Finite Group Databasebeta

← #10830130

Subgroups — #10830130

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic
2#82480C211yesmaximal, characteristic, complemented
3C2×#4849C211yesmaximal, characteristic, complemented
4C2×#4710C211yesmaximal, characteristic
5C3×#1985C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic
7#69822C211yesmaximal, characteristic, complemented
8#68685C212yesmaximal, complemented
9#68683C212yesmaximal, complemented
10#68677C212yesmaximal
11#68630C212yesmaximal
12#24201—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C22×Dic12C2212yes—
16C22×#204C2211yescharacteristic
17#5584—24no—
18#5584—24no—
19C2×#898—22no—
20C2×#892—21no—
21C2×#875C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5230—21no—
29C2×#821C2211yescharacteristic
30C2×#821C2212yes—
31C2×#818C2211yescharacteristic
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2211yescharacteristic
35C2×#816C2212yes—
36#4985—21no—
37#4978—22no—
38#4936C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41#4849—24no—
42#4710—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×#440—31no—
49#2581—31no—
50#2042—32no—
51#2015—32no—
52#1985D311yescharacteristic, complemented
53#1960—31no—
54#1878—32no—
55#1863—32no—
56#1804—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⋊C4D412yes—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71#946—28no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—21no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—21no—
85C2×Dic12—24no—
86C2×Dic12—24no—
87C2×Dic12—42no—
88C2×#204—22no—
89C2×#204—22no—
90C2×#204—22no—
91#898—216no—
92#892—28no—
93#875—24no—
94#875—24no—
95C4×D12—22no—
96C4×D12D414yes—
97#869—22no—
98#869D414yes—
99C4×Dic6—24no—
100C4×Dic6—24no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×C4⋊C8—22no—
104C3×#97—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#821—22no—
111#821—42no—
112#818—22no—
113#816—24no—
114#816—24no—
115#816—22no—
116#816—42no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×#117—61no—
130#462—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97C2×D311yescharacteristic
141#400—61no—
142#395—61no—
143#391—32no—
144#389—32no—
145#387—62no—
146#382—31no—
147#340—32no—
148#336C2×D312yes—
149#336C2×D312yes—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—42no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12C2×D412yes—
169C2×C4×D3—24no—
170C2×C4×D3—44no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—22no—
174C2×Dic6—21no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6—21no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—21no—
181C2×Dic6C2×D414yes—
182C2×Dic6#4112yes—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8#4112yes—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—24no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—24no—
201C4⋊Dic3—24no—
202C4⋊Dic3—24no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3C2×D412yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—44no—
211Dic3⋊C4—42no—
212C4×Dic3—28no—
213C4×Dic3—42no—
214Dic12—48no—
215#204—28no—
216C22×Q8—32no—
217C22×Q8—31no—
218C22×Q8D1211yescharacteristic
219C22×D4—31no—
220C23×C4—61no—
221C2×Q16—64no—
222C2×Q16—64no—
223C2×Q16—122no—
224C2×SD16—62no—
225C2×SD16—62no—
226C2×SD16—62no—
227C22×C8C22×D312yes—
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D311yescharacteristic
230C4⋊Q8—34no—
231C4⋊Q8—62no—
232#126—616no—
233#124—64no—
234#124—64no—
235C22⋊Q8—68no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—64no—
239C4×Q8—64no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×D4—34no—
243C4×D4—62no—
244#117—68no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4D1211yescharacteristic
254C2×C4⋊C4D1212yes—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—62no—
258C2×C42—62no—
259C2×C42—61no—
260C2×C42C22×D312yes—
261C2×C42C22×D311yescharacteristic
262#107—62no—
263C4⋊C8—22no—
264#103—62no—
265#103—64no—
266#103—64no—
267#103—122no—
268#102—62no—
269#102—122no—
270#97—22no—
271#95—61no—
272#95—61no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3C22×D411yescharacteristic
277C22×D3—22no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—42no—
281C22×D3—42no—
282C3×Q8—44no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4#12012yes—
291C2×C3×C4#12012yes—
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4H(5,𝔽2)11yescharacteristic
297C2×C3×C42−1+411yescharacteristic
298C2×Dic3—22no—
299C2×Dic3—24no—
300C2×Dic3—22no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311D12—22no—
312D12—22no—
313D12—41no—
314D12—41no—
315C4×D3—48no—
316Dic6—22no—
317Dic6—22no—
318Dic6—24no—
319Dic6—24no—
320Dic6—42no—
321Dic6—41no—
322Dic6—42no—
323Dic6—41no—
324C3×C8—22no—
325C3×C8—41no—
326C3×C8—42no—
327C24—61no—
328C2×Q8—22no—
329C2×Q8—34no—
330C2×Q8—32no—
331C2×Q8—42no—
332C2×Q8—62no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—62no—
337C2×Q8—61no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—61no—
342C2×D4—32no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—64no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—124no—
355C22×C4C2×D1212yes—
356C22×C4C2×D1212yes—
357C22×C4C2×D1211yescharacteristic
358C22×C4C2×D1211yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—128no—
361SD16—68no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8#23912yes—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—32no—
374C4⋊C4—68no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—124no—
383C4⋊C4—122no—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4—64no—
387C42—22no—
388C42—22no—
389C42—68no—
390C42—122no—
391C42D3×D412yes—
392C42D3×D412yes—
393C22×C3C2×#12011yescharacteristic
394C22×C3#54811yescharacteristic
395C22×C3#55011yescharacteristic
396C22×C3#57512yes—
397C22×C3#57812yes—
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—82no—
402C2×D3—82no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4#52511yescharacteristic
408C3×C4#53711yescharacteristic
409C3×C4#54611yescharacteristic
410C3×C4#54711yescharacteristic
411Dic3—42no—
412Dic3—44no—
413Dic3—42no—
414Dic3—82no—
415Dic3—81no—
416C23—62no—
417C23—122no—
418C23—122no—
419C23—121no—
420C23—121no—
421C23C22×D1211yescharacteristic
422Q8—44no—
423Q8—62no—
424Q8—64no—
425Q8—64no—
426Q8—62no—
427Q8—121no—
428Q8—122no—
429Q8—122no—
430Q8—121no—
431D4—62no—
432D4—62no—
433D4—121no—
434D4—121no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—62no—
443C2×C4—64no—
444C2×C4—62no—
445C2×C4—128no—
446C2×C4—121no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4#88212yes—
457C2×C4#88212yes—
458C2×C4C2×D3×D411yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4C2×#23911yescharacteristic
461C2×C4#100711yescharacteristic
462C2×C4D3×#4111yescharacteristic
463C2×C4#100911yescharacteristic
464C8—22no—
465C8—42no—
466C8—41no—
467C2×C3#240711yescharacteristic
468C2×C3#300511yescharacteristic
469C2×C3#301111yescharacteristic
470C2×C3#337712yes—
471C2×C3#340212yes—
472D3—82no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22—242no—
477C22—242no—
478C22C2×#88211yescharacteristic
479C22#557611yescharacteristic
480C22#557711yescharacteristic
481C22#574812yes—
482C22#575312yes—
483C4—22no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—122no—
488C4—124no—
489C4—122no—
490C4—241no—
491C4—242no—
492C4#562711yescharacteristic
493C4#562911yescharacteristic
494C4#567611yescharacteristic
495C4#572611yescharacteristic
496C3#2420111yescharacteristic, complemented, Sylow
497C2—242no—
498C2#7661811yescharacteristic
499C2#7863811yescharacteristic
500C2#7867911yescharacteristic
501C2#7995512yes—
502C2#7998512yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1985D3
hypocenterC3#24201
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #82480C2semidirect
3. C2×#4849C2semidirect
5. C3×#1985C2semidirect
7. #69822C2semidirect
8. #68685C2semidirect
9. #68683C2semidirect
52. #1985D3semidirect
496. C3#24201semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
#824802
C2×#48492
C2×#47102
C3×#19852
#704802
#698222
#686852
#686832
#686772
#686302
#242013