Finite Group Databasebeta

← #10800522

Subgroups — #10800522

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86273C211yesmaximal, characteristic
2#79039C211yesmaximal, characteristic, complemented
3C4×#872C211yesmaximal, characteristic, complemented
4C3×#1994C211yesmaximal, characteristic, complemented
5#71840C211yesmaximal, characteristic, complemented
6#70513C211yesmaximal, characteristic, complemented
7#69395C211yesmaximal, characteristic, complemented
8#68757C212yesmaximal, complemented
9#68756C212yesmaximal, complemented
10#68729C212yesmaximal, complemented
11#68728C212yesmaximal, complemented
12#21419—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14#5733C414yescomplemented
15#5698C412yescomplemented
16#5697C412yescomplemented
17#5645—22no—
18#5644—22no—
19#5580—24no—
20#5524C2212yes—
21#5522C2212yes—
22#5477C2211yescharacteristic
23C2×#872C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C3×C43C2211yescharacteristic
28C2×#825C2212yes—
29C2×#825C2212yes—
30#5126C2212yes—
31Dic3×M16C2212yes—
32C4×(C2×D3)⋊C4—22no—
33#4932C2211yescharacteristic
34#4932C2211yescharacteristic
35C4×Dic3⋊C4—22no—
36C42×Dic3C2211yescharacteristic
37C2×#805C2212yes—
38C2×#805C2212yes—
39#4924C2211yescharacteristic
40#4840—24no—
41C3×#339C2212yes—
42C3×#339C2212yes—
43#4532C2212yes—
44#4532C2212yes—
45#3455—31no—
46#2921—31no—
47C4×#117—31no—
48#1995—31no—
49#1994D311yescharacteristic, complemented
50#1963—31no—
51#1962—31no—
52#1897—32no—
53#1896—32no—
54#1759—32no—
55C4×#104—32no—
56C2×#235C2312yes—
57C22×C4×D3—21no—
58C22×C4×D3C2311yescharacteristic
59C2×C3×M16C2312yes—
60C3×C4⋊Q8C2×C412yescomplemented
61C3×#125C2×C412yescomplemented
62C3×C22⋊Q8—22no—
63C3×C4×Q8—22no—
64C3×#117C2312yes—
65C2×C3×C42—22no—
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69#947—24no—
70#946—24no—
71#930—22no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4D412yes—
74C2×Dic3⋊C4—21no—
75C2×Dic3⋊C4D412yes—
76C2×C4×Dic3—21no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3C2311yescharacteristic
79C2×C4×Dic3C2311yescharacteristic
80C2×C4×Dic3C2311yescharacteristic
81C2×#208C2312yes—
82#898—24no—
83#898—24no—
84#897C2×C414yes—
85#895—24no—
86#895C2×C414yes—
87#894—24no—
88#894—24no—
89#893C2×C414yescomplemented
90#890—24no—
91#887C2×C414yescomplemented
92#886—22no—
93#882—22no—
94#67⋊C4—24no—
95#875C2×C412yescomplemented
96#874C2×C411yescharacteristic, complemented
97C4×D12—24no—
98C4×D12—22no—
99#872—22no—
100#872—22no—
101#872—22no—
102#872C2×C412yes—
103#872C2×C412yes—
104#872D414yes—
105C42×D3—22no—
106C42×D3—22no—
107C42×D3—22no—
108#869C2×C411yescharacteristic, complemented
109C3×C4⋊C8—22no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112#825—24no—
113#825—24no—
114(C2×D3)⋊C8—24no—
115#815—22no—
116C8×Dic3—22no—
117#805—24no—
118#805—24no—
119C4⋊(C3⋊C8)—22no—
120#542—34no—
121#541—32no—
122#540—32no—
123#539—62no—
124#538—62no—
125#533C2×D311yescharacteristic
126#532—31no—
127#529—64no—
128#525—32no—
129C4×#41—32no—
130C2×#117—31no—
131C2×#117—31no—
132C22×C42—31no—
133#433—64no—
134#423—31no—
135#423C2×D311yescharacteristic
136C2×#104—32no—
137C2×#104—32no—
138C2×#104—32no—
139C2×#104—32no—
140#407—32no—
141C4×M16—32no—
142C4×C4⋊C4—62no—
143C4×C22⋊C4—62no—
144#376—31no—
145#376—31no—
146C43—31no—
147C43C2×D311yescharacteristic
148#339—32no—
149#339—32no—
150#339C2×D312yes—
151#339C2×D312yes—
152C23×D3C2×D411yescharacteristic
153C2×C3×Q8—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2411yescharacteristic
158C2×#67—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D412yes—
163#235—22no—
164#235—22no—
165#235—22no—
166#235—22no—
167C2×D12—24no—
168C2×D12—22no—
169C2×D12C22×C412yes—
170C2×C4×D3—21no—
171C2×C4×D3—21no—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—24no—
175C2×C4×D3—22no—
176C2×C4×D3—24no—
177C2×C4×D3—44no—
178C2×C4×D3C22×C412yes—
179C2×C4×D3C2×D412yes—
180C2×Dic6C22×C412yes—
181C3×M16—22no—
182C3×M16—22no—
183C2×C3×C8—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4C22×C414yes—
187C3×C22⋊C4—22no—
188C3×C42—22no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42—44no—
192C3×C42C22×C411yescharacteristic
193C3×C42C22×C411yescharacteristic
194C3×C42C2×D412yes—
195C3×C42#4112yes—
196C3×C42#4112yes—
197(C2×D3)⋊C4—24no—
198(C2×D3)⋊C4—24no—
199(C2×D3)⋊C4—24no—
200(C2×D3)⋊C4—24no—
201(C2×D3)⋊C4—24no—
202(C2×D3)⋊C4C22⋊C414yes—
203(C2×D3)⋊C4C22⋊C414yes—
204C4⋊Dic3—22no—
205C4⋊Dic3—24no—
206Dic3⋊C4—24no—
207Dic3⋊C4—24no—
208Dic3⋊C4C22⋊C414yes—
209Dic3⋊C4C22⋊C414yes—
210C4×Dic3—24no—
211C4×Dic3—21no—
212C4×Dic3—21no—
213C4×Dic3—24no—
214C4×Dic3—44no—
215C4×Dic3C22×C412yes—
216C4×Dic3C2×D412yes—
217C4×Dic3#4112yes—
218C4×Dic3#4112yes—
219#208—22no—
220#208—22no—
221C2×C3⋊C8—22no—
222C2×#41—32no—
223C23×C4—31no—
224C23×C4—61no—
225C2×M16—32no—
226C2×M16C22×D312yes—
227C4⋊Q8—31no—
228C4⋊Q8C4×D312yescomplemented
229#127—31no—
230#126—64no—
231#126—64no—
232#125—64no—
233#125C4×D312yescomplemented
234#124—32no—
235#124—64no—
236#123—62no—
237#123—64no—
238#123—64no—
239C22⋊Q8—22no—
240C22⋊Q8—34no—
241C22⋊Q8—64no—
242#121—34no—
243#121—64no—
244#121—62no—
245#120—62no—
246C4×Q8—22no—
247C4×Q8—34no—
248C4×D4—34no—
249C4×D4—64no—
250C4×D4—64no—
251C4×D4—62no—
252#117—32no—
253#117—32no—
254#117—34no—
255#117—62no—
256#117—62no—
257#117—62no—
258#117C22×D312yes—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C22⋊C4—32no—
262C2×C22⋊C4—61no—
263C2×C42—22no—
264C2×C42—31no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—62no—
268C2×C42—62no—
269C2×C42—61no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42C22×D311yescharacteristic
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—62no—
277#104—64no—
278#104—64no—
279#104—64no—
280#104—64no—
281C22⋊C8—64no—
282#97—62no—
283C4×C8—62no—
284#95—62no—
285#95—62no—
286C23×C3C22×D411yescharacteristic
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—22no—
290C22×D3—42no—
291C22×D3C2×C22⋊C412yes—
292C3×Q8—82no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4C2×C22⋊C412yes—
300C2×C3×C4C2×C22⋊C412yes—
301C2×C3×C4C23×C411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307#67—42no—
308#67—42no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3C2×C22⋊C414yes—
318C2×Dic3C2×C22⋊C412yes—
319D12—22no—
320D12—22no—
321D12—82no—
322D12—84no—
323C4×D3—22no—
324C4×D3—22no—
325C4×D3—22no—
326C4×D3—22no—
327C4×D3—48no—
328C4×D3—42no—
329C4×D3—42no—
330C4×D3—84no—
331Dic6—22no—
332Dic6—22no—
333C3×C8—42no—
334C3⋊C8—42no—
335C24—31no—
336#41—62no—
337#41—62no—
338#41—62no—
339#41—62no—
340C2×Q8—22no—
341C2×Q8—32no—
342C2×D4—32no—
343C2×D4—64no—
344C2×D4—62no—
345C2×D4—62no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—31no—
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—64no—
353C22×C4—64no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4—124no—
362C22×C4D3×D412yes—
363C22×C4C23×D311yescharacteristic
364M16—22no—
365M16—22no—
366M16—62no—
367M16—62no—
368C2×C8—22no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—34no—
373C4⋊C4—34no—
374C4⋊C4—64no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—62no—
378C4⋊C4C2×C4×D314yes—
379C22⋊C4—22no—
380C22⋊C4—34no—
381C22⋊C4—34no—
382C22⋊C4—64no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C22⋊C4—64no—
386C22⋊C4—64no—
387C42—22no—
388C42—21no—
389C42—21no—
390C42—32no—
391C42—32no—
392C42—32no—
393C42—32no—
394C42—44no—
395C42—64no—
396C42—64no—
397C42—61no—
398C42—61no—
399C42—124no—
400C42C2×C4×D311yescharacteristic
401C42C2×C4×D311yescharacteristic
402C42D3×D412yes—
403C42#23912yes—
404C42#23912yes—
405C22×C3—22no—
406C22×C3C22×C22⋊C411yescharacteristic
407C22×C3#52511yescharacteristic
408C22×C3#52511yescharacteristic
409C2×D3—24no—
410C2×D3—22no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—44no—
414C2×D3—42no—
415C3×C4—24no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—44no—
420C3×C4C4×#4111yescharacteristic
421C3×C4C4×#4111yescharacteristic
422C3×C4#53211yescharacteristic
423C3×C4#53211yescharacteristic
424Dic3—24no—
425Dic3—22no—
426Dic3—42no—
427Dic3—42no—
428Dic3—42no—
429C23—32no—
430C23—61no—
431C23—61no—
432C23—62no—
433C23—122no—
434C23C2×D3×D411yescharacteristic
435Q8—62no—
436Q8—62no—
437Q8—82no—
438D4—62no—
439D4—62no—
440D4—122no—
441D4—122no—
442D4—242no—
443D4—244no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—34no—
449C2×C4—32no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—128no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—244no—
468C2×C4D3×C22⋊C412yes—
469C2×C4D3×C22⋊C412yes—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×#23911yescharacteristic
473C2×C4C2×#23911yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C8—42no—
477C8—122no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3#229711yescharacteristic
481C2×C3#290112yes—
482D3—42no—
483D3—82no—
484C22—22no—
485C22—64no—
486C22—62no—
487C22—122no—
488C22—122no—
489C22—124no—
490C22—122no—
491C22C2×D3×C22⋊C411yescharacteristic
492C22#562711yescharacteristic
493C22#562711yescharacteristic
494C4—24no—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—44no—
499C4—64no—
500C4—62no—
501C4—122no—
502C4—122no—
503C4—122no—
504C4C4×#23911yescharacteristic
505C4C4×#23911yescharacteristic
506C4#566811yescharacteristic
507C4#572511yescharacteristic
508C3#2141911yescharacteristic, complemented, Sylow
509C2—21no—
510C2—21no—
511C2—122no—
512C2—242no—
513C2#7647811yescharacteristic
514C2#7898312yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1994C2
socleC22×C3C22×C22⋊C4
prime radical#86273C2
cosocleC22×C3×C4C24
hypercenter#1994D3
hypocenterC3#21419
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
2. #79039C2semidirect
3. C4×#872C2semidirect
4. C3×#1994C2semidirect
5. #71840C2semidirect
6. #70513C2semidirect
7. #69395C2semidirect
8. #68757C2semidirect
9. #68756C2semidirect
10. #68729C2semidirect
11. #68728C2semidirect
14. #5733C4semidirect
15. #5698C4semidirect
16. #5697C4semidirect
49. #1994D3semidirect
60. C3×C4⋊Q8C2×C4semidirect
61. C3×#125C2×C4semidirect
89. #893C2×C4semidirect
91. #887C2×C4semidirect
95. #875C2×C4semidirect
96. #874C2×C4semidirect
108. #869C2×C4semidirect
228. C4⋊Q8C4×D3semidirect
233. #125C4×D3semidirect
508. C3#21419semidirect

Maximal subgroups

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

subgroupindex
#862732
#790392
C4×#8722
C3×#19942
#718402
#705132
#693952
#687572
#687562
#687292
#687282
#214193