Finite Group Databasebeta

← #10829960

Subgroups — #10829960

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic, complemented
2#82500C211yesmaximal, characteristic
3C2×#4841C211yesmaximal, characteristic
4C2×C4×#204C211yesmaximal, characteristic
5C3×#1984C211yesmaximal, characteristic, complemented
6#70485C211yesmaximal, characteristic, complemented
7#69771C211yesmaximal, characteristic, complemented
8#68687C212yesmaximal
9#68686C212yesmaximal
10#68684C212yesmaximal, complemented
11#68631C212yesmaximal, complemented
12#24187—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C2×Q8×Dic3—21no—
16C22×#204C2211yescharacteristic
17C22×#204C2212yes—
18#5570—24no—
19#5570—24no—
20C2×#890—21no—
21C2×#890—22no—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2212yes—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2212yes—
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4994—22no—
37#4987—21no—
38#4939C2212yes—
39#4935C2211yescharacteristic
40#4928C2212yes—
41#4841—24no—
42C4×#204—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#557—31no—
47C2×#461—31no—
48C2×C4×SD16—31no—
49#2538—31no—
50#2043—32no—
51#2013—32no—
52#1984D311yescharacteristic, complemented
53#1956—31no—
54#1880—32no—
55#1865—32no—
56#1803—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6C2312yes—
61C22×Dic6C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D412yes—
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71#946—44no—
72Q8×Dic3—28no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—21no—
90C2×#204—22no—
91C2×#204—22no—
92C2×#204—22no—
93C2×#204—24no—
94C2×#204—24no—
95C2×#204—42no—
96#890—216no—
97#890—28no—
98C4×D12—22no—
99C4×D12—22no—
100#869—24no—
101#869—24no—
102#869—22no—
103#869D414yes—
104C4×Dic6—24no—
105C4×Dic6—22no—
106C4×Dic6—24no—
107C4×Dic6D414yes—
108C3×C4⋊C8—22no—
109C3×C4×C8—22no—
110C3⋊#95—22no—
111#821—22no—
112#817—22no—
113#816—22no—
114#816—24no—
115#816—24no—
116#816—42no—
117#816—42no—
118C22×SD16—31no—
119C22×SD16—32no—
120#557—64no—
121#557—64no—
122C2×C4⋊Q8—31no—
123C2×C4⋊Q8—32no—
124C2×#125—61no—
125C2×#125—62no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—32no—
129C2×C4×Q8—61no—
130C2×C4×Q8C2×D311yescharacteristic
131C2×C4×D4—31no—
132#461—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—32no—
137C2×#103—32no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#102—31no—
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336C2×D312yes—
150#336C2×D312yes—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8SD1614yescomplemented
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12#4112yes—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—42no—
174C2×Dic6—22no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—22no—
178C2×Dic6—21no—
179C2×Dic6—22no—
180C2×Dic6—21no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—21no—
184C2×Dic6C2×D414yes—
185C2×Dic6C2×D412yes—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8#4112yes—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201(C2×D3)⋊C4—42no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—82no—
204C4⋊Dic3—28no—
205C4⋊Dic3—28no—
206C4⋊Dic3—24no—
207C4⋊Dic3—24no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—42no—
214C4⋊Dic3—81no—
215C4⋊Dic3C2×D412yes—
216Dic3⋊C4—28no—
217Dic3⋊C4—24no—
218Dic3⋊C4—44no—
219C4×Dic3—28no—
220C4×Dic3—24no—
221#204—28no—
222#204—48no—
223C22×Q8—31no—
224C22×Q8—32no—
225C22×Q8D1211yescharacteristic
226C22×D4—31no—
227C23×C4—61no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—62no—
233C2×SD16—122no—
234C22×C8C22×D312yes—
235C22×C8C22×D311yescharacteristic
236C22×C8C22×D311yescharacteristic
237C4⋊Q8—34no—
238C4⋊Q8—64no—
239C4⋊Q8—64no—
240C4⋊Q8—62no—
241#125—616no—
242#125—68no—
243C22⋊Q8—124no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—34no—
247C4×Q8—64no—
248C4×Q8—64no—
249C4×Q8—68no—
250C4×Q8—62no—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4D1211yescharacteristic
266C2×C4⋊C4D1212yes—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—62no—
271C2×C42—61no—
272C2×C42C22×D312yes—
273C2×C42C22×D311yescharacteristic
274#106—62no—
275C4⋊C8—22no—
276#103—62no—
277#103—64no—
278#103—64no—
279#103—122no—
280#103—122no—
281#102—62no—
282C4×C8—22no—
283#95—62no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C3×Q8—28no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4#12012yes—
301C2×C3×C4#12012yes—
302C2×C3×C4C2×SD1614yes—
303C2×C3×C4C2×SD1612yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—22no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326D12—22no—
327D12—22no—
328D12—41no—
329D12—41no—
330C4×D3—84no—
331Dic6—22no—
332Dic6—22no—
333Dic6—24no—
334Dic6—24no—
335Dic6—42no—
336Dic6—42no—
337Dic6—41no—
338Dic6—41no—
339C3×C8—22no—
340C3×C8—41no—
341C3×C8—42no—
342C24—61no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—32no—
346C2×Q8—34no—
347C2×Q8—61no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8—61no—
351C2×Q8—62no—
352C2×Q8—61no—
353C2×Q8—62no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—62no—
357C2×Q8#20414yescomplemented
358C2×D4—32no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C22×C4—61no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—122no—
370C22×C4—122no—
371C22×C4—122no—
372C22×C4C2×D1212yes—
373C22×C4C2×D1211yescharacteristic
374C22×C4C2×D1212yes—
375C22×C4C2×D1211yescharacteristic
376C22×C4C23×D311yescharacteristic
377SD16—68no—
378SD16—128no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—22no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8#23912yes—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—32no—
391C4⋊C4—68no—
392C4⋊C4—68no—
393C4⋊C4—68no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—64no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—124no—
403C4⋊C4—122no—
404C4⋊C4—241no—
405C22⋊C4—122no—
406C22⋊C4—124no—
407C22⋊C4—242no—
408C42—22no—
409C42—22no—
410C42—68no—
411C42—64no—
412C42D3×D412yes—
413C42D3×D412yes—
414C22×C3C2×#12011yescharacteristic
415C22×C3#54811yescharacteristic
416C22×C3#55011yescharacteristic
417C22×C3C22×SD1611yescharacteristic
418C22×C3C2×#13711yescharacteristic
419C22×C3#57711yescharacteristic
420C22×C3#57811yescharacteristic
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—81no—
427C2×D3—81no—
428C3×C4—24no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4#44814yes—
432C3×C4#52511yescharacteristic
433C3×C4#53711yescharacteristic
434C3×C4#54611yescharacteristic
435C3×C4#54711yescharacteristic
436Dic3—42no—
437Dic3—44no—
438Dic3—42no—
439Dic3—82no—
440Dic3—81no—
441C23—62no—
442C23—121no—
443C23—121no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23—121no—
448C23C22×D1211yescharacteristic
449Q8—28no—
450Q8—64no—
451Q8—64no—
452Q8—62no—
453Q8—62no—
454Q8—121no—
455Q8—121no—
456Q8—122no—
457Q8—122no—
458D4—62no—
459D4—62no—
460D4—121no—
461D4—121no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—62no—
470C2×C4—64no—
471C2×C4—62no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—122no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—121no—
478C2×C4—122no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—244no—
485C2×C4#88212yes—
486C2×C4#88212yes—
487C2×C4C2×#20414yes—
488C2×C4C2×#20412yes—
489C2×C4#90912yes—
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×D3×D411yescharacteristic
492C2×C4C2×#23911yescharacteristic
493C2×C4#100711yescharacteristic
494C2×C4D3×#4111yescharacteristic
495C2×C4#100911yescharacteristic
496C8—22no—
497C8—42no—
498C8—41no—
499C2×C3#240711yescharacteristic
500C2×C3C2×#44811yescharacteristic
501C2×C3#301311yescharacteristic
502C2×C3#335911yescharacteristic
503C2×C3#337211yescharacteristic
504C2×C3#339211yescharacteristic
505C2×C3#339811yescharacteristic
506D3—82no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—241no—
511C22—241no—
512C22—241no—
513C22—241no—
514C22C2×#88211yescharacteristic
515C22#557611yescharacteristic
516C22#557711yescharacteristic
517C22C22×#20411yescharacteristic
518C22C2×#90911yescharacteristic
519C22#575111yescharacteristic
520C22#575311yescharacteristic
521C4—24no—
522C4—22no—
523C4—42no—
524C4—124no—
525C4—122no—
526C4—122no—
527C4—241no—
528C4—242no—
529C4#473814yes—
530C4#562711yescharacteristic
531C4#562911yescharacteristic
532C4#567611yescharacteristic
533C4#572611yescharacteristic
534C3#2418711yescharacteristic, complemented, Sylow
535C2—242no—
536C2#7661811yescharacteristic
537C2C2×#473811yescharacteristic
538C2#7868011yescharacteristic
539C2#7991111yescharacteristic
540C2#7992511yescharacteristic
541C2#7995111yescharacteristic
542C2#7997211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1984D3
hypocenterC3#24187
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
1. C2×#5570C2semidirect
5. C3×#1984C2semidirect
6. #70485C2semidirect
7. #69771C2semidirect
10. #68684C2semidirect
11. #68631C2semidirect
52. #1984D3semidirect
154. C2×C3×Q8SD16semidirect
357. C2×Q8#204semidirect
534. C3#24187semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
#825002
C2×#48412
C2×C4×#2042
C3×#19842
#704852
#697712
#686872
#686862
#686842
#686312
#241873