Finite Group Databasebeta

← #10830141

Subgroups — #10830141

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2#82480C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic
4C2×#4716C211yesmaximal, characteristic, complemented
5C3×#1985C211yesmaximal, characteristic, complemented
6#70486C211yesmaximal, characteristic, complemented
7#69823C211yesmaximal, characteristic
8#68677C212yesmaximal, complemented
9#68672C212yesmaximal, complemented
10#68671C212yesmaximal, complemented
11#68630C212yesmaximal, complemented
12#24202—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15C22×Dic12C2211yescharacteristic
16C22×#204C2212yes—
17#5585—24no—
18#5585—24no—
19C2×#894—22no—
20C2×#893—21no—
21C2×#875C2212yes—
22C2×#874C2211yescharacteristic
23C2×C4×D12C2212yes—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5230—21no—
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2212yes—
36#4985—21no—
37#4978—22no—
38#4936C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41#4848—24no—
42#4716—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#559—31no—
47C2×#460—31no—
48C2×#441—31no—
49#2581—31no—
50#2042—32no—
51#2015—32no—
52#1985D311yescharacteristic, complemented
53#1961—31no—
54#1877—32no—
55#1864—32no—
56#1805—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×Dic6C2311yescharacteristic
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C42C2312yes—
72C2×C3×C42C2311yescharacteristic
73#947—28no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—21no—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—21no—
85C2×Dic12—22no—
86C2×Dic12—22no—
87C2×Dic12—22no—
88C2×#204—24no—
89C2×#204—24no—
90C2×#204—42no—
91#894—216no—
92#893—28no—
93#875—24no—
94#875—24no—
95#874—22no—
96#874D414yes—
97C4×D12—24no—
98C4×D12—24no—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6D414yes—
103C3×C4⋊C8—22no—
104C3×#97—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#821—24no—
111#821—24no—
112#821—42no—
113#817—22no—
114#816—22no—
115#816—22no—
116#816—42no—
117C22×Q16—31no—
118C22×SD16—32no—
119#559—64no—
120#559—64no—
121C2×#127—31no—
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—31no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130#460—64no—
131#441—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—32no—
138C2×#102—32no—
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—22no—
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—21no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—24no—
168C2×D12—22no—
169C2×D12—24no—
170C2×D12—44no—
171C2×D12—82no—
172C2×D12C2×D414yes—
173C2×C4×D3—28no—
174C2×C4×D3—84no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C2×D412yes—
186C2×Dic6#4112yes—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8#4112yes—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—24no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—28no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—24no—
212Dic3⋊C4—44no—
213Dic3⋊C4—42no—
214C4×Dic3—24no—
215C4×Dic3—42no—
216Dic12—28no—
217#204—48no—
218C22×Q8—31no—
219C22×Q8—31no—
220C22×Q8D1211yescharacteristic
221C22×D4—32no—
222C22×D4—61no—
223C22×D4—61no—
224C23×C4—62no—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—122no—
231C22×C8C22×D312yes—
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D311yescharacteristic
234#127—34no—
235#127—62no—
236#124—64no—
237#124—64no—
238#124—68no—
239#123—616no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—34no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8—62no—
246C4×D4—64no—
247C4×D4—64no—
248C4×D4—68no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4D1211yescharacteristic
256C2×C4⋊C4D1212yes—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C42—61no—
261C2×C42—61no—
262C2×C42C22×D312yes—
263C2×C42C22×D311yescharacteristic
264#106—62no—
265C4⋊C8—22no—
266#103—62no—
267#103—62no—
268#103—122no—
269#102—64no—
270#102—64no—
271#102—122no—
272#97—22no—
273#95—61no—
274#95—61no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3C22×D411yescharacteristic
279C22×D3—24no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—44no—
283C22×D3—44no—
284C3×Q8—44no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4#12012yes—
293C2×C3×C4#12012yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4H(5,𝔽2)11yescharacteristic
299C2×C3×C42−1+411yescharacteristic
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313D12—24no—
314D12—24no—
315D12—48no—
316D12—42no—
317D12—42no—
318D12—84no—
319C4×D3—88no—
320Dic6—22no—
321Dic6—22no—
322Dic6—22no—
323Dic6—22no—
324Dic6—41no—
325Dic6—41no—
326Dic6—41no—
327Dic6—41no—
328C3×C8—22no—
329C3×C8—41no—
330C3×C8—42no—
331C24—62no—
332C2×Q8—22no—
333C2×Q8—32no—
334C2×Q8—32no—
335C2×Q8—42no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×D4—34no—
347C2×D4—64no—
348C2×D4—64no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—124no—
355C2×D4—242no—
356C22×C4—68no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—244no—
363C22×C4C2×D1212yes—
364C22×C4C2×D1212yes—
365C22×C4C2×D1211yescharacteristic
366C22×C4C2×D1211yescharacteristic
367C22×C4C23×D311yescharacteristic
368Q16—68no—
369SD16—128no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8#23912yes—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—32no—
382C4⋊C4—64no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—124no—
389C4⋊C4—122no—
390C22⋊C4—68no—
391C22⋊C4—68no—
392C22⋊C4—68no—
393C42—22no—
394C42—22no—
395C42—64no—
396C42—122no—
397C42D3×D412yes—
398C42D3×D412yes—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54811yescharacteristic
401C22×C3#55011yescharacteristic
402C22×C3#57512yes—
403C22×C3#57712yes—
404C2×D3—44no—
405C2×D3—44no—
406C2×D3—44no—
407C2×D3—84no—
408C2×D3—84no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4#52511yescharacteristic
414C3×C4#53711yescharacteristic
415C3×C4#54611yescharacteristic
416C3×C4#54711yescharacteristic
417Dic3—42no—
418Dic3—42no—
419Dic3—42no—
420Dic3—82no—
421Dic3—81no—
422C23—64no—
423C23—124no—
424C23—124no—
425C23—122no—
426C23—122no—
427C23C22×D1211yescharacteristic
428Q8—44no—
429Q8—62no—
430Q8—62no—
431Q8—62no—
432Q8—62no—
433Q8—121no—
434Q8—121no—
435Q8—121no—
436Q8—121no—
437D4—64no—
438D4—64no—
439D4—128no—
440D4—122no—
441D4—122no—
442D4—244no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—248no—
464C2×C4#88212yes—
465C2×C4#88212yes—
466C2×C4C2×D3×D411yescharacteristic
467C2×C4C2×D3×D411yescharacteristic
468C2×C4C2×#23911yescharacteristic
469C2×C4#100711yescharacteristic
470C2×C4D3×#4111yescharacteristic
471C2×C4#100911yescharacteristic
472C8—22no—
473C8—42no—
474C8—41no—
475C2×C3#240711yescharacteristic
476C2×C3#300511yescharacteristic
477C2×C3#300911yescharacteristic
478C2×C3#337112yes—
479C2×C3#340112yes—
480D3—84no—
481C22—124no—
482C22—124no—
483C22—124no—
484C22—244no—
485C22—244no—
486C22C2×#88211yescharacteristic
487C22#557611yescharacteristic
488C22#557711yescharacteristic
489C22#574812yes—
490C22#575112yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—122no—
496C4—122no—
497C4—122no—
498C4—241no—
499C4—242no—
500C4#562711yescharacteristic
501C4#562911yescharacteristic
502C4#567611yescharacteristic
503C4#572611yescharacteristic
504C3#2420211yescharacteristic, complemented, Sylow
505C2—244no—
506C2#7661811yescharacteristic
507C2#7863811yescharacteristic
508C2#7867111yescharacteristic
509C2#7995412yes—
510C2#7998212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1985D3
hypocenterC3#24202
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
4. C2×#4716C2semidirect
5. C3×#1985C2semidirect
6. #70486C2semidirect
8. #68677C2semidirect
9. #68672C2semidirect
10. #68671C2semidirect
11. #68630C2semidirect
52. #1985D3semidirect
504. C3#24202semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#824802
C2×#48482
C2×#47162
C3×#19852
#704862
#698232
#686772
#686722
#686712
#686302
#242023