Finite Group Databasebeta

← #10800524

Subgroups — #10800524

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#86236C211yesmaximal, characteristic
2#79039C211yesmaximal, characteristic, complemented
3#73382C211yesmaximal, characteristic, complemented
4C3×#1963C211yesmaximal, characteristic, complemented
5#71815C211yesmaximal, characteristic, complemented
6#70514C211yesmaximal, characteristic, complemented
7#70383C212yesmaximal, complemented
8#70382C212yesmaximal, complemented
9#70354C212yesmaximal, complemented
10Dic3×#104C212yesmaximal, complemented
11#69441C211yesmaximal, characteristic, complemented
12#21419—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14#5828C2212yes—
15Dic3×#41C2212yes—
16#5731C412yescomplemented
17#5719C412yescomplemented
18#5682C414yescomplemented
19#5617—22no—
20#5602—22no—
21#5538—24no—
22C2×#872C2211yescharacteristic
23C2×#872C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25#5470C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C2×C3×#104C2212yes—
28C2×C3×#104C2212yes—
29C3×#376C2211yescharacteristic
30C2×#837C2212yes—
31C2×#837C2212yes—
32#5126C2212yes—
33Dic3×M16C2212yes—
34#4932C2211yescharacteristic
35C42×Dic3C2211yescharacteristic
36C42×Dic3C2211yescharacteristic
37#4924C2211yescharacteristic
38#4840—24no—
39#4666—22no—
40#4649—22no—
41#4555C2212yes—
42#4555C2212yes—
43#4523C2212yes—
44#4523C2212yes—
45#3455—31no—
46#2921—31no—
47C4×#117—31no—
48#1995—31no—
49#1994—31no—
50#1963D311yescharacteristic, complemented
51#1962—31no—
52#1897—32no—
53#1896—32no—
54#1759—32no—
55C4×#104—32no—
56C2×C3×#41C2312yes—
57C22×C4×D3—21no—
58C22×C4×D3C2311yescharacteristic
59C2×C3×M16C2312yes—
60C3×C4⋊Q8C2×C411yescharacteristic, complemented
61C3×#127C2×C411yescharacteristic, complemented
62C3×#124C2×C412yescomplemented
63C3×#121—22no—
64C3×C4×D4—22no—
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68#946C2×C414yes—
69Q8×Dic3C2×C414yescomplemented
70#939—24no—
71#938C2×C414yes—
72#937—22no—
73#935—22no—
74#935—24no—
75D4×Dic3—24no—
76D4×Dic3C2×C414yescomplemented
77C4×#67—24no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4D412yes—
80#926—22no—
81#926C2312yes—
82#924—22no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4D412yes—
85C2×C4×Dic3—21no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3C2311yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C2×#208C2312yes—
93#898—24no—
94#889—24no—
95#886—24no—
96#885—24no—
97#879—24no—
98#878—24no—
99#872—22no—
100#872—22no—
101#872C2×C412yes—
102#872C2×C412yes—
103#872D414yes—
104C42×D3—22no—
105C42×D3—22no—
106#870C2×C412yescomplemented
107#869C2×C412yescomplemented
108C4×Dic6—22no—
109C3×C4⋊C8—22no—
110C3×#104—24no—
111C3×#104—24no—
112C3×#95—22no—
113#837—24no—
114#837—24no—
115C3⋊#95—22no—
116(C2×D3)⋊C8—24no—
117#815—22no—
118C8×Dic3—22no—
119C4⋊(C3⋊C8)—22no—
120#542—34no—
121#541—32no—
122#540—32no—
123#539—62no—
124#538—62no—
125#533—31no—
126#532C2×D311yescharacteristic
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×#104C2×D312yes—
139C2×#104C2×D312yes—
140#407—32no—
141C4×M16—32no—
142C4×C4⋊C4—62no—
143C4×C22⋊C4—62no—
144#376—31no—
145#376C2×D311yescharacteristic
146C43—31no—
147C43—31no—
148#339—32no—
149#339—32no—
150#339—32no—
151#339—32no—
152C23×D3C2×D411yescharacteristic
153C3×#41—22no—
154C3×#41—22no—
155C3×#41—22no—
156C3×#41—22no—
157C2×C3×Q8C22×C412yes—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4C22×C412yes—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C2×#67—24no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—24no—
170C22×Dic3C2×D411yescharacteristic
171C22×Dic3C2×D412yes—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—24no—
177C2×C4×D3—44no—
178C2×C4×D3C22×C412yes—
179C2×C4×D3C2×D412yes—
180C2×Dic6—22no—
181C3×M16—22no—
182C3×M16—22no—
183C2×C3×C8—22no—
184C3×C4⋊C4—22no—
185C3×C22⋊C4—24no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42C22×C411yescharacteristic
189C3×C42C22×C411yescharacteristic
190C3×C42C2×D412yes—
191C3×C42#4112yes—
192C3×C42#4112yes—
193#217—22no—
194#217—24no—
195#217—24no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—24no—
198(C2×D3)⋊C4C22⋊C414yes—
199(C2×D3)⋊C4C22⋊C414yes—
200C4⋊Dic3—22no—
201C4⋊Dic3—24no—
202C4⋊Dic3C22×C414yes—
203Dic3⋊C4—24no—
204Dic3⋊C4—24no—
205Dic3⋊C4—24no—
206Dic3⋊C4C22⋊C414yes—
207Dic3⋊C4C22⋊C414yes—
208C4×Dic3—24no—
209C4×Dic3—22no—
210C4×Dic3—21no—
211C4×Dic3—21no—
212C4×Dic3—24no—
213C4×Dic3—44no—
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×#41C22×D312yes—
223C23×C4—31no—
224C23×C4—61no—
225C2×M16—32no—
226C2×M16C22×D312yes—
227C4⋊Q8—32no—
228C4⋊Q8C4×D311yescharacteristic, complemented
229#127C4×D311yescharacteristic, complemented
230#126—64no—
231#126—64no—
232#125—32no—
233#125—64no—
234#124—64no—
235#124C4×D312yescomplemented
236#123—64no—
237#123—64no—
238#123—62no—
239C22⋊Q8—34no—
240C22⋊Q8—64no—
241C22⋊Q8—62no—
242#121—22no—
243#121—34no—
244#121—64no—
245#120—62no—
246C4×Q8—34no—
247C4×Q8—62no—
248C4×D4—22no—
249C4×D4—34no—
250C4×D4—64no—
251C4×D4—64no—
252#117—32no—
253#117—32no—
254#117—34no—
255#117—32no—
256#117—62no—
257#117—62no—
258#117—62no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C22⋊C4—32no—
262C2×C22⋊C4—61no—
263C2×C42—31no—
264C2×C42—31no—
265C2×C42—31no—
266C2×C42—62no—
267C2×C42—62no—
268C2×C42—61no—
269C2×C42—62no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42C22×D311yescharacteristic
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—62no—
277#104—24no—
278#104—24no—
279#104—64no—
280#104—64no—
281C22⋊C8—64no—
282#97—62no—
283C4×C8—62no—
284#95—22no—
285#95—62no—
286C23×C3—22no—
287C23×C3C22×D411yescharacteristic
288C22×D3—21no—
289C22×D3—21no—
290C22×D3—42no—
291C22×D3C2×C22⋊C412yes—
292C3×Q8—22no—
293C3×Q8—22no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—82no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4C2×C22⋊C412yes—
308C2×C3×C4C2×C22⋊C412yes—
309C2×C3×C4C23×C411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315#67—84no—
316C2×Dic3—24no—
317C2×Dic3—21no—
318C2×Dic3—22no—
319C2×Dic3—21no—
320C2×Dic3—24no—
321C2×Dic3—22no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—84no—
325C2×Dic3C2×C22⋊C414yes—
326C2×Dic3C2×C22⋊C412yes—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—22no—
330C4×D3—22no—
331C4×D3—48no—
332Dic6—82no—
333C3×C8—42no—
334C3⋊C8—42no—
335C24—31no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340C2×Q8—62no—
341C2×Q8C2×C4×D312yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—64no—
345C2×D4C2×C4×D312yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—31no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—64no—
354C22×C4—64no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
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—22no—
371C4⋊C4—34no—
372C4⋊C4—34no—
373C4⋊C4—34no—
374C4⋊C4—64no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—62no—
379C22⋊C4—24no—
380C22⋊C4—34no—
381C22⋊C4—34no—
382C22⋊C4—64no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C22⋊C4—64no—
386C22⋊C4—62no—
387C42—21no—
388C42—21no—
389C42—32no—
390C42—32no—
391C42—32no—
392C42—32no—
393C42—64no—
394C42—64no—
395C42—62no—
396C42—61no—
397C42—61no—
398C42—124no—
399C42—124no—
400C42C2×C4×D311yescharacteristic
401C42C2×C4×D311yescharacteristic
402C42D3×D412yes—
403C42#23712yes—
404C42#23712yes—
405C22×C3—22no—
406C22×C3—44no—
407C22×C3—42no—
408C22×C3C22×C22⋊C411yescharacteristic
409C22×C3#52511yescharacteristic
410C22×C3#52511yescharacteristic
411C2×D3—24no—
412C2×D3—22no—
413C2×D3—42no—
414C2×D3—42no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4C4×#4111yescharacteristic
421C3×C4C4×#4111yescharacteristic
422C3×C4#53211yescharacteristic
423C3×C4#53211yescharacteristic
424Dic3—24no—
425Dic3—24no—
426Dic3—22no—
427Dic3—42no—
428Dic3—44no—
429C23—22no—
430C23—32no—
431C23—61no—
432C23—61no—
433C23—122no—
434C23C2×D3×D411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—242no—
438D4—22no—
439D4—22no—
440D4—42no—
441D4—42no—
442D4—82no—
443D4—244no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—34no—
448C2×C4—32no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—64no—
455C2×C4—62no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—64no—
463C2×C4—62no—
464C2×C4—128no—
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×#23711yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C8—42no—
477C8—122no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—82no—
481C2×C3#229711yescharacteristic
482C2×C3#290112yes—
483D3—42no—
484C22—22no—
485C22—42no—
486C22—44no—
487C22—62no—
488C22—64no—
489C22—122no—
490C22—122no—
491C22C2×D3×C22⋊C411yescharacteristic
492C22#560611yescharacteristic
493C22#560611yescharacteristic
494C4—22no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—64no—
500C4—62no—
501C4—64no—
502C4—122no—
503C4—124no—
504C4C4×#23711yescharacteristic
505C4C4×#23711yescharacteristic
506C4#566811yescharacteristic
507C4#571611yescharacteristic
508C3#2141911yescharacteristic, complemented, Sylow
509C2—21no—
510C2—21no—
511C2—82no—
512C2—122no—
513C2#7629011yescharacteristic
514C2#7899212yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1963C2
socleC22×C3C22×C22⋊C4
prime radical#86236C2
cosocleC22×C3×C4C24
hypercenter#1963D3
hypocenterC3#21419
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #79039C2semidirect
3. #73382C2semidirect
4. C3×#1963C2semidirect
5. #71815C2semidirect
6. #70514C2semidirect
7. #70383C2semidirect
8. #70382C2semidirect
9. #70354C2semidirect
10. Dic3×#104C2semidirect
11. #69441C2semidirect
16. #5731C4semidirect
17. #5719C4semidirect
18. #5682C4semidirect
50. #1963D3semidirect
60. C3×C4⋊Q8C2×C4semidirect
61. C3×#127C2×C4semidirect
62. C3×#124C2×C4semidirect
69. Q8×Dic3C2×C4semidirect
76. D4×Dic3C2×C4semidirect
106. #870C2×C4semidirect
107. #869C2×C4semidirect
228. C4⋊Q8C4×D3semidirect
229. #127C4×D3semidirect
235. #124C4×D3semidirect
508. C3#21419semidirect

Maximal subgroups

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

subgroupindex
#862362
#790392
#733822
C3×#19632
#718152
#705142
#703832
#703822
#703542
Dic3×#1042
#694412
#214193