Finite Group Databasebeta

← #10690195

Subgroups — #10690195

Order 768.

1296 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74005C211yesmaximal, characteristic, complemented
2C2×#4671C211yesmaximal, characteristic
3#69959C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic, complemented
5#69521C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9017—31nomaximal, Sylow
9C23×D12—21no—
10C23×D12C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C2×D3×C22⋊C4—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5212C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#821—22no—
20#4989—22no—
21C2×#809—22no—
22#4942—21no—
23#4671—22no—
24#4671C414yescomplemented
25#4668—22no—
26#4468—22no—
27C2×#392—31no—
28#2315—31no—
29#1786—31no—
30#1786—31no—
31C2×#323—31no—
32#1294—31no—
33#1294D311yescharacteristic, complemented
34C24×D3—21no—
35C24×D3D411yescharacteristic
36C23×C3×C4—21no—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—24no—
44C22×D12—21no—
45C22×D12—44no—
46C22×D12—44no—
47C22×D12—44no—
48C22×D12—42no—
49C22×D12—42no—
50C22×D12C2×C412yescomplemented
51C22×C4×D3—22no—
52C22×C3×C8—21no—
53C2×C3×C4⋊C4D412yescomplemented
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—24no—
62C2×(C2×D3)⋊C4—44no—
63C2×(C2×D3)⋊C4C2×C412yes—
64C2×(C2×D3)⋊C4D412yes—
65C2×C4⋊Dic3D412yescomplemented
66C2×Dic3⋊C4—22no—
67C22×C3⋊C8—21no—
68D3×C22⋊C4—42no—
69D3×C22⋊C4—42no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95C2×C412yescomplemented
75C3⋊#95—22no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#821—44no—
80#809—44no—
81C23×D4—31no—
82C23×D4—61no—
83C22×C22⋊C4—31no—
84C22×C22⋊C4—61no—
85C22×C22⋊C4—61no—
86C2×#102—62no—
87C2×#102—62no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8C2×D311yescharacteristic
90#392—34no—
91#392—62no—
92#385—62no—
93#385—62no—
94#380—31no—
95#380—61no—
96#380C2×D311yescharacteristic
97C2×#95C2×D311yescharacteristic
98#323—62no—
99C24×C3C2×D411yescharacteristic
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—24no—
105C23×D3—22no—
106C23×D3—42no—
107C23×D3—42no—
108C23×D3—42no—
109C23×D3—42no—
110C23×D3—42no—
111C23×D3—41no—
112C23×D3—41no—
113C23×D3—41no—
114C23×D3—41no—
115C23×D3—41no—
116C23×D3—41no—
117C23×D3C22⋊C412yes—
118C23×D3D812yes—
119C23×D3SD1612yes—
120C22×C3×C4—24no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C22×Dic3—42no—
137C22×Dic3C22⋊C412yes—
138C2×D12—24no—
139C2×D12—24no—
140C2×D12—22no—
141C2×D12—22no—
142C2×D12—44no—
143C2×D12—44no—
144C2×D12—44no—
145C2×D12—42no—
146C2×D12—42no—
147C2×D12—48no—
148C2×D12—44no—
149C2×D12—48no—
150C2×D12—416no—
151C2×D12—416no—
152C2×D12—48no—
153C2×D12—41no—
154C2×D12—41no—
155C2×C4×D3—44no—
156C2×C4×D3—82no—
157C2×C4×D3—82no—
158C2×C3×C8—42no—
159C2×C3×C8—41no—
160C3×C4⋊C4—42no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163#217—41no—
164#217—41no—
165(C2×D3)⋊C4—22no—
166(C2×D3)⋊C4—22no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—48no—
170(C2×D3)⋊C4—44no—
171C4⋊Dic3—42no—
172Dic3⋊C4—82no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—41no—
175C25—31no—
176C25—61no—
177C22×D4—32no—
178C22×D4—64no—
179C22×D4—61no—
180C22×D4—61no—
181C22×D4—61no—
182C22×D4—61no—
183C22×D4—61no—
184C22×D4—124no—
185C22×D4—124no—
186C22×D4—122no—
187C22×D4—122no—
188C22×D4—124no—
189C23×C4—21no—
190C23×C4—31no—
191C23×C4—62no—
192C23×C4C22×D311yescharacteristic
193C22×C8—21no—
194C22×C8—61no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4D1212yescomplemented
198C2×C22⋊C4—21no—
199C2×C22⋊C4—32no—
200C2×C22⋊C4—32no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—64no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—122no—
209C2×C22⋊C4—122no—
210C2×C22⋊C4—124no—
211#102—124no—
212#102—124no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217#95—21no—
218#95—21no—
219#95—62no—
220#95—61no—
221#95C4×D312yescomplemented
222C23×C3—22no—
223C23×C3—22no—
224C23×C3C2×C22⋊C411yescharacteristic
225C23×C3#12011yescharacteristic
226C23×C3#12311yescharacteristic
227C23×C3C2×D811yescharacteristic
228C23×C3C2×SD1611yescharacteristic
229C23×C3AGL(1,ℤ/8)11yescharacteristic
230C23×C3#13711yescharacteristic
231C22×D3—22no—
232C22×D3—22no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—22no—
236C22×D3—22no—
237C22×D3—24no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—44no—
243C22×D3—44no—
244C22×D3—42no—
245C22×D3—44no—
246C22×D3—44no—
247C22×D3—42no—
248C22×D3—42no—
249C22×D3—44no—
250C22×D3—44no—
251C22×D3—44no—
252C22×D3—42no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—44no—
256C22×D3—82no—
257C22×D3—82no—
258C22×D3—81no—
259C22×D3—81no—
260C22×D3—82no—
261C22×D3—82no—
262C22×D3—82no—
263C22×D3—82no—
264C22×D3—82no—
265C22×D3—81no—
266C22×D3—81no—
267C22×D3—82no—
268C22×D3—81no—
269C22×D3—81no—
270C22×D3#10214yescomplemented
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4C4×D411yescharacteristic
286C2×C3×C4C4×D411yescharacteristic
287C2×C3×C4#12111yescharacteristic
288C2×C3×C4#12111yescharacteristic
289C2×Dic3—22no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—81no—
296D12—24no—
297D12—24no—
298D12—44no—
299D12—44no—
300D12—42no—
301D12—42no—
302D12—432no—
303C4×D3—88no—
304C3×C8—81no—
305C3⋊C8—81no—
306C24—32no—
307C24—32no—
308C24—32no—
309C24—64no—
310C24—61no—
311C24—61no—
312C24—61no—
313C24—61no—
314C24—62no—
315C24—122no—
316C24—122no—
317C24—122no—
318C24—122no—
319C24—122no—
320C24—121no—
321C24—121no—
322C24—121no—
323C24—121no—
324C24—121no—
325C24—121no—
326C24D3×D411yescharacteristic
327C2×D4—64no—
328C2×D4—64no—
329C2×D4—62no—
330C2×D4—62no—
331C2×D4—122no—
332C2×D4—122no—
333C2×D4—128no—
334C2×D4—124no—
335C2×D4—124no—
336C2×D4—124no—
337C2×D4—128no—
338C2×D4—124no—
339C2×D4—128no—
340C2×D4—1216no—
341C2×D4—1216no—
342C2×D4—121no—
343C2×D4—121no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—32no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—124no—
356C22×C4—122no—
357C22×C4—242no—
358C22×C4—242no—
359C22×C4C2×C4×D311yescharacteristic
360C22×C4C2×D1211yescharacteristic
361C22×C4#23511yescharacteristic
362C22×C4D3×D411yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4#23911yescharacteristic
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—122no—
368C2×C8—121no—
369C4⋊C4—42no—
370C4⋊C4—122no—
371C4⋊C4—242no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—62no—
375C22⋊C4—62no—
376C22⋊C4—62no—
377C22⋊C4—62no—
378C22⋊C4—128no—
379C22⋊C4—124no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3#38611yescharacteristic
391C22×C3C2×#10211yescharacteristic
392C22×C3#41711yescharacteristic
393C22×C3#44711yescharacteristic
394C22×C3#44811yescharacteristic
395C22×C3#45312yes—
396C22×C3B(4,𝔽2)12yes—
397C22×C3#48011yescharacteristic
398C22×C3#48111yescharacteristic
399C2×D3—24no—
400C2×D3—44no—
401C2×D3—44no—
402C2×D3—42no—
403C2×D3—42no—
404C2×D3—44no—
405C2×D3—44no—
406C2×D3—44no—
407C2×D3—44no—
408C2×D3—82no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—84no—
414C2×D3—84no—
415C2×D3—84no—
416C2×D3—84no—
417C2×D3—82no—
418C2×D3—82no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—44no—
424C3×C4—82no—
425Dic3—42no—
426Dic3—82no—
427C23—22no—
428C23—22no—
429C23—34no—
430C23—64no—
431C23—62no—
432C23—62no—
433C23—62no—
434C23—62no—
435C23—62no—
436C23—62no—
437C23—122no—
438C23—122no—
439C23—124no—
440C23—124no—
441C23—122no—
442C23—122no—
443C23—124no—
444C23—122no—
445C23—124no—
446C23—124no—
447C23—124no—
448C23—124no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—124no—
453C23—122no—
454C23—122no—
455C23—241no—
456C23—241no—
457C23—242no—
458C23—242no—
459C23—242no—
460C23—242no—
461C23—242no—
462C23—242no—
463C23—242no—
464C23—241no—
465C23—241no—
466C23—241no—
467C23—242no—
468C23—241no—
469C23D3×C22⋊C411yescharacteristic
470C23#88211yescharacteristic
471C23#89411yescharacteristic
472C23D3×D811yescharacteristic
473C23D3×SD1611yescharacteristic
474C23#91411yescharacteristic
475C23#91811yescharacteristic
476D4—64no—
477D4—64no—
478D4—124no—
479D4—124no—
480D4—122no—
481D4—122no—
482D4—1232no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—62no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—248no—
504C2×C4—241no—
505C2×C4C4×D1211yescharacteristic
506C2×C4#88411yescharacteristic
507C2×C4#89311yescharacteristic
508C2×C4#89511yescharacteristic
509C8—81no—
510C8—241no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3#159512yes—
516C2×C3#159612yes—
517C2×C3#187411yescharacteristic
518C2×C3#188711yescharacteristic
519C2×C3#188911yescharacteristic
520D3—44no—
521D3—84no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—64no—
531C22—124no—
532C22—124no—
533C22—124no—
534C22—122no—
535C22—122no—
536C22—124no—
537C22—124no—
538C22—124no—
539C22—242no—
540C22—242no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22—244no—
545C22—244no—
546C22—242no—
547C22—242no—
548C22—244no—
549C22—244no—
550C22#466511yescharacteristic
551C22#474112yes—
552C22D3×#10211yescharacteristic
553C22#477311yescharacteristic
554C22#480311yescharacteristic
555C22#480611yescharacteristic
556C22#482212yes—
557C22#486311yescharacteristic
558C22#488211yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—44no—
564C4—82no—
565C4—122no—
566C4—242no—
567C3#901711yescharacteristic, complemented, Sylow
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—124no—
573C2—244no—
574C2#6828412yes—
575C2#6831612yes—
576C2#6876011yescharacteristic
577C2#6888011yescharacteristic
578C2#6896211yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC2×#4671C2
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9017
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
1. #74005C2semidirect
3. #69959C2semidirect
4. C2×#4468C2semidirect
5. #69521C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
24. #4671C4semidirect
33. #1294D3semidirect
50. C22×D12C2×C4semidirect
53. C2×C3×C4⋊C4D4semidirect
65. C2×C4⋊Dic3D4semidirect
74. C3×#95C2×C4semidirect
197. C2×C4⋊C4D12semidirect
221. #95C4×D3semidirect
270. C22×D3#102semidirect
567. C3#9017semidirect

Maximal subgroups

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

subgroupindex
#740052
C2×#46712
#699592
C2×#44682
#695212
C3×#12942
#676122
#90173