Finite Group Databasebeta

← #10950918

Subgroups — #10950918

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#467C211yesmaximal, characteristic, complemented
2C2×#5115C211yesmaximal, characteristic
3#72063C212yesmaximal, complemented
4C2×#4549C211yesmaximal, characteristic, complemented
5#71085C212yesmaximal, complemented
6#71082C214yesmaximal, complemented
7#71078C212yesmaximal, complemented
8#70600C212yesmaximal, complemented
9#33753—31nomaximal, Sylow
10C22×C3×Q16C2211yescharacteristic, complemented
11C23×C3×C8C2211yescharacteristic
12C2×C3×C22⋊Q8—21no—
13C2×#930—21no—
14C22×D24C2211yescharacteristic
15C3×#467—21no—
16C3×#467—21no—
17C3×#467C2212yes—
18C3×#467C2212yescomplemented
19C2×C3×#107C2211yescharacteristic, complemented
20C2×C3×#103—21no—
21C2×#828C2214yescomplemented
22#5115—21no—
23#5115—21no—
24#5115C2212yes—
25#5115C2212yes—
26C2×#821—21no—
27C2×#818C2211yescharacteristic
28C22×C3⋊C16C2212yescomplemented
29#4564C2214yescomplemented
30#4549C2212yes—
31#4549C2212yescomplemented
32#4491C2214yescomplemented
33#4490C2214yescomplemented
34C2×#467D311yescharacteristic, complemented
35C2×#466—31no—
36#2235—32no—
37#2233—32no—
38#2217—34no—
39#2187—32no—
40#2185—32no—
41C2×C22⋊C16—31no—
42C22×C3×Q8—21no—
43C23×C3×C4D411yescharacteristic
44C22×#67—41no—
45C22×D12—21no—
46C2×C3×Q16—21no—
47C2×C3×Q16—42no—
48C2×C3×Q16C2312yes—
49C22×C3×C8—42no—
50C22×C3×C8D411yescharacteristic
51C22×C3×C8D411yescharacteristic
52C22×C3×C8D411yescharacteristic
53C22×C3×C8C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×C22⋊Q8—22no—
57C3×C22⋊Q8—22no—
58C3×C22⋊Q8—41no—
59C3×C22⋊Q8—41no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—41no—
62C2×C3×C22⋊C4—41no—
63#930—22no—
64#930—22no—
65#930—41no—
66#930—41no—
67C2×(C2×D3)⋊C4—41no—
68C2×C4⋊Dic3—21no—
69C2×D24—21no—
70C2×D24—42no—
71C2×D24C2312yes—
72C3×#107—21no—
73C3×#107C2312yes—
74C3×#103—22no—
75C3×#103—22no—
76#828—44no—
77#821—22no—
78#821—22no—
79#818—21no—
80#818C2312yes—
81C2×C3⋊C16—42no—
82C2×C3⋊C16C2314yes—
83C22×Q16C2×D311yescharacteristic, complemented
84C22×D8—31no—
85C23×C8C2×D311yescharacteristic
86C2×C22⋊Q8—21no—
87C2×#121—61no—
88C2×SD32—34no—
89C22×C16—32no—
90#467—21no—
91#467—21no—
92#467C2×D312yes—
93#467C2×D312yescomplemented
94#466—32no—
95#466—32no—
96#466—61no—
97#466—61no—
98C2×#107—31no—
99C2×#107C2×D311yescharacteristic, complemented
100C2×#103—21no—
101C2×#102—61no—
102#367—34no—
103#358—34no—
104#357—34no—
105C22⋊C16—32no—
106C22⋊C16—32no—
107C24×C3D811yescharacteristic
108C23×D3—41no—
109C2×C3×Q8—22no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4D811yescharacteristic
118C22×C3×C4D811yescharacteristic
119C22×C3×C4D811yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—42no—
124C2×#67—42no—
125C2×#67—84no—
126C22×Dic3—41no—
127C2×D12—22no—
128C2×D12—42no—
129C2×D12—42no—
130C2×D12—41no—
131C3×Q16—24no—
132C3×Q16—42no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8C2×D411yescharacteristic
140C2×C3×C8C2×D411yescharacteristic
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8C2411yescharacteristic
143C3×C4⋊C4—22no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—42no—
146C3×C4⋊C4—42no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—42no—
149(C2×D3)⋊C4—42no—
150(C2×D3)⋊C4—42no—
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153D24—24no—
154D24—42no—
155C3⋊C16—24no—
156C22×Q8—21no—
157C22×D4—61no—
158C22×D4—121no—
159C23×C4#6711yescharacteristic
160C2×Q16—21no—
161C2×Q16—42no—
162C2×Q16C22×D312yes—
163C2×D8—32no—
164C2×D8—61no—
165C2×D8—122no—
166C22×C8—42no—
167C22×C8#6711yescharacteristic
168C22×C8#6711yescharacteristic
169C22×C8#6711yescharacteristic
170C22×C8C22×D311yescharacteristic
171C22×C8C22×D311yescharacteristic
172C22×C8C22×D311yescharacteristic
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177#121—62no—
178#121—62no—
179#121—121no—
180#121—121no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—121no—
186SD32—124no—
187C2×C16—34no—
188C2×C16—122no—
189#107—21no—
190#107—32no—
191#107—61no—
192#107C22×D312yes—
193#103—22no—
194#103—22no—
195#102—62no—
196#102—62no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3SD3214yescomplemented
200C23×C3C2×D811yescharacteristic
201C23×C3C2×D811yescharacteristic
202C23×C3C2×D811yescharacteristic
203C22×D3—42no—
204C22×D3—82no—
205C22×D3—82no—
206C22×D3—82no—
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—82no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—82no—
222C2×C3×C4—82no—
223C2×C3×C4C2×D811yescharacteristic
224C2×C3×C4C2×D811yescharacteristic
225C2×C3×C4C2×D811yescharacteristic
226C2×C3×C4C22×D411yescharacteristic
227#67—84no—
228#67—84no—
229C2×Dic3—42no—
230C2×Dic3—82no—
231D12—42no—
232D12—42no—
233D12—82no—
234C3×C8—21no—
235C3×C8—21no—
236C3×C8—21no—
237C3×C8H(5,𝔽2)12yes—
238C24—121no—
239C24#21311yescharacteristic
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×D4—62no—
245C2×D4—122no—
246C2×D4—122no—
247C2×D4—122no—
248C2×D4—122no—
249C2×D4—121no—
250C2×D4—244no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—121no—
256C22×C4#21311yescharacteristic
257C22×C4#21311yescharacteristic
258C22×C4#21311yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—24no—
263Q16—42no—
264D8—64no—
265D8—122no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8C2×#6711yescharacteristic
273C2×C8C2×#6711yescharacteristic
274C2×C8C2×#6711yescharacteristic
275C2×C8C23×D311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—62no—
281C4⋊C4—121no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—122no—
285C22⋊C4—122no—
286C16—64no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—44no—
294C22×C3C2×SD3212yes—
295C22×C3C2×SD3212yes—
296C22×C3C2×SD3212yes—
297C22×C3C22×D811yescharacteristic
298C2×D3—82no—
299C2×D3—82no—
300C2×D3—82no—
301C2×D3—164no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—82no—
306C3×C4—82no—
307C3×C4#53411yescharacteristic
308C3×C4#57511yescharacteristic
309Dic3—82no—
310C23—41no—
311C23—41no—
312C23—122no—
313C23—242no—
314C23—242no—
315C23—242no—
316C23#82814yescomplemented
317C23C2×#21311yescharacteristic
318C23C2×#21311yescharacteristic
319C23C2×#21311yescharacteristic
320Q8—42no—
321Q8—42no—
322Q8—82no—
323D4—122no—
324D4—122no—
325D4—242no—
326D4—244no—
327D4—244no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—122no—
342C2×C4—242no—
343C2×C4C2×#21311yescharacteristic
344C2×C4C2×#21311yescharacteristic
345C2×C4C2×#21311yescharacteristic
346C2×C4C22×#6711yescharacteristic
347C8—21no—
348C8—21no—
349C8—21no—
350C8#100412yes—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—22no—
354C2×C3#318611yescharacteristic
355C2×C3C22×SD3211yescharacteristic
356C2×C3#341411yescharacteristic
357D3—162no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—242no—
366C22—242no—
367C22—242no—
368C22—484no—
369C22C2×#82812yes—
370C22C2×#82812yes—
371C22C2×#82812yes—
372C22C22×#21311yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—82no—
377C4—82no—
378C4—242no—
379C4#580411yescharacteristic
380C4#582211yescharacteristic
381C3#3375311yescharacteristic, complemented, Sylow
382C2—22no—
383C2—22no—
384C2—22no—
385C2—482no—
386C2#8262511yescharacteristic
387C2C22×#82811yescharacteristic
388C2#8389411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#467C2
socleC22×C3C22×D8
prime radicalC2×#5115C2
hypercenterC2×#467D3
hypocenterC3#33753
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×C3×#467C2semidirect
3. #72063C2semidirect
4. C2×#4549C2semidirect
5. #71085C2semidirect
6. #71082C2semidirect
7. #71078C2semidirect
8. #70600C2semidirect
10. C22×C3×Q16C22semidirect
18. C3×#467C22semidirect
19. C2×C3×#107C22semidirect
21. C2×#828C22semidirect
28. C22×C3⋊C16C22semidirect
29. #4564C22semidirect
31. #4549C22semidirect
32. #4491C22semidirect
33. #4490C22semidirect
34. C2×#467D3semidirect
83. C22×Q16C2×D3semidirect
93. #467C2×D3semidirect
99. C2×#107C2×D3semidirect
199. C23×C3SD32semidirect
316. C23#828semidirect
381. C3#33753semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4672
C2×#51152
#720632
C2×#45492
#710852
#710822
#710782
#706002
#337533