Finite Group Databasebeta

← #10698518

Subgroups — #10698518

Order 768.

602 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2639C211yesmaximal, characteristic
2#71692C211yesmaximal, characteristic, complemented
3C2×#4538C211yesmaximal, characteristic, complemented
4#71677C211yesmaximal, characteristic
5C2×#4537C211yesmaximal, characteristic
6#69415C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#10404—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#400—21no—
13C3×#397—21no—
14C3×#394—21no—
15C3×#393—21no—
16C3×#386—21no—
17C3×#386—21no—
18C2×C3×#95C2211yescharacteristic
19C2×#835—21no—
20C2×#832—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95—21no—
23C2×C3⋊#95C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#4538—22no—
27#4537—22no—
28#4529—21no—
29#4529—21no—
30#2639D311yescharacteristic
31#1874—31no—
32#1872—31no—
33C2×#328—31no—
34C2×#327—31no—
35#1442—31no—
36#1280—31no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—42no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73#835—42no—
74#832—42no—
75C3⋊#95—21no—
76C3⋊#95—42no—
77C3⋊#95—42no—
78C3⋊#95D412yes—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83C2×C22⋊Q8C2×D311yescharacteristic
84C2×#121C2×D311yescharacteristic
85C22×C4⋊C4—31no—
86C2×#103—61no—
87C2×#102—61no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8—31no—
90#400—21no—
91#397—21no—
92#394—21no—
93#393—21no—
94#386—21no—
95#386—21no—
96C2×#95—31no—
97C2×#95—61no—
98C2×#95—61no—
99C2×#95C2×D311yescharacteristic
100#340—61no—
101#340—61no—
102#328—62no—
103#327—62no—
104C24×C3—21no—
105C24×C3C2×D411yescharacteristic
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—81no—
166C3×C22⋊C4—81no—
167C3×C22⋊C4—81no—
168C3×C22⋊C4—81no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—41no—
172C4⋊Dic3—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C22×Q8#6711yescharacteristic
180C22×D4—21no—
181C22×D4#6711yescharacteristic
182C23×C4—61no—
183C23×C4—61no—
184C23×C4#6711yescharacteristic
185C23×C4#6711yescharacteristic
186C23×C4C22×D311yescharacteristic
187C22×C8—61no—
188C22×C8—61no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196#121—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—122no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209#103—122no—
210#102—122no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—32no—
222#95—61no—
223#95—122no—
224#95—122no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3#12011yescharacteristic
237C23×C3#12111yescharacteristic
238C23×C3#12411yescharacteristic
239C23×C3#13511yescharacteristic
240C23×C3#13511yescharacteristic
241C23×C3AGL(1,ℤ/8)11yescharacteristic
242C23×C3#13711yescharacteristic
243C3×Q8—42no—
244C3×Q8—81no—
245C3×D4—42no—
246C3×D4—81no—
247C3×D4—81no—
248C3×D4—81no—
249C3×D4—81no—
250C3×D4—81no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—81no—
277C2×C3×C4—81no—
278C2×C3×C4#12111yescharacteristic
279C2×C3×C4#12111yescharacteristic
280C2×C3×C4#12311yescharacteristic
281C2×C3×C4#12311yescharacteristic
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—81no—
289C2×Dic3—81no—
290C2×Dic3—81no—
291C2×Dic3—81no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C3⋊C8—81no—
297C3⋊C8—81no—
298C24—21no—
299C24C2×#6711yescharacteristic
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4D3×D411yescharacteristic
344C22×C4#23711yescharacteristic
345C22×C4#23711yescharacteristic
346C22×C4#23711yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C2×C8—121no—
350C2×C8—121no—
351C2×C8—121no—
352C2×C8—121no—
353C2×C8—121no—
354C2×C8—121no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—81no—
370C22⋊C4—81no—
371C22⋊C4—81no—
372C22⋊C4—81no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3#39411yescharacteristic
393C22×C3#44911yescharacteristic
394C22×C3#45211yescharacteristic
395C22×C3#45411yescharacteristic
396C22×C3#45511yescharacteristic
397C22×C3B(4,𝔽2)11yescharacteristic
398C22×C3#45811yescharacteristic
399C22×C3#46311yescharacteristic
400C22×C3#46411yescharacteristic
401C22×C3#48811yescharacteristic
402C22×C3#48911yescharacteristic
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4—81no—
412Dic3—82no—
413Dic3—82no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23#93611yescharacteristic
426C23#93811yescharacteristic
427C23#94911yescharacteristic
428C23#95011yescharacteristic
429C23#95011yescharacteristic
430C23#95111yescharacteristic
431C23#95311yescharacteristic
432Q8—42no—
433Q8—81no—
434D4—42no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—81no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—241no—
474C2×C4—241no—
475C2×C4—241no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4—241no—
480C2×C4—241no—
481C2×C4#93511yescharacteristic
482C2×C4#93511yescharacteristic
483C2×C4#93911yescharacteristic
484C2×C4#93911yescharacteristic
485C8—241no—
486C8—241no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—161no—
492C2×C3#159711yescharacteristic
493C2×C3#160611yescharacteristic
494C2×C3#161211yescharacteristic
495C2×C3#161511yescharacteristic
496C2×C3#204011yescharacteristic
497C2×C3#204211yescharacteristic
498C2×C3#204411yescharacteristic
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22#505411yescharacteristic
519C22#505511yescharacteristic
520C22#505611yescharacteristic
521C22#505711yescharacteristic
522C22#522211yescharacteristic
523C22#523811yescharacteristic
524C22#523911yescharacteristic
525C22#524211yescharacteristic
526C22#524311yescharacteristic
527C22#524411yescharacteristic
528C22#524511yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—242no—
539C4—242no—
540C3#1040411yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2—161no—
546C2#7172211yescharacteristic
547C2#7185011yescharacteristic
548C2#7187711yescharacteristic
549C2#7188211yescharacteristic
550C2#7188511yescharacteristic
551C2#7188711yescharacteristic
552C2#7188811yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2639C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenter#2639D3
hypocenterC3#10404
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2639

Complements

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

NH ≅ G/Ntype
2. #71692C2semidirect
3. C2×#4538C2semidirect
6. #69415C2semidirect
7. #67613C2semidirect
540. C3#10404semidirect

Maximal subgroups

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

subgroupindex
C3×#26392
#716922
C2×#45382
#716772
C2×#45372
#694152
#676132
#104043