Finite Group Databasebeta

← #10823672

Subgroups — #10823672

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85777C211yesmaximal, characteristic
2C3×#2326C211yesmaximal, characteristic, complemented
3C4×#942C212yesmaximal, complemented
4#71731C212yesmaximal, complemented
5C4×#833C211yesmaximal, characteristic, complemented
6#69694C211yesmaximal, characteristic, complemented
7#69673C211yesmaximal, characteristic, complemented
8#69665C212yesmaximal, complemented
9#69593C211yesmaximal, characteristic, complemented
10#69573C211yesmaximal, characteristic, complemented
11#69430C212yesmaximal, complemented
12#23593—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#942C2212yes—
17#5805C2212yes—
18#5791—22no—
19#5562—22no—
20#5561—22no—
21#5557—22no—
22#5548—22no—
23#5510C2211yescharacteristic
24#5500C2211yescharacteristic
25#5494—22no—
26C4×#235C2212yes—
27C3×#391C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#388—22no—
30C3×#385—22no—
31C3×#384C2211yescharacteristic
32C3×#376C2211yescharacteristic
33#5225C2212yes—
34C2×#833C2212yes—
35#5030—22no—
36C4×#216—22no—
37#5026—22no—
38C4×#215—22no—
39#4966C2212yes—
40#4964C2212yes—
41C2×#805C2212yes—
42C2×#805C2212yes—
43C4×#208C2212yes—
44#4530C2211yescharacteristic
45#4530C2211yescharacteristic
46#4523C2212yes—
47#4523C2212yes—
48#3460—31no—
49C4×#137—32no—
50#2326D311yescharacteristic, complemented
51#2060—31no—
52#2039—31no—
53#2037—32no—
54#2028—31no—
55#2007—31no—
56#2004—32no—
57#1759—32no—
58C4×#100—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4D412yes—
62C2×#235—22no—
63C2×#235C2312yes—
64C3×#124—22no—
65C3×#124—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81#953—24no—
82C2×#216—22no—
83#942—24no—
84#942—24no—
85C2×#215—22no—
86#940—22no—
87#938—22no—
88#938—22no—
89#936—22no—
90#930—22no—
91#930—22no—
92#929—24no—
93C4×#67—24no—
94C4×#67—22no—
95C4×#67—22no—
96#926—22no—
97#926C2312yes—
98#924—22no—
99#924—22no—
100C2×#208C2312yes—
101C2×#208C2312yes—
102#897—22no—
103#895—22no—
104#890—22no—
105#888—22no—
106#885—24no—
107#884—24no—
108#883—24no—
109#879—24no—
110C4×D12D412yes—
111C4×D12D412yes—
112#872—22no—
113C42×D3—22no—
114C4×Dic6D412yes—
115C4×Dic6D412yes—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—22no—
119#835—22no—
120#835—22no—
121#833—24no—
122#810—22no—
123#809—22no—
124#808—22no—
125#807—22no—
126#805—24no—
127#805—24no—
128#803—22no—
129C4×C3⋊C8—22no—
130C2×#137—32no—
131#544—31no—
132#542—62no—
133#540—62no—
134#538—62no—
135#537—31no—
136#535—32no—
137#532—62no—
138#529—62no—
139C2×#124C2×D311yescharacteristic
140#525—62no—
141C4×#41—32no—
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4C2×D311yescharacteristic
144#441—62no—
145#440—62no—
146C4×Q16—62no—
147C4×SD16—62no—
148#428—32no—
149C2×#104—32no—
150C2×#104—32no—
151#419—32no—
152#417—32no—
153C2×#100—32no—
154C4×M16—32no—
155#391C2×D311yescharacteristic
156#390C2×D311yescharacteristic
157#388—22no—
158#385—22no—
159#384C2×D311yescharacteristic
160#376C2×D311yescharacteristic
161#343—31no—
162#343—31no—
163#339—32no—
164#339—32no—
165C24×C3C2×D412yes—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8#4112yescomplemented
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—82no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—22no—
186C2×#67—24no—
187C2×#67—24no—
188#235—22no—
189#235—22no—
190#235—22no—
191#235—22no—
192#235—84no—
193C2×D12—22no—
194C2×D12C2×D412yes—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×Dic6—22no—
199C2×Dic6C2×D412yes—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4—82no—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—82no—
208C3×C42—21no—
209C3×C42—22no—
210C3×C42—42no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213C3×C42#4112yescomplemented
214#217—24no—
215#217—24no—
216#217—22no—
217#216—28no—
218#215—28no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3C2×D412yes—
225C4⋊Dic3C2×D412yes—
226Dic3⋊C4—24no—
227Dic3⋊C4—22no—
228Dic3⋊C4—22no—
229C4×Dic3—22no—
230C4×Dic3—22no—
231C4×Dic3—22no—
232#208—22no—
233#208—22no—
234#208—22no—
235#208#4114yescomplemented
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×#41—32no—
239C2×#41—62no—
240C22×Q8C22×D311yescharacteristic
241C22×D4C22×D311yescharacteristic
242C23×C4#6712yes—
243#137—64no—
244#137—64no—
245C2×Q16—62no—
246C2×SD16—62no—
247C2×M16—32no—
248C2×M16—32no—
249C4⋊Q8—62no—
250#127—62no—
251#126—64no—
252#125—62no—
253#124—22no—
254#124—22no—
255#124—64no—
256#124—62no—
257#123—64no—
258#123—64no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262#121—64no—
263#121—62no—
264#121—62no—
265#121—62no—
266#121—62no—
267#121—62no—
268#120—64no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—32no—
272C4×Q8—32no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—32no—
276C4×D4—32no—
277C4×D4—64no—
278C4×D4—62no—
279C4×D4—62no—
280#117—32no—
281#117—62no—
282#117—62no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4#6712yes—
290C2×C42—21no—
291C2×C42—62no—
292C2×C42C22×D311yescharacteristic
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—62no—
297#104—64no—
298#104—64no—
299#103—62no—
300#103—62no—
301#103—62no—
302#102—62no—
303#100—64no—
304#97—62no—
305C4×C8—62no—
306#95—21no—
307#95—21no—
308#95—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3C22×D411yescharacteristic
315C22×D3—22no—
316C3×Q8—24no—
317C3×Q8—42no—
318C3×D4—42no—
319C3×D4—42no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—44no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—82no—
337C2×C3×C4#12011yescharacteristic
338C2×C3×C4#12012yes—
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4C22×D412yes—
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4112yes—
344#67—42no—
345#67—42no—
346#67—84no—
347#67—84no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351D12—22no—
352D12—22no—
353D12—82no—
354C4×D3—44no—
355C4×D3—42no—
356C4×D3—42no—
357C4×D3—82no—
358Dic6—22no—
359Dic6—22no—
360Dic6—82no—
361C3⋊C8—24no—
362C3⋊C8—42no—
363C24C2×#6712yes—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41—244no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—32no—
373C2×Q8—62no—
374C2×Q8#23512yescomplemented
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—32no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—64no—
383C2×D4—64no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—82no—
394C22×C4C2×#6712yes—
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C23×D311yescharacteristic
398Q16—68no—
399SD16—68no—
400M16—34no—
401M16—62no—
402M16—62no—
403M16—62no—
404C2×C8—62no—
405C2×C8—62no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—32no—
409C4⋊C4—32no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—64no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—82no—
418C22⋊C4—28no—
419C22⋊C4—24no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—62no—
424C22⋊C4—62no—
425C22⋊C4—62no—
426C22⋊C4—82no—
427C42—21no—
428C42—22no—
429C42—42no—
430C42—62no—
431C42—62no—
432C42—62no—
433C42#23512yescomplemented
434C42D3×D412yes—
435C42D3×D412yes—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—82no—
443C22×C3C2×#12011yescharacteristic
444C22×C3#52511yescharacteristic
445C22×C3#53211yescharacteristic
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—42no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—81no—
456C3×C4—81no—
457C3×C4#52512yes—
458C3×C4#53212yes—
459Dic3—42no—
460Dic3—42no—
461Dic3—42no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—62no—
468C23C22×#6711yescharacteristic
469Q8—24no—
470Q8—42no—
471Q8—62no—
472Q8—62no—
473Q8—242no—
474D4—42no—
475D4—42no—
476D4—62no—
477D4—62no—
478D4—122no—
479D4—122no—
480D4—242no—
481D4—244no—
482D4—244no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—82no—
503C2×C4—124no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—242no—
507C2×C4#93712yes—
508C2×C4#93711yescharacteristic
509C2×C4#93711yescharacteristic
510C2×C4C2×#23511yescharacteristic
511C2×C4C2×#23511yescharacteristic
512C2×C4C2×D3×D412yes—
513C2×C4D3×#4112yes—
514C8—64no—
515C8—122no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—82no—
519C2×C3#238811yescharacteristic
520C2×C3#301911yescharacteristic
521C2×C3#302211yescharacteristic
522D3—82no—
523C22—21no—
524C22—21no—
525C22—44no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—82no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22#579111yescharacteristic
534C22C2×#93711yescharacteristic
535C22#583211yescharacteristic
536C4—21no—
537C4—21no—
538C4—22no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—81no—
543C4—81no—
544C4—122no—
545C4—122no—
546C4—122no—
547C4#554712yes—
548C4#554812yes—
549C3#2359311yescharacteristic, complemented, Sylow
550C2—21no—
551C2—21no—
552C2—82no—
553C2—242no—
554C2#8240011yescharacteristic
555C2#8339411yescharacteristic
556C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2326C2
socleC22×C3C2×#120
prime radical#85777C2
hypercenter#2326D3
hypocenterC3#23593
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C3×#2326C2semidirect
3. C4×#942C2semidirect
4. #71731C2semidirect
5. C4×#833C2semidirect
6. #69694C2semidirect
7. #69673C2semidirect
8. #69665C2semidirect
9. #69593C2semidirect
10. #69573C2semidirect
11. #69430C2semidirect
50. #2326D3semidirect
169. C2×C3×Q8#41semidirect
213. C3×C42#41semidirect
235. #208#41semidirect
374. C2×Q8#235semidirect
433. C42#235semidirect
549. C3#23593semidirect

Maximal subgroups

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

subgroupindex
#857772
C3×#23262
C4×#9422
#717312
C4×#8332
#696942
#696732
#696652
#695932
#695732
#694302
#235933