Finite Group Databasebeta

← #10830108

Subgroups — #10830108

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic, complemented
2#82501C211yesmaximal, characteristic
3C2×#4841C211yesmaximal, characteristic
4C2×#4710C211yesmaximal, characteristic
5C3×#1985C211yesmaximal, characteristic, complemented
6#70486C211yesmaximal, characteristic, complemented
7#69822C211yesmaximal, characteristic, complemented
8#68687C212yesmaximal
9#68686C212yesmaximal
10#68683C212yesmaximal, complemented
11#68631C212yesmaximal, complemented
12#24199—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×Q8×Dic3—21no—
15C22×#204C2211yescharacteristic
16C22×#204C2212yes—
17#5570—24no—
18#5570—24no—
19C2×#890—22no—
20C2×#890—21no—
21C2×C4×D12C2211yescharacteristic
22C2×#869C2212yes—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5231—21no—
29C2×#821C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4993—21no—
37#4989—22no—
38#4939C2212yes—
39#4936C2211yescharacteristic
40#4928C2212yes—
41#4841—24no—
42#4710—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#557—31no—
47C2×#461—31no—
48C2×#440—31no—
49#2580—31no—
50#2043—32no—
51#2013—32no—
52#1985D311yescharacteristic, complemented
53#1960—31no—
54#1880—32no—
55#1863—32no—
56#1805—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
71Q8×Dic3—28no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2312yes—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—21no—
87C2×#204—22no—
88C2×#204—22no—
89C2×#204—22no—
90C2×#204—24no—
91C2×#204—24no—
92C2×#204—42no—
93#890—216no—
94#890—28no—
95C4×D12—22no—
96C4×D12—22no—
97#869—24no—
98#869—24no—
99#869—22no—
100#869D414yes—
101C4×Dic6—24no—
102C4×Dic6—22no—
103C4×Dic6—24no—
104C4×Dic6D414yes—
105C3×C4⋊C8—22no—
106C3×#97—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#821—22no—
111#818—22no—
112#816—22no—
113#816—24no—
114#816—24no—
115#816—42no—
116#816—42no—
117C22×SD16—31no—
118C22×SD16—32no—
119#557—64no—
120#557—64no—
121C2×C4⋊Q8—31no—
122C2×C4⋊Q8—32no—
123C2×#125—62no—
124C2×#125—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—61no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—31no—
130#461—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#102—31no—
140C2×#97C2×D311yescharacteristic
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336C2×D312yes—
149#336C2×D312yes—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—24no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12#4112yes—
169C2×C4×D3—42no—
170C2×C4×D3—82no—
171C2×Dic6—22no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6—22no—
175C2×Dic6—21no—
176C2×Dic6—22no—
177C2×Dic6—21no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—21no—
181C2×Dic6C2×D414yes—
182C2×Dic6C2×D412yes—
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—24no—
193C3×C4⋊C4—22no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198(C2×D3)⋊C4—42no—
199(C2×D3)⋊C4—44no—
200(C2×D3)⋊C4—82no—
201C4⋊Dic3—28no—
202C4⋊Dic3—28no—
203C4⋊Dic3—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—28no—
212Dic3⋊C4—24no—
213Dic3⋊C4—82no—
214C4×Dic3—28no—
215C4×Dic3—24no—
216#204—28no—
217#204—48no—
218C22×Q8—31no—
219C22×Q8—32no—
220C22×Q8D1211yescharacteristic
221C22×D4—31no—
222C23×C4—61no—
223C2×SD16—62no—
224C2×SD16—62no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—62no—
228C2×SD16—122no—
229C22×C8C22×D312yes—
230C22×C8C22×D311yescharacteristic
231C22×C8C22×D311yescharacteristic
232C4⋊Q8—34no—
233C4⋊Q8—64no—
234C4⋊Q8—64no—
235C4⋊Q8—62no—
236#125—616no—
237#125—68no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—34no—
241C4×Q8—64no—
242C4×Q8—64no—
243C4×Q8—68no—
244C4×Q8—62no—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4D1212yes—
258C2×C4⋊C4D1211yescharacteristic
259C2×C22⋊C4—61no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C42—62no—
263C2×C42—61no—
264C2×C42C22×D312yes—
265C2×C42C22×D311yescharacteristic
266#107—62no—
267C4⋊C8—22no—
268#103—62no—
269#103—64no—
270#103—64no—
271#103—122no—
272#103—122no—
273#102—62no—
274#97—22no—
275#95—61no—
276#95—61no—
277#95—62no—
278C23×C3C22×D411yescharacteristic
279C22×D3—22no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—42no—
284C3×Q8—28no—
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—22no—
292C2×C3×C4#12012yes—
293C2×C3×C4#12012yes—
294C2×C3×C4#13714yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4H(5,𝔽2)11yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301C2×Dic3—22no—
302C2×Dic3—24no—
303C2×Dic3—22no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314D12—22no—
315D12—22no—
316D12—41no—
317D12—41no—
318C4×D3—84no—
319Dic6—22no—
320Dic6—22no—
321Dic6—24no—
322Dic6—24no—
323Dic6—42no—
324Dic6—42no—
325Dic6—41no—
326Dic6—41no—
327C3×C8—22no—
328C3×C8—41no—
329C3×C8—42no—
330C24—61no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—32no—
334C2×Q8—34no—
335C2×Q8—61no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—61no—
339C2×Q8—62no—
340C2×Q8—61no—
341C2×Q8—62no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—62no—
345C2×D4—32no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C22×C4—61no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—122no—
357C22×C4—242no—
358C22×C4C2×D1212yes—
359C22×C4C2×D1211yescharacteristic
360C22×C4C2×D1211yescharacteristic
361C22×C4C2×D1212yes—
362C22×C4C23×D311yescharacteristic
363SD16—68no—
364SD16—128no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8#23912yes—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—32no—
377C4⋊C4—68no—
378C4⋊C4—68no—
379C4⋊C4—68no—
380C4⋊C4—64no—
381C4⋊C4—64no—
382C4⋊C4—64no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—242no—
389C22⋊C4—122no—
390C22⋊C4—124no—
391C22⋊C4—242no—
392C42—22no—
393C42—22no—
394C42—68no—
395C42—64no—
396C42D3×D412yes—
397C42D3×D412yes—
398C22×C3C2×#12011yescharacteristic
399C22×C3#54811yescharacteristic
400C22×C3#55011yescharacteristic
401C22×C3C2×#13712yes—
402C22×C3#57712yes—
403C2×D3—44no—
404C2×D3—42no—
405C2×D3—82no—
406C2×D3—81no—
407C2×D3—81no—
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4#52511yescharacteristic
413C3×C4#53711yescharacteristic
414C3×C4#54611yescharacteristic
415C3×C4#54711yescharacteristic
416Dic3—42no—
417Dic3—44no—
418Dic3—42no—
419Dic3—82no—
420Dic3—81no—
421C23—62no—
422C23—121no—
423C23—122no—
424C23—122no—
425C23—121no—
426C23C22×D1211yescharacteristic
427Q8—28no—
428Q8—64no—
429Q8—64no—
430Q8—62no—
431Q8—62no—
432Q8—121no—
433Q8—121no—
434Q8—122no—
435Q8—122no—
436D4—62no—
437D4—62no—
438D4—121no—
439D4—121no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—62no—
448C2×C4—64no—
449C2×C4—62no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—244no—
461C2×C4#88212yes—
462C2×C4#88212yes—
463C2×C4#90914yes—
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×D3×D411yescharacteristic
466C2×C4C2×#23911yescharacteristic
467C2×C4#100711yescharacteristic
468C2×C4D3×#4111yescharacteristic
469C2×C4#100911yescharacteristic
470C8—22no—
471C8—42no—
472C8—41no—
473C2×C3#240711yescharacteristic
474C2×C3#300411yescharacteristic
475C2×C3#301011yescharacteristic
476C2×C3#337212yes—
477C2×C3#340012yes—
478D3—82no—
479C22—122no—
480C22—124no—
481C22—241no—
482C22—242no—
483C22—241no—
484C22C2×#88211yescharacteristic
485C22#557611yescharacteristic
486C22#557711yescharacteristic
487C22C2×#90912yes—
488C22#575112yes—
489C4—22no—
490C4—24no—
491C4—22no—
492C4—42no—
493C4—124no—
494C4—122no—
495C4—122no—
496C4—242no—
497C4—241no—
498C4#562711yescharacteristic
499C4#562911yescharacteristic
500C4#567611yescharacteristic
501C4#572611yescharacteristic
502C3#2419911yescharacteristic, complemented, Sylow
503C2—242no—
504C2#7661811yescharacteristic
505C2#7864511yescharacteristic
506C2#7867811yescharacteristic
507C2#7995312yes—
508C2#7997212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1985D3
hypocenterC3#24199
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
1. C2×#5570C2semidirect
5. C3×#1985C2semidirect
6. #70486C2semidirect
7. #69822C2semidirect
10. #68683C2semidirect
11. #68631C2semidirect
52. #1985D3semidirect
502. C3#24199semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
#825012
C2×#48412
C2×#47102
C3×#19852
#704862
#698222
#686872
#686862
#686832
#686312
#241993