Finite Group Databasebeta

← #79792

Subgroups — #79792

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5578C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic
3C3×#440C211yesmaximal, characteristic, complemented
4#5175C211yesmaximal, characteristic, complemented
5#5174C211yesmaximal, characteristic, complemented
6#5026C211yesmaximal, characteristic, complemented
7#5018C211yesmaximal, characteristic, complemented
8#4886C211yesmaximal, characteristic, complemented
9#4874C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4792C211yesmaximal, characteristic, complemented
12#4780C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4710C211yesmaximal, characteristic, complemented
15#4707C211yesmaximal, characteristic, complemented
16#3182—31nomaximal, Sylow
17C2×#235—21no—
18C2×C3×SD16C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#946C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#215C2211yescharacteristic
24#940C2211yescharacteristic
25#938C2211yescharacteristic
26C2×#214C2211yescharacteristic
27C4×#67—21no—
28C2×#204C2211yescharacteristic
29C2×ΓL(1,𝔽25)C2211yescharacteristic
30C2×ΓL(1,𝔽25)C2211yescharacteristic
31#897C2211yescharacteristic
32#896—21no—
33#895C2211yescharacteristic
34#889—21no—
35#888C2211yescharacteristic
36#888C2211yescharacteristic
37#885—21no—
38#884—21no—
39C4×D12C2211yescharacteristic
40#872—21no—
41C42×D3C2211yescharacteristic
42C4×Dic6—21no—
43C4×Dic6C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47C3×#97C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#821C2211yescharacteristic
51#818C2211yescharacteristic
52#816C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#533—31no—
61#532—31no—
62#500—31no—
63#492—31no—
64#487—31no—
65#486—31no—
66#468—31no—
67#468—31no—
68#468—31no—
69#468—31no—
70#440—31no—
71#440—31no—
72#440—31no—
73#440D311yescharacteristic, complemented
74C4×M16—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81#235—44no—
82C2×D12—21no—
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×C4×D3D411yescharacteristic
86C2×C4×D3D411yescharacteristic
87C2×C4×D3C2311yescharacteristic
88C2×Dic6—21no—
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C3×SD16—22no—
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C42—21no—
99C3×C42C2311yescharacteristic
100#217—21no—
101#215—22no—
102#214—22no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116C4×Dic3—21no—
117C4×Dic3D411yescharacteristic
118C4×Dic3D411yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122#204—22no—
123ΓL(1,𝔽25)—22no—
124ΓL(1,𝔽25)—22no—
125C2×#41—61no—
126C2×SD16—31no—
127C2×SD16—31no—
128C2×SD16—31no—
129C2×SD16C2×D311yescharacteristic
130C2×M16—31no—
131C2×M16—31no—
132C4⋊Q8—31no—
133C4⋊Q8—31no—
134C4⋊Q8—31no—
135#127—31no—
136#125—61no—
137#124—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141#121—31no—
142#121—31no—
143#121—61no—
144C4×Q8—31no—
145C4×Q8—61no—
146C4×Q8C2×D311yescharacteristic
147C4×D4—31no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150#117—61no—
151C2×C42—31no—
152#107—31no—
153#107—31no—
154#107—31no—
155#107C2×D311yescharacteristic
156#103—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic
160#102—31no—
161#102—31no—
162#102—31no—
163#102C2×D311yescharacteristic
164#97—31no—
165#97C2×D311yescharacteristic
166C4×C8—31no—
167C4×C8—31no—
168C23×C3—21no—
169C22×D3—21no—
170C22×D3C2×D411yescharacteristic
171C3×Q8—21no—
172C3×Q8—21no—
173C3×D4—21no—
174C3×D4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—42no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2411yescharacteristic
182#67—42no—
183#67—42no—
184#67—42no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191D12—21no—
192D12—21no—
193D12—42no—
194C4×D3—21no—
195C4×D3—21no—
196C4×D3—22no—
197C4×D3—22no—
198C4×D3—42no—
199Dic6—21no—
200Dic6—21no—
201Dic6—42no—
202Dic6—42no—
203C3×C8—21no—
204C3×C8#4112yescomplemented
205C3⋊C8—21no—
206C3⋊C8#4112yescomplemented
207#41—124no—
208C2×Q8—31no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8C22×D311yescharacteristic
212C2×D4—31no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4C22×D311yescharacteristic
218C22×C4—21no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—61no—
223SD16—22no—
224SD16—62no—
225SD16—62no—
226SD16—62no—
227M16—62no—
228M16—62no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8C22×D311yescharacteristic
232C2×C8C22×D311yescharacteristic
233C4⋊C4—21no—
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4C22×D311yescharacteristic
244C4⋊C4C22×D311yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C42—21no—
252C42—31no—
253C42—31no—
254C42—31no—
255C42—61no—
256C42C22×D311yescharacteristic
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3C22×D411yescharacteristic
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4C2×#4111yescharacteristic
273C3×C4C2×#4111yescharacteristic
274Dic3—21no—
275Dic3—41no—
276Dic3—41no—
277Dic3—41no—
278Dic3AGL(1,ℤ/8)12yes—
279Dic3#13712yes—
280C23—21no—
281C23—31no—
282C23—61no—
283Q8—21no—
284Q8—21no—
285Q8—61no—
286Q8—61no—
287Q8—122no—
288Q8—122no—
289D4—21no—
290D4—21no—
291D4—61no—
292D4—61no—
293D4—122no—
294D4—122no—
295D4—122no—
296D4—122no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—42no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—122no—
312C2×C4D3×D411yescharacteristic
313C2×C4D3×D411yescharacteristic
314C2×C4C23×D311yescharacteristic
315C8—21no—
316C8—32no—
317C8—61no—
318C8#23512yescomplemented
319C2×C3—81no—
320C2×C3#53211yescharacteristic
321C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
322C2×C3C2×#13711yescharacteristic
323D3—41no—
324D3—81no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—61no—
329C22—61no—
330C22—61no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22C2×D3×D411yescharacteristic
335C4—21no—
336C4—21no—
337C4—32no—
338C4—32no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—61no—
343C4—121no—
344C4—121no—
345C4—121no—
346C4C2×#23511yescharacteristic
347C4D3×#4111yescharacteristic
348C3#318211yescharacteristic, complemented, Sylow
349C2—81no—
350C2—121no—
351C2—241no—
352C2#554811yescharacteristic
353C2D3×AGL(1,ℤ/8)11yescharacteristic
354C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
prime radical#5548C2
hypercenter#440D3
hypocenterC3#3182
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #5578C2semidirect
3. C3×#440C2semidirect
4. #5175C2semidirect
5. #5174C2semidirect
6. #5026C2semidirect
7. #5018C2semidirect
8. #4886C2semidirect
9. #4874C2semidirect
10. #4812C2semidirect
11. #4792C2semidirect
12. #4780C2semidirect
13. #4760C2semidirect
14. #4710C2semidirect
15. #4707C2semidirect
73. #440D3semidirect
204. C3×C8#41semidirect
206. C3⋊C8#41semidirect
318. C8#235semidirect
348. C3#3182semidirect

Maximal subgroups

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

subgroupindex
#55782
#55482
C3×#4402
#51752
#51742
#50262
#50182
#48862
#48742
#48122
#47922
#47802
#47602
#47102
#47072
#31823