Finite Group Databasebeta

← #10767975

Subgroups — #10767975

Order 768.

1106 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
1#86280C211yesmaximal, characteristic, complemented
2#83353C211yesmaximal, characteristic, complemented
3#79344C211yesmaximal, characteristic, complemented
4#73418C211yesmaximal, characteristic
5C3×#1506C211yesmaximal, characteristic, complemented
6#70827C211yesmaximal, characteristic, complemented
7#69433C211yesmaximal, characteristic, complemented
8#68322C214yesmaximal, complemented
9#68316C214yesmaximal, complemented
10#18180—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12#5730—21no—
13#5727—21no—
14#5690—22no—
15#5634C2214yes—
16D3×C22⋊Q8—22no—
17#5579—22no—
18#5579—21no—
19Q8×D12—21no—
20#5521—21no—
21#5519—21no—
22C2×C4×D12—22no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25#5470C2211yescharacteristic
26C3×#432C2211yescharacteristic
27C3×#376C2211yescharacteristic
28#5087—21no—
29#5083—21no—
30#5046C2212yes—
31#5044C2212yes—
32#5026—22no—
33#4926C2211yescharacteristic
34#4846—21no—
35#4841—21no—
36#4731C2212yes—
37#4722C2212yes—
38#4710—22no—
39#4677—21no—
40#4677C2212yes—
41#4671—21no—
42#4671C2212yes—
43C3×#328C2214yes—
44#4468C2214yes—
45#4452C2214yes—
46#3501—31no—
47#3110—31no—
48#3110—31no—
49#2397—31no—
50#1614—34no—
51#1596—34no—
52#1506—31no—
53#1506D311yescharacteristic, complemented
54#1466—31no—
55C2×D3×Q8—22no—
56C22×D12—22no—
57C22×D12D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×C4×D3D412yes—
61C3×C4⋊Q8D411yescharacteristic, complemented
62C3×C4⋊Q8D411yescharacteristic, complemented
63C3×#125—21no—
64C3×C22⋊Q8—21no—
65C3×C22⋊Q8C2312yes—
66C3×C4×Q8—21no—
67C3×C4×Q8D412yescomplemented
68C3×#117—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71#946—22no—
72#946—24no—
73C2×#215—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4C2312yes—
78#926—21no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82#924—21no—
83#924C2312yes—
84C2×#204—22no—
85#897—22no—
86#897—24no—
87#896—24no—
88#896—24no—
89#896—24no—
90#893—22no—
91#893—22no—
92D3×C4⋊C4—24no—
93D3×C4⋊C4—22no—
94#890—22no—
95#888—22no—
96D3×C22⋊C4—24no—
97#878—24no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—21no—
101C4×D12—21no—
102C4×D12—21no—
103C4×D12—21no—
104C4×D12—21no—
105C4×D12—21no—
106C4×D12—21no—
107C4×D12—21no—
108C4×D12—48no—
109#870—22no—
110#870—21no—
111#869—22no—
112#869D411yescharacteristic, complemented
113#869D411yescharacteristic, complemented
114C4×Dic6—21no—
115C4×Dic6D412yescomplemented
116C3×C4⋊C8—21no—
117C3×C22⋊C8C2312yes—
118C3×#97—21no—
119C3×#95—21no—
120C3×#95C2312yes—
121#835—22no—
122#830C2312yes—
123#821—22no—
124#818—22no—
125#816—22no—
126#809—22no—
127#807—22no—
128C4⋊(C3⋊C8)—21no—
129#803—21no—
130#554—61no—
131#554—62no—
132#554—61no—
133#554—62no—
134D4×Q8—61no—
135D4×Q8—61no—
136#544—61no—
137#543—34no—
138#533—31no—
139#533C2×D311yescharacteristic
140C2×C22⋊Q8—62no—
141#518—61no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—62no—
145#484—32no—
146#484—32no—
147#474—61no—
148#474—61no—
149#461—61no—
150#461—61no—
151#450—32no—
152#450—32no—
153#440—62no—
154#440—62no—
155#432—31no—
156#432C2×D311yescharacteristic
157#399—32no—
158#399—61no—
159#392—32no—
160#392—61no—
161#376C2×D311yescharacteristic
162#328—34no—
163#328C2×D314yes—
164#323—34no—
165C23×D3—21no—
166C23×D3C2×D412yes—
167C2×C3×Q8—21no—
168C2×C3×Q8C2×D412yes—
169C22×C3×C4—22no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—21no—
174C22×Dic3C2×D412yes—
175D3×Q8—84no—
176C2×D12—24no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—42no—
182C2×D12—44no—
183C2×D12—44no—
184C2×D12—44no—
185C2×C4×D3—22no—
186C2×C4×D3—24no—
187C2×C4×D3—24no—
188C2×C4×D3—22no—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×C4×D3—41no—
192C2×C4×D3—41no—
193C2×Dic6—21no—
194C2×Dic6—24no—
195C2×Dic6C2×D412yes—
196C2×C3×C8—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4C2×D412yes—
203C3×C22⋊C4—22no—
204C3×C42—24no—
205C3×C42—22no—
206C3×C42C2×D411yescharacteristic
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209C3×C42C2×D411yescharacteristic
210#217—22no—
211#215—44no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—22no—
216(C2×D3)⋊C4—44no—
217(C2×D3)⋊C4—44no—
218(C2×D3)⋊C4—41no—
219(C2×D3)⋊C4—41no—
220C4⋊Dic3—24no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—44no—
229C4⋊Dic3C2×D412yes—
230Dic3⋊C4—24no—
231Dic3⋊C4—24no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234C4×Dic3—22no—
235C2×C3⋊C8—22no—
236#204—44no—
237C22×Q8—62no—
238C22×D4—31no—
239C22×D4—31no—
240C22×D4—62no—
241C23×C4—32no—
242C23×C4—61no—
243C2×SD16—62no—
244C2×SD16—62no—
245C4⋊Q8—31no—
246C4⋊Q8—31no—
247C4⋊Q8—62no—
248C4⋊Q8—62no—
249C4⋊Q8D1211yescharacteristic, complemented
250C4⋊Q8D1211yescharacteristic, complemented
251#125—21no—
252#125—62no—
253#125—62no—
254#125—61no—
255C22⋊Q8—21no—
256C22⋊Q8—32no—
257C22⋊Q8—64no—
258C22⋊Q8—64no—
259C22⋊Q8—64no—
260C22⋊Q8—62no—
261C22⋊Q8—64no—
262C22⋊Q8—62no—
263C22⋊Q8—64no—
264C22⋊Q8—61no—
265C22⋊Q8—64no—
266C22⋊Q8C22×D312yes—
267C4×Q8—21no—
268C4×Q8—32no—
269C4×Q8—61no—
270C4×Q8D1212yescomplemented
271C4×D4—62no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—61no—
276C4×D4—61no—
277C4×D4—61no—
278C4×D4—61no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4—61no—
283C4×D4—128no—
284#117—21no—
285#117—61no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—64no—
290C2×C4⋊C4—62no—
291C2×C22⋊C4—32no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—61no—
296C2×C42—21no—
297C2×C42C22×D311yescharacteristic
298#107—62no—
299#107—62no—
300C4⋊C8—21no—
301C4⋊C8—61no—
302#103—62no—
303#103—62no—
304#102—62no—
305#102—62no—
306C22⋊C8—32no—
307C22⋊C8C22×D312yes—
308#97—21no—
309#97—61no—
310#95—21no—
311#95C22×D312yes—
312C23×C3C22×D411yescharacteristic
313C22×D3—22no—
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—42no—
322C22×D3#13714yes—
323C3×Q8—42no—
324C3×Q8—42no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12011yescharacteristic
336C2×C3×C4#12011yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—41no—
345D12—24no—
346D12—24no—
347D12—22no—
348D12—22no—
349D12—41no—
350D12—41no—
351D12—416no—
352C4×D3—44no—
353C4×D3—82no—
354C4×D3—84no—
355C4×D3—84no—
356C4×D3—82no—
357C4×D3—82no—
358Dic6—42no—
359Dic6—42no—
360Dic6—84no—
361C3×C8—42no—
362C3⋊C8—42no—
363C24—32no—
364C24—61no—
365C2×Q8—21no—
366C2×Q8—32no—
367C2×Q8—64no—
368C2×Q8—61no—
369C2×Q8—244no—
370C2×Q8C2×D1212yes—
371C2×D4—64no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—61no—
375C2×D4—61no—
376C2×D4—124no—
377C2×D4—122no—
378C2×D4—124no—
379C2×D4—124no—
380C22×C4—22no—
381C22×C4—32no—
382C22×C4—64no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—124no—
388C22×C4—124no—
389C22×C4—121no—
390C22×C4—121no—
391C22×C4D3×D411yescharacteristic
392C22×C4D3×D411yescharacteristic
393C22×C4C23×D311yescharacteristic
394SD16—124no—
395SD16—124no—
396C2×C8—22no—
397C2×C8—62no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—32no—
404C4⋊C4—64no—
405C4⋊C4—64no—
406C4⋊C4—64no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—62no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—62no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C4⋊C4—124no—
417C4⋊C4C2×D1212yes—
418C22⋊C4—22no—
419C22⋊C4—64no—
420C22⋊C4—64no—
421C22⋊C4—62no—
422C22⋊C4—62no—
423C22⋊C4—62no—
424C22⋊C4—124no—
425C22⋊C4—124no—
426C22⋊C4—121no—
427C22⋊C4—121no—
428C42—24no—
429C42—22no—
430C42—62no—
431C42C2×D1211yescharacteristic
432C42C2×D1211yescharacteristic
433C42D3×D411yescharacteristic
434C42D3×D411yescharacteristic
435C22×C3—21no—
436C22×C3—21no—
437C22×C3C2×#12011yescharacteristic
438C22×C3C2×#13712yes—
439C2×D3—24no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—44no—
443C2×D3—44no—
444C2×D3—42no—
445C2×D3—82no—
446C2×D3—82no—
447C2×D3—81no—
448C2×D3—81no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—44no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4#45312yes—
458Dic3—42no—
459Dic3—42no—
460Dic3—42no—
461Dic3—81no—
462C23—34no—
463C23—62no—
464C23—62no—
465C23—62no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—121no—
470C23—121no—
471C23—122no—
472C23C2×D3×D411yescharacteristic
473Q8—42no—
474Q8—42no—
475Q8—122no—
476Q8—122no—
477Q8—244no—
478D4—64no—
479D4—64no—
480D4—62no—
481D4—62no—
482D4—121no—
483D4—121no—
484D4—1216no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—124no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—121no—
500C2×C4—242no—
501C2×C4—242no—
502C2×C4—244no—
503C2×C4—244no—
504C2×C4—242no—
505C2×C4#88211yescharacteristic
506C2×C4#88211yescharacteristic
507C2×C4#88211yescharacteristic
508C2×C4#88211yescharacteristic
509C2×C4C22×D1211yescharacteristic
510C2×C4C2×D3×D411yescharacteristic
511C8—42no—
512C8—122no—
513C2×C3—22no—
514C2×C3#300411yescharacteristic
515C2×C3C2×#45311yescharacteristic
516C2×C3#302511yescharacteristic
517D3—44no—
518D3—82no—
519C22—21no—
520C22—21no—
521C22—64no—
522C22—124no—
523C22—124no—
524C22—122no—
525C22—122no—
526C22—122no—
527C22—242no—
528C22—242no—
529C22—241no—
530C22—241no—
531C22C2×#88211yescharacteristic
532C22D3×#13712yes—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—44no—
539C4—42no—
540C4—42no—
541C4—122no—
542C4—122no—
543C4—122no—
544C4—241no—
545C4#482212yes—
546C3#1818011yescharacteristic, complemented, Sylow
547C2—22no—
548C2—124no—
549C2—242no—
550C2#7869011yescharacteristic
551C2#7895811yescharacteristic
552C2C2×#482211yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1506C2
socleC22×C3C2×#120
prime radical#73418C2
hypercenter#1506D3
hypocenterC3#18180
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
1. #86280C2semidirect
2. #83353C2semidirect
3. #79344C2semidirect
5. C3×#1506C2semidirect
6. #70827C2semidirect
7. #69433C2semidirect
8. #68322C2semidirect
9. #68316C2semidirect
53. #1506D3semidirect
61. C3×C4⋊Q8D4semidirect
62. C3×C4⋊Q8D4semidirect
67. C3×C4×Q8D4semidirect
112. #869D4semidirect
113. #869D4semidirect
115. C4×Dic6D4semidirect
249. C4⋊Q8D12semidirect
250. C4⋊Q8D12semidirect
270. C4×Q8D12semidirect
546. C3#18180semidirect

Maximal subgroups

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

subgroupindex
#862802
#833532
#793442
#734182
C3×#15062
#708272
#694332
#683222
#683162
#181803