Finite Group Databasebeta

← #10800521

Subgroups — #10800521

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#85738C211yesmaximal, characteristic
2C3×C4×#117C211yesmaximal, characteristic, complemented
3#82209C211yesmaximal, characteristic, complemented
4#71591C212yesmaximal, complemented
5#71532C212yesmaximal, complemented
6#69441C211yesmaximal, characteristic, complemented
7#69440C211yesmaximal, characteristic, complemented
8#69430C212yesmaximal, complemented
9#69396C211yesmaximal, characteristic, complemented
10#69395C211yesmaximal, characteristic, complemented
11C4×#805C212yesmaximal, complemented
12#21419—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16#5791C2212yes—
17#5535C412yescomplemented
18#5534C414yescomplemented
19#5528—24no—
20#5518C412yescomplemented
21#5515—22no—
22#5494—22no—
23#5477C2211yescharacteristic
24C4×#235C2212yes—
25#5470C2211yescharacteristic
26C3×C4×C4⋊C4—22no—
27C3×C4×C22⋊C4—22no—
28C3×#376C2211yescharacteristic
29C3×#376C2211yescharacteristic
30C3×C43C2211yescharacteristic
31C3×C43C2211yescharacteristic
32#5207C2212yes—
33#5000—24no—
34C2×#805C2212yes—
35C2×#805C2212yes—
36C2×#805C2212yes—
37C2×#805C2212yes—
38#4924C2211yescharacteristic
39#4924C2211yescharacteristic
40C4×#208C2212yes—
41#4523C2212yes—
42#4523C2212yes—
43#4523C2212yes—
44#4523C2212yes—
45#3455—31no—
46#2921—31no—
47C4×#117D311yescharacteristic, complemented
48#1995—31no—
49#1994—31no—
50#1963—31no—
51#1962—31no—
52#1897—32no—
53#1896—32no—
54#1759—32no—
55C4×#104—32no—
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C2×#235C2312yes—
59C3×#117—22no—
60C3×#117—22no—
61C3×#117C2×C412yes—
62C3×#117C2×C412yes—
63C3×#117D414yes—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D412yes—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4D412yes—
68C2×C3×C42—21no—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79#953—22no—
80#930—22no—
81#930C2×C414yes—
82#929—22no—
83C4×#67—24no—
84C4×#67—24no—
85#926C2312yes—
86#924—22no—
87#924C2×C414yes—
88C2×#208C2312yes—
89C2×#208C2312yes—
90#898—24no—
91#896—24no—
92#894—24no—
93#889—24no—
94#885—24no—
95#884—24no—
96#883—24no—
97#879—24no—
98#875C2×C412yescomplemented
99#874C2×C411yescharacteristic, complemented
100C4×D12—22no—
101C4×D12C2×C414yescomplemented
102#872—22no—
103C42×D3—22no—
104#870C2×C412yescomplemented
105#869C2×C411yescharacteristic, complemented
106#869C2×C412yescomplemented
107C4×Dic6—22no—
108C4×Dic6C2×C414yescomplemented
109C3×#95—22no—
110C3×#95—22no—
111#830—24no—
112#805—24no—
113#805—24no—
114#805—24no—
115#805—24no—
116C4⋊(C3⋊C8)—22no—
117C4⋊(C3⋊C8)—22no—
118#803—22no—
119C4×C3⋊C8—22no—
120#542—34no—
121#541—32no—
122#540—32no—
123#539—62no—
124#538—62no—
125#533—31no—
126#532—31no—
127#529—64no—
128#525—32no—
129C4×#41—32no—
130C2×#117C2×D311yescharacteristic
131C2×#117C2×D311yescharacteristic
132C22×C42C2×D311yescharacteristic
133#433—64no—
134#423—31no—
135#423—31no—
136C2×#104—32no—
137C2×#104—32no—
138C2×#104—32no—
139C2×#104—32no—
140#407—32no—
141C4×M16—32no—
142C4×C4⋊C4—22no—
143C4×C22⋊C4—22no—
144#376C2×D311yescharacteristic
145#376C2×D311yescharacteristic
146C43C2×D311yescharacteristic
147C43C2×D311yescharacteristic
148#339—32no—
149#339—32no—
150#339—32no—
151#339—32no—
152C24×C3C2×D411yescharacteristic
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—44no—
163C22×C3×C4C22×C412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2411yescharacteristic
169C2×#67—22no—
170C2×#67—24no—
171#235—22no—
172#235—22no—
173#235—22no—
174#235—22no—
175C2×D12—22no—
176C2×D12C22×C412yes—
177C2×C4×D3—22no—
178C2×C4×D3—24no—
179C2×Dic6—22no—
180C2×Dic6C22×C412yes—
181C3×C4⋊C4—24no—
182C3×C4⋊C4C22⋊C414yes—
183C3×C4⋊C4C22⋊C414yes—
184C3×C22⋊C4—24no—
185C3×C22⋊C4C22⋊C414yes—
186C3×C22⋊C4C22⋊C414yes—
187C3×C42—24no—
188C3×C42—21no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42—21no—
192C3×C42—44no—
193C3×C42—44no—
194C3×C42C22×C412yes—
195C3×C42C22×C411yescharacteristic
196C3×C42C22×C411yescharacteristic
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199C3×C42#4112yes—
200C3×C42#4112yes—
201C3×C42#4112yes—
202C3×C42#4112yes—
203#217—24no—
204#217—22no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3C22×C414yes—
211Dic3⋊C4—24no—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214C4×Dic3—24no—
215C4×Dic3—22no—
216#208—22no—
217#208—22no—
218#208—22no—
219#208—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×#41—32no—
223C23×C4—21no—
224C23×C4C22×D311yescharacteristic
225C2×M16—32no—
226C2×M16—32no—
227C4⋊Q8—32no—
228C4⋊Q8—31no—
229#127—31no—
230#126—64no—
231#126—64no—
232#125—32no—
233#125—64no—
234#124—32no—
235#124—64no—
236#123—62no—
237#123—64no—
238#123—64no—
239C22⋊Q8—34no—
240C22⋊Q8—62no—
241C22⋊Q8—64no—
242#121—34no—
243#121—62no—
244#121—64no—
245#120—62no—
246C4×Q8—34no—
247C4×Q8—62no—
248C4×D4—34no—
249C4×D4—64no—
250C4×D4—64no—
251C4×D4—62no—
252#117—22no—
253#117—22no—
254#117—32no—
255#117—62no—
256#117C4×D312yes—
257#117C4×D312yes—
258#117D1214yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4#6712yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4#6712yes—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—62no—
269C2×C42C22×D311yescharacteristic
270C2×C42C22×D311yescharacteristic
271C2×C42C22×D311yescharacteristic
272C2×C42C22×D311yescharacteristic
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275C4⋊C8—62no—
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—22no—
285#95—22no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—42no—
289C23×C3C2×C22⋊C412yes—
290C23×C3C22×D411yescharacteristic
291C22×D3—22no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—48no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4C2×C22⋊C414yes—
310C2×C3×C4C2×C22⋊C412yes—
311C2×C3×C4C2×C22⋊C412yes—
312C2×C3×C4C2×C22⋊C412yes—
313C2×C3×C4C23×C411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319#67—42no—
320#67—42no—
321#67—84no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324D12—22no—
325D12—22no—
326D12—82no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—84no—
330Dic6—22no—
331Dic6—22no—
332Dic6—82no—
333C3⋊C8—42no—
334C3⋊C8—42no—
335C24C2×#6711yescharacteristic
336#41—62no—
337#41—62no—
338#41—62no—
339#41—62no—
340C2×Q8—32no—
341C2×Q8—62no—
342C2×D4—32no—
343C2×D4—62no—
344C2×D4—64no—
345C2×D4—62no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—44no—
356C22×C4—62no—
357C22×C4—64no—
358C22×C4C2×C4×D312yes—
359C22×C4C2×D1212yes—
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6712yes—
362C22×C4C2×#6712yes—
363C22×C4C23×D311yescharacteristic
364M16—62no—
365M16—62no—
366M16—62no—
367M16—62no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—34no—
372C4⋊C4—64no—
373C4⋊C4—62no—
374C4⋊C4—64no—
375C4⋊C4—64no—
376C4⋊C4—62no—
377C4⋊C4(C2×D3)⋊C414yes—
378C4⋊C4(C2×D3)⋊C414yes—
379C22⋊C4—24no—
380C22⋊C4—64no—
381C22⋊C4—62no—
382C22⋊C4—64no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C22⋊C4(C2×D3)⋊C414yes—
386C22⋊C4(C2×D3)⋊C414yes—
387C42—24no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—44no—
393C42—44no—
394C42—64no—
395C42—62no—
396C42C2×C4×D312yes—
397C42C2×C4×D311yescharacteristic
398C42C2×C4×D311yescharacteristic
399C42C2×D1212yes—
400C42C2×D1212yes—
401C42#23512yes—
402C42#23512yes—
403C42#23512yes—
404C42#23512yes—
405C22×C3—22no—
406C22×C3—24no—
407C22×C3—22no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3C22×C22⋊C411yescharacteristic
411C22×C3#52511yescharacteristic
412C22×C3#52511yescharacteristic
413C2×D3—44no—
414C2×D3—42no—
415C3×C4—24no—
416C3×C4—22no—
417C3×C4—24no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4C4×#4111yescharacteristic
424C3×C4C4×#4111yescharacteristic
425C3×C4#53211yescharacteristic
426C3×C4#53211yescharacteristic
427Dic3—44no—
428Dic3—42no—
429C23—21no—
430C23—21no—
431C23—42no—
432C23—62no—
433C23C2×(C2×D3)⋊C412yes—
434C23C22×#6711yescharacteristic
435Q8—62no—
436Q8—62no—
437Q8—242no—
438D4—62no—
439D4—62no—
440D4—122no—
441D4—122no—
442D4—242no—
443D4—244no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—64no—
462C2×C4—62no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—244no—
466C2×C4C2×(C2×D3)⋊C414yes—
467C2×C4C2×(C2×D3)⋊C412yes—
468C2×C4C2×(C2×D3)⋊C412yes—
469C2×C4C2×(C2×D3)⋊C412yes—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C22×D1211yescharacteristic
472C2×C4C2×#23511yescharacteristic
473C2×C4C2×#23511yescharacteristic
474C2×C4C2×#23511yescharacteristic
475C2×C4C2×#23511yescharacteristic
476C8—122no—
477C8—122no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—42no—
481C2×C3#229711yescharacteristic
482C2×C3#290112yes—
483D3—82no—
484C22—22no—
485C22—24no—
486C22—22no—
487C22—42no—
488C22—42no—
489C22—124no—
490C22—122no—
491C22C22×(C2×D3)⋊C411yescharacteristic
492C22#579111yescharacteristic
493C22#579111yescharacteristic
494C4—24no—
495C4—24no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—124no—
503C4—122no—
504C4C4×#23511yescharacteristic
505C4C4×#23511yescharacteristic
506C4#547711yescharacteristic
507C4#547711yescharacteristic
508C3#2141911yescharacteristic, complemented, Sylow
509C2—21no—
510C2—21no—
511C2—42no—
512C2—242no—
513C2#8207011yescharacteristic
514C2#8253612yes—
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#117C2
socleC22×C3C22×C22⋊C4
prime radical#85738C2
cosocleC22×C3×C4C24
hypercenterC4×#117D3
hypocenterC3#21419
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. C3×C4×#117C2semidirect
3. #82209C2semidirect
4. #71591C2semidirect
5. #71532C2semidirect
6. #69441C2semidirect
7. #69440C2semidirect
8. #69430C2semidirect
9. #69396C2semidirect
10. #69395C2semidirect
11. C4×#805C2semidirect
17. #5535C4semidirect
18. #5534C4semidirect
20. #5518C4semidirect
47. C4×#117D3semidirect
98. #875C2×C4semidirect
99. #874C2×C4semidirect
101. C4×D12C2×C4semidirect
104. #870C2×C4semidirect
105. #869C2×C4semidirect
106. #869C2×C4semidirect
108. C4×Dic6C2×C4semidirect
508. C3#21419semidirect

Maximal subgroups

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

subgroupindex
#857382
C3×C4×#1172
#822092
#715912
#715322
#694412
#694402
#694302
#693962
#693952
C4×#8052
#214193