Finite Group Databasebeta

← #10952012

Subgroups — #10952012

Order 768.

654 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
1#79614C211yesmaximal, characteristic, complemented
2D8×D12C211yesmaximal, characteristic, complemented
3C3×C4×D16C211yesmaximal, characteristic, complemented
4#71467C212yesmaximal, complemented
5#70889C211yesmaximal, characteristic, complemented
6#70886C211yesmaximal, characteristic, complemented
7#69330C212yesmaximal, complemented
8#69195C212yesmaximal, complemented
9D3×#357C212yesmaximal, complemented
10#69058C211yesmaximal, characteristic, complemented
11C16×D12C211yesmaximal, characteristic, complemented
12#33861—31nomaximal, Sylow
13C2×D3×D8C2212yes—
14#5551—21no—
15D4×D12—21no—
16C2×C3×D16C2211yescharacteristic
17C3×C4×D8C2211yescharacteristic
18C3×C4×D8C2211yescharacteristic, complemented
19#5157C2212yescomplemented
20C2×#826C2212yescomplemented
21#5015—21no—
22#5007—21no—
23C2×D3×C16C2212yescomplemented
24#4883C2212yes—
25D3×#107C2212yescomplemented
26#4774—22no—
27#4773—22no—
28#4695C2211yescharacteristic
29C8×D12C2211yescharacteristic
30#4682C2211yescharacteristic, complemented
31C3×#366C2211yescharacteristic, complemented
32C3×#357C2211yescharacteristic
33C3×#357C2211yescharacteristic, complemented
34C3×C4×C16C2211yescharacteristic, complemented
35#4562C2212yes—
36#4509C2212yescomplemented
37#4509C2212yescomplemented
38(C2×D3)⋊C16C2212yescomplemented
39#4491C2212yescomplemented
40#4489C2212yescomplemented
41C4⋊(C3⋊C16)C2211yescharacteristic, complemented
42#3334—31no—
43D4×D8—31no—
44#2240—31no—
45#2232—32no—
46#2224—31no—
47#2215—32no—
48#2206—31no—
49#2184—32no—
50C4×D16D311yescharacteristic, complemented
51D4×C16—31no—
52C2×#357—32no—
53C2×D3×D4—22no—
54C2×#235—41no—
55C22×D12—41no—
56C2×C3×D8C2311yescharacteristic
57C2×C3×D8C2311yescharacteristic
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60#938—22no—
61C2×#213—22no—
62#930—41no—
63#924—41no—
64D3×D8—24no—
65D3×D8—42no—
66C2×D24—21no—
67C2×D24C2312yes—
68C2×D3×C8C2312yes—
69#895—22no—
70D3×C4⋊C4—22no—
71#883—42no—
72#882—42no—
73#874—21no—
74C4×D12D411yescharacteristic
75#869—21no—
76C3×D16—22no—
77C2×C3×C16C2311yescharacteristic
78C2×C3×C16C2311yescharacteristic
79C3×#107C2311yescharacteristic
80C3×#107C2311yescharacteristic
81C3×#102—21no—
82C3×#102—21no—
83C3×C4×C8C2311yescharacteristic
84#832—22no—
85#826—42no—
86(C2×D3)⋊C8D412yes—
87#818—21no—
88#818C2312yes—
89C2×C3⋊C16C2312yes—
90#809—22no—
91#807—22no—
92C4⋊(C3⋊C8)D411yescharacteristic
93D3×C16—42no—
94C22×D8—32no—
95#547—61no—
96D42—61no—
97C2×D16—32no—
98C2×D16C2×D311yescharacteristic
99C22×C16—32no—
100#500—31no—
101#493—31no—
102#480—62no—
103#474—61no—
104#466—32no—
105#466—32no—
106#459—61no—
107#447—62no—
108C4×D8C2×D311yescharacteristic
109C4×D8C2×D311yescharacteristic, complemented
110C8×D4—31no—
111C2×#107—32no—
112#366—32no—
113#366C2×D311yescharacteristic, complemented
114C4⋊C16—31no—
115#357—32no—
116#357—32no—
117#357—32no—
118#357—32no—
119#357C2×D311yescharacteristic
120#357C2×D311yescharacteristic, complemented
121C22⋊C16—32no—
122C4×C16C2×D311yescharacteristic, complemented
123C23×D3—42no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C2×#67—42no—
129C2×#67—42no—
130D3×D4—44no—
131D3×D4—82no—
132#235—84no—
133C2×D12—22no—
134C2×D12—42no—
135C2×D12—41no—
136C2×D12—41no—
137C2×D12—84no—
138C2×D12D811yescharacteristic
139C2×C4×D3—42no—
140C2×C4×D3C2×D412yes—
141C2×Dic6—41no—
142C3×D8—21no—
143C3×D8—21no—
144C3×D8—21no—
145C3×D8#4112yescomplemented
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2411yescharacteristic
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42C2×D411yescharacteristic
153#217—42no—
154#213—44no—
155(C2×D3)⋊C4—42no—
156(C2×D3)⋊C4D812yes—
157C4⋊Dic3—22no—
158C4⋊Dic3—41no—
159C4⋊Dic3D811yescharacteristic
160Dic3⋊C4—42no—
161C2×C3⋊C8C2×D412yes—
162D24—22no—
163D24—22no—
164D24—81no—
165D3×C8—22no—
166D3×C8—22no—
167C3×C16—21no—
168C3×C16—21no—
169C3⋊C16—22no—
170C2×#41—121no—
171C22×D4—62no—
172C22×D4—121no—
173C2×D8—32no—
174C2×D8—62no—
175C2×D8—64no—
176C2×D8—61no—
177C2×D8—122no—
178C2×D8C22×D311yescharacteristic
179C2×D8C22×D311yescharacteristic
180C22×C8—32no—
181C4⋊Q8—61no—
182#127—61no—
183#123—122no—
184C22⋊Q8—121no—
185#121—62no—
186#121—62no—
187#121—121no—
188#120—122no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—31no—
192C2×C4⋊C4—62no—
193D16—22no—
194D16—122no—
195C2×C16—32no—
196C2×C16—122no—
197C2×C16C22×D311yescharacteristic
198C2×C16C22×D311yescharacteristic
199#107—32no—
200#107—62no—
201#107—61no—
202#107C22×D311yescharacteristic
203#107C22×D311yescharacteristic
204C4⋊C8—31no—
205#102—21no—
206#102—21no—
207#102—62no—
208#102—62no—
209C22⋊C8—32no—
210C4×C8C22×D311yescharacteristic
211C23×C3—41no—
212C23×C3—41no—
213C22×D3—42no—
214C22×D3—84no—
215C22×D3—84no—
216C22×D3—162no—
217C22×D3C2×D812yes—
218C3×D4—22no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—82no—
225C2×C3×C4—82no—
226C2×C3×C4C2×D811yescharacteristic
227C2×C3×C4C2×D811yescharacteristic
228C2×C3×C4C22×D411yescharacteristic
229#67—84no—
230#67—84no—
231C2×Dic3—42no—
232C2×Dic3C2×D812yes—
233D12—42no—
234D12—42no—
235D12—84no—
236D12—82no—
237D12—161no—
238D12D1614yescomplemented
239C4×D3—22no—
240C4×D3—22no—
241C4×D3—84no—
242Dic6—82no—
243C3×C8—21no—
244C3×C8C2×#4111yescharacteristic
245C3×C8H(5,𝔽2)11yescharacteristic
246C3⋊C8—22no—
247C24—122no—
248#41—244no—
249C2×Q8—121no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—31no—
253C2×D4—62no—
254C2×D4—122no—
255C2×D4—122no—
256C2×D4—124no—
257C2×D4—122no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—242no—
261C2×D4—244no—
262C22×C4—32no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—122no—
266D8—21no—
267D8—21no—
268D8—21no—
269D8—62no—
270D8—62no—
271D8—124no—
272D8—241no—
273D8#23512yescomplemented
274C2×C8—32no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8D3×D411yescharacteristic
278C2×C8C23×D311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—31no—
282C4⋊C4—62no—
283C4⋊C4—122no—
284C4⋊C4—121no—
285C22⋊C4—32no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—122no—
289C22⋊C4—122no—
290C42D3×D411yescharacteristic
291C16—21no—
292C16—21no—
293C16—62no—
294C22×C3—42no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3C22×D811yescharacteristic
301C2×D3—22no—
302C2×D3—22no—
303C2×D3—82no—
304C2×D3—82no—
305C2×D3—82no—
306C2×D3—168no—
307C2×D3C2×D1614yes—
308C3×C4—21no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4C2×D1612yes—
312C3×C4#54611yescharacteristic
313C3×C4#57711yescharacteristic
314Dic3—22no—
315Dic3—82no—
316C23—32no—
317C23—41no—
318C23—41no—
319C23—122no—
320C23—244no—
321C23—244no—
322C23—482no—
323Q8—242no—
324D4—22no—
325D4—34no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—122no—
330D4—122no—
331D4—244no—
332D4—244no—
333D4—244no—
334D4—242no—
335D4—481no—
336C2×C4—32no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—122no—
344C2×C4—244no—
345C2×C4D3×D811yescharacteristic
346C2×C4D3×D811yescharacteristic
347C2×C4C2×D3×D411yescharacteristic
348C8—21no—
349C8—62no—
350C8C2×#23511yescharacteristic
351C8#100911yescharacteristic
352C2×C3—82no—
353C2×C3—161no—
354C2×C3#329411yescharacteristic
355C2×C3C22×D1611yescharacteristic
356C2×C3#341611yescharacteristic
357D3—24no—
358D3—162no—
359C22—34no—
360C22—42no—
361C22—62no—
362C22—62no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—242no—
369C22—242no—
370C22—242no—
371C22—488no—
372C22C2×D3×D811yescharacteristic
373C4—21no—
374C4—62no—
375C4—81no—
376C4—81no—
377C4—242no—
378C4D3×D1612yes—
379C4#555011yescharacteristic
380C4#577011yescharacteristic
381C3#3386111yescharacteristic, complemented, Sylow
382C2—64no—
383C2—82no—
384C2—161no—
385C2—482no—
386C2#7960811yescharacteristic
387C2C2×D3×D1611yescharacteristic
388C2#8203211yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×C4×D16C2
socleC22×C3C22×D8
hypercenterC4×D16D3
hypocenterC3#33861
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D16

Complements

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

NH ≅ G/Ntype
1. #79614C2semidirect
2. D8×D12C2semidirect
3. C3×C4×D16C2semidirect
4. #71467C2semidirect
5. #70889C2semidirect
6. #70886C2semidirect
7. #69330C2semidirect
8. #69195C2semidirect
9. D3×#357C2semidirect
10. #69058C2semidirect
11. C16×D12C2semidirect
18. C3×C4×D8C22semidirect
19. #5157C22semidirect
20. C2×#826C22semidirect
23. C2×D3×C16C22semidirect
25. D3×#107C22semidirect
30. #4682C22semidirect
31. C3×#366C22semidirect
33. C3×#357C22semidirect
34. C3×C4×C16C22semidirect
36. #4509C22semidirect
37. #4509C22semidirect
38. (C2×D3)⋊C16C22semidirect
39. #4491C22semidirect
40. #4489C22semidirect
41. C4⋊(C3⋊C16)C22semidirect
50. C4×D16D3semidirect
109. C4×D8C2×D3semidirect
113. #366C2×D3semidirect
120. #357C2×D3semidirect
122. C4×C16C2×D3semidirect
145. C3×D8#41semidirect
238. D12D16semidirect
273. D8#235semidirect
381. C3#33861semidirect

Maximal subgroups

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

subgroupindex
#796142
D8×D122
C3×C4×D162
#714672
#708892
#708862
#693302
#691952
D3×#3572
#690582
C16×D122
#338613