Finite Group Databasebeta

← #78805

Subgroups — #78805

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic, complemented
2#5507C211yesmaximal, characteristic
3C2×C3×#103C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#5114C211yesmaximal, characteristic, complemented
6C8×#67C211yesmaximal, characteristic, complemented
7#4971C211yesmaximal, characteristic, complemented
8#4818C211yesmaximal, characteristic, complemented
9#4807C211yesmaximal, characteristic, complemented
10#4806C211yesmaximal, characteristic, complemented
11#4805C211yesmaximal, characteristic, complemented
12D3×#103C211yesmaximal, characteristic, complemented
13#4792C211yesmaximal, characteristic, complemented
14#4791C211yesmaximal, characteristic, complemented
15#4788C211yesmaximal, characteristic, complemented
16#3298—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#235—21no—
20C22×C3×C8C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22#946—21no—
23#946C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25#944—21no—
26#944C2211yescharacteristic
27C2×#215C2211yescharacteristic
28C2×#215C2211yescharacteristic
29#930C2211yescharacteristic
30#929—21no—
31C4×#67—21no—
32C4×#67C2211yescharacteristic
33#924—21no—
34#924C2211yescharacteristic
35C2×#204C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#896—21no—
38#895C2211yescharacteristic
39#894—21no—
40#893C2211yescharacteristic
41D3×C4⋊C4C2211yescharacteristic
42#888C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#830C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#817C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60D4×Q8—31no—
61#547—31no—
62#487—31no—
63#483—31no—
64#483—31no—
65#475—31no—
66#465—31no—
67#465—31no—
68#461—31no—
69#448—31no—
70#448—31no—
71C4×SD16—31no—
72C8×D4—31no—
73C2×#103—31no—
74C2×#103D311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67D411yescharacteristic
85D3×Q8—22no—
86D3×Q8—41no—
87#235—44no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×Dic6—21no—
95C2×Dic6—21no—
96C2×Dic6C2311yescharacteristic
97C2×C3×C8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103#217—21no—
104#217D411yescharacteristic
105#215—22no—
106#215—41no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204—41no—
127D3×C8—41no—
128C2×#41—61no—
129C22×Q8—31no—
130C22×Q8C2×D311yescharacteristic
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C22×C8—31no—
135C22×C8C2×D311yescharacteristic
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138C4⋊Q8—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146#121—31no—
147#121—31no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4C2×D311yescharacteristic
154#106—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#103C2×D311yescharacteristic
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167#102—31no—
168C22⋊C8—31no—
169C22⋊C8—31no—
170C4×C8—31no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—42no—
178C3×Q8#4112yescomplemented
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—42no—
190#67SD1614yescomplemented
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—21no—
198D12—21no—
199D12—42no—
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—42no—
203C4×D3—42no—
204C4×D3—42no—
205Dic6—21no—
206Dic6—21no—
207Dic6—42no—
208Dic6—42no—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—124no—
214C2×Q8—21no—
215C2×Q8—22no—
216C2×Q8—31no—
217C2×Q8—41no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—62no—
221C2×Q8—121no—
222C2×Q8C22×D311yescharacteristic
223C2×Q8C22×D311yescharacteristic
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4—61no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—31no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235SD16—62no—
236SD16—121no—
237SD16—121no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—41no—
241C2×C8—121no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4C22×D311yescharacteristic
256C4⋊C4C22×D311yescharacteristic
257C22⋊C4—31no—
258C22⋊C4—31no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—31no—
265C42—61no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3C2×SD1612yes—
269C22×C3C22×D411yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3C2×SD1612yes—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4C2×#4111yescharacteristic
282C3×C4H(5,𝔽2)11yescharacteristic
283Dic3—21no—
284Dic3—21no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3C2×SD1612yes—
289C23—31no—
290C23—61no—
291C23D3×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—61no—
297Q8—61no—
298Q8—122no—
299Q8—122no—
300Q8#23512yescomplemented
301D4—34no—
302D4—61no—
303D4—61no—
304D4—122no—
305D4—122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4D3×D411yescharacteristic
326C2×C4C23×D311yescharacteristic
327C8—21no—
328C8—21no—
329C8—61no—
330C8—61no—
331C2×C3—22no—
332C2×C3#54611yescharacteristic
333C2×C3C22×SD1611yescharacteristic
334C2×C3#57811yescharacteristic
335D3—22no—
336D3—81no—
337C22—21no—
338C22—21no—
339C22—32no—
340C22—61no—
341C22—61no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22D3×SD1612yes—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—22no—
349C4—32no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—61no—
354C4—61no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4C2×#23511yescharacteristic
359C4#100411yescharacteristic
360C3#329811yescharacteristic, complemented, Sylow
361C2—22no—
362C2—62no—
363C2—241no—
364C2#549011yescharacteristic
365C2C2×D3×SD1611yescharacteristic
366C2#576611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#103C2
socleC22×C3C22×D4
prime radical#5507C2
hypercenterC2×#103D3
hypocenterC3#3298
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. Q8×#67C2semidirect
3. C2×C3×#103C2semidirect
4. #5227C2semidirect
5. #5114C2semidirect
6. C8×#67C2semidirect
7. #4971C2semidirect
8. #4818C2semidirect
9. #4807C2semidirect
10. #4806C2semidirect
11. #4805C2semidirect
12. D3×#103C2semidirect
13. #4792C2semidirect
14. #4791C2semidirect
15. #4788C2semidirect
74. C2×#103D3semidirect
178. C3×Q8#41semidirect
190. #67SD16semidirect
300. Q8#235semidirect
360. C3#3298semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
#55072
C2×C3×#1032
#52272
#51142
C8×#672
#49712
#48182
#48072
#48062
#48052
D3×#1032
#47922
#47912
#47882
#32983