesmaximal, characteristic
8#11136—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#5229—21no—
20#5229—21no—
21#5229—22no—
22C2×#836—22no—
23C2×#836—22no—
24C2×#836C414yescomplemented
25#5211—22no—
26#4964—22no—
27#4964—22no—
28C2×#808—21no—
29C2×#807—21no—
30#4529—22no—
31#4523—22no—
32#4523—22no—
33C2×#428—31no—
34C22×#101—31no—
35#2310—31no—
36C2×#391D311yescharacteristic
37#1736—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×Q8—21no—
52C22×C3×Q8C2×C412yes—
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⋊C4D412yes—
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#926—21no—
79#926—21no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3D411yescharacteristic
82C2×C4⋊Dic3D412yes—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208C2×C412yes—
94C2×#208C2×C412yes—
95C2×#208D412yes—
96C2×#208D412yes—
97C22×C3⋊C8—21no—
98C22×C3⋊C8—21no—
99C3×#95—28no—
100#836—28no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103#808—24no—
104#807—24no—
105C23×Q8C2×D311yescharacteristic
106C22×M16—31no—
107C22×M16—31no—
108C2×#117—31no—
109C2×#117—31no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C42C2×D311yescharacteristic
112#428—62no—
113#428—62no—
114C2×#107—61no—
115C2×#106—61no—
116C2×#101—34no—
117C2×#101—62no—
118C2×#101—62no—
119#391—22no—
120#391—22no—
121#391—62no—
122#391—61no—
123#391—61no—
124#381—62no—
125C2×#95—22no—
126#340—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—41no—
143C2×C3×Q8—41no—
144C2×C3×Q8—41no—
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×Q8—41no—
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—22no—
167C22×C3×C4—42no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—44no—
171C22×C3×C4—41no—
172C22×C3×C4—41no—
173C22×C3×C4C4⋊C414yes—
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—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—41no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—41no—
198C3×C42—41no—
199#217—41no—
200#217—41no—
201#217—41no—
202#217—41no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4×Dic3—41no—
210C4×Dic3—41no—
211C4×Dic3—41no—
212C4×Dic3—41no—
213#208—22no—
214#208—22no—
215#208—22no—
216#208—22no—
217#208—22no—
218#208—41no—
219#208—41no—
220#208C22⋊C414yes—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8—41no—
227C22×Q8—21no—
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—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4#6712yes—
244C23×C4C22×D311yescharacteristic
245C2×M16—32no—
246C2×M16—32no—
247C2×M16—32no—
248C2×M16—32no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C22×C8—61no—
258C22×C8—61no—
259#117—61no—
260#117—61no—
261#117—61no—
262#117—61no—
263#117—61no—
264#117—61no—
265#117—61no—
266#117—61no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4#6712yes—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—61no—
284C2×C42—61no—
285C2×C42#6712yes—
286#107—64no—
287#106—64no—
288#101—68no—
289#95—28no—
290#95—64no—
291#95—64no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3C2×C4⋊C411yescharacteristic
295C23×C3#12011yescharacteristic
296C23×C3#12011yescharacteristic
297C23×C3#12111yescharacteristic
298C23×C3#12111yescharacteristic
299C23×C3#12311yescharacteristic
300C23×C3#12311yescharacteristic
301C3×Q8—42no—
302C3×Q8—42no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—42no—
306C3×Q8—42no—
307C3×Q8—42no—
308C3×Q8—42no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—44no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—82no—
324C2×C3×C4—82no—
325C2×C3×C4C2×C22⋊C411yescharacteristic
326C2×C3×C4#11711yescharacteristic
327C2×C3×C4C4×D411yescharacteristic
328C2×C3×C4C4×D411yescharacteristic
329C2×C3×C4C4×D411yescharacteristic
330C2×C3×C4C4×D411yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12111yescharacteristic
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4#12111yescharacteristic
335C2×C3×C4#12111yescharacteristic
336C2×C3×C4#12111yescharacteristic
337C2×C3×C4#12111yescharacteristic
338C2×C3×C4C22⋊Q811yescharacteristic
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#12411yescharacteristic
348C2×C3×C4#12411yescharacteristic
349C2×C3×C4#12411yescharacteristic
350C2×C3×C4#12611yescharacteristic
351C2×C3×C4#12611yescharacteristic
352C2×C3×C4#12711yescharacteristic
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C3⋊C8—24no—
361C3⋊C8—42no—
362C24C2×#6711yescharacteristic
363C2×Q8—22no—
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—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
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—
391C22×C4—24no—
392C22×C4—22no—
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—44no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4Dic3⋊C414yes—
410C22×C4C2×Dic611yescharacteristic
411C22×C4C2×C4×D311yescharacteristic
412C22×C4#23511yescharacteristic
413C22×C4#23511yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4#23711yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4#23711yescharacteristic
421C22×C4#23711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424M16—34no—
425M16—62no—
426M16—62no—
427M16—62no—
428M16—62no—
429M16—62no—
430M16—121no—
431M16—121no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—121no—
437C2×C8—121no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C22⋊C4—121no—
449C22⋊C4—121no—
450C22⋊C4—121no—
451C22⋊C4—121no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42—41no—
456C42—121no—
457C42—121no—
458C42—121no—
459C42—121no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3#38011yescharacteristic
469C22×C3#38511yescharacteristic
470C22×C3#38511yescharacteristic
471C22×C3#39211yescharacteristic
472C22×C3#39411yescharacteristic
473C22×C3#39411yescharacteristic
474C22×C3#39711yescharacteristic
475C22×C3#45412yes—
476C22×C3#45612yes—
477C3×C4—44no—
478C3×C4—44no—
479C3×C4—82no—
480C3×C4#38611yescharacteristic
481C3×C4#38811yescharacteristic
482C3×C4#38811yescharacteristic
483C3×C4#39011yescharacteristic
484C3×C4#39311yescharacteristic
485C3×C4#39611yescharacteristic
486C3×C4#39611yescharacteristic
487C3×C4#39911yescharacteristic
488Dic3—81no—
489Dic3—81no—
490Dic3—82no—
491C23—22no—
492C23—22no—
493C23C2×Dic3⋊C411yescharacteristic
494C23#92911yescharacteristic
495C23#93511yescharacteristic
496C23#93711yescharacteristic
497C23#93911yescharacteristic
498C23#93911yescharacteristic
499C23#95311yescharacteristic
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508C2×C4—24no—
509C2×C4—24no—
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—44no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4#87711yescharacteristic
532C2×C4#87811yescharacteristic
533C2×C4#87811yescharacteristic
534C2×C4#87811yescharacteristic
535C2×C4#87811yescharacteristic
536C2×C4#87911yescharacteristic
537C2×C4#87911yescharacteristic
538C2×C4D3×C22⋊C411yescharacteristic
539C2×C4#67⋊C411yescharacteristic
540C2×C4#67⋊C411yescharacteristic
541C2×C4#88311yescharacteristic
542C2×C4#88311yescharacteristic
543C2×C4#88411yescharacteristic
544C2×C4#88411yescharacteristic
545C2×C4#88511yescharacteristic
546C2×C4#88511yescharacteristic
547C2×C4#92411yescharacteristic
548C2×C4#92411yescharacteristic
549C2×C4C4×#6711yescharacteristic
550C2×C4C4×#6711yescharacteristic
551C2×C4#93511yescharacteristic
552C2×C4#93611yescharacteristic
553C2×C4#93711yescharacteristic
554C2×C4#93811yescharacteristic
555C2×C4#93811yescharacteristic
556C2×C4#93911yescharacteristic
557C2×C4#93911yescharacteristic
558C2×C4#94011yescharacteristic
559C8—64no—
560C8—122no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3#143911yescharacteristic
566C2×C3#185712yes—
567C2×C3#200311yescharacteristic
568C2×C3#200511yescharacteristic
569C2×C3#203812yes—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—21no—
577C22—21no—
578C22#494211yescharacteristic
579C22#494611yescharacteristic
580C22#495211yescharacteristic
581C22#495911yescharacteristic
582C22#506212yes—
583C22#509512yes—
584C22#521011yescharacteristic
585C22#522211yescharacteristic
586C22#522311yescharacteristic
587C4—44no—
588C4—44no—
589C4—82no—
590C4—241no—
591C4—242no—
592C4—241no—
593C4#494411yescharacteristic
594C4#494511yescharacteristic
595C4#494711yescharacteristic
596C4#494811yescharacteristic
597C4#494811yescharacteristic
598C4#494911yescharacteristic
599C4#495611yescharacteristic
600C4#495711yescharacteristic
601C3#1113611yescharacteristic, complemented, Sylow
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2#6967212yes—
607C2#6969512yes—
608C2#7154711yescharacteristic
609C2#7173011yescharacteristic
610C2#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#11136
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
24. C2×#836C4semidirect
601. C3#11136semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
#840622
C22×#8362
C2×#49642
#715582
C2×#45232
#111363