Finite Group Databasebeta

← #81122

Subgroups — #81122

Order 384.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5732C211yesmaximal, characteristic, complemented
2#5576C211yesmaximal, characteristic
3C3×#477C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5030C211yesmaximal, characteristic, complemented
6#4888C211yesmaximal, characteristic, complemented
7#4887C211yesmaximal, characteristic, complemented
8#4886C211yesmaximal, characteristic, complemented
9#4885C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4809C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4804C211yesmaximal, characteristic, complemented
15#4802C211yesmaximal, characteristic, complemented
16#3313—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21#947—21no—
22Q8×Dic3—21no—
23C2×#216C2211yescharacteristic
24C2×Dic12C2211yescharacteristic
25C2×#204C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic
28#898—21no—
29#898—21no—
30#897—21no—
31#897C2211yescharacteristic
32#897C2211yescharacteristic
33#897C2211yescharacteristic
34#895—21no—
35#895C2211yescharacteristic
36#892C2211yescharacteristic
37#892C2211yescharacteristic
38#875—21no—
39#875C2211yescharacteristic
40C4×D12—21no—
41C4×D12C2211yescharacteristic
42C4×Dic6C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×C4⋊C8C2211yescharacteristic
46C3×#103C2211yescharacteristic
47C3×#103C2211yescharacteristic
48#835C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#816C2211yescharacteristic
53#810C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#803C2211yescharacteristic
60#556—31no—
61#548—31no—
62#489—31no—
63#485—31no—
64#482—31no—
65#477D311yescharacteristic, complemented
66#468—31no—
67#467—31no—
68#464—31no—
69#452—31no—
70#451—31no—
71#441—31no—
72#435—31no—
73#419—31no—
74#416—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77#239—22no—
78#239—41no—
79D3×Q8—22no—
80D3×Q8—41no—
81C2×D12—21no—
82C2×D12—21no—
83C2×D12D411yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3C2311yescharacteristic
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C42—21no—
100C3×C42C2311yescharacteristic
101#216—22no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4D411yescharacteristic
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3D411yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123Dic12—41no—
124#204—41no—
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8C2×D311yescharacteristic
135#126—61no—
136#126—61no—
137#124—31no—
138#124—61no—
139#124—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144#121—31no—
145#121—61no—
146C4×Q8—31no—
147C4×Q8—61no—
148C4×Q8C2×D311yescharacteristic
149C4×D4—31no—
150C4×D4—61no—
151#117—31no—
152#117—31no—
153#107C2×D311yescharacteristic
154#107C2×D311yescharacteristic
155#106—31no—
156C4⋊C8C2×D311yescharacteristic
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—42no—
175C3×Q8#4112yes—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2411yescharacteristic
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188D12—21no—
189D12—21no—
190D12—41no—
191D12—42no—
192D12—42no—
193C4×D3—21no—
194C4×D3—21no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—42no—
198C4×D3—42no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—62no—
209#41—121no—
210C2×Q8—21no—
211C2×Q8—31no—
212C2×Q8—31no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—121no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—61no—
220C2×D4—61no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—61no—
224C22×C4—61no—
225Q16—62no—
226Q16—121no—
227SD16—121no—
228M16—121no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—121no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4C22×D311yescharacteristic
243C4⋊C4C22×D311yescharacteristic
244C4⋊C4C22×D311yescharacteristic
245C22⋊C4—31no—
246C22⋊C4—31no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—21no—
256C42—61no—
257C42—61no—
258C42—61no—
259C42C22×D311yescharacteristic
260C22×C3C22×D411yescharacteristic
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4C2×#4111yescharacteristic
277C3×C4H(5,𝔽2)11yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283C23—31no—
284C23—31no—
285C23—61no—
286Q8—21no—
287Q8—42no—
288Q8—61no—
289Q8—61no—
290Q8—61no—
291Q8—61no—
292Q8—122no—
293Q8#23912yes—
294D4—61no—
295D4—61no—
296D4—121no—
297D4—122no—
298D4—122no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4D3×D411yescharacteristic
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3#54611yescharacteristic
322C2×C3#57511yescharacteristic
323C2×C3#57811yescharacteristic
324D3—41no—
325D3—41no—
326D3—81no—
327C22—61no—
328C22—61no—
329C22—61no—
330C22—61no—
331C22—61no—
332C22—61no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—61no—
344C4—61no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4C2×#23911yescharacteristic
349C4#100911yescharacteristic
350C3#331311yescharacteristic, complemented, Sylow
351C2—121no—
352C2—121no—
353C2—241no—
354C2#563011yescharacteristic
355C2#576611yescharacteristic
356C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#477C2
socleC22×C3C22×D4
prime radical#5576C2
hypercenter#477D3
hypocenterC3#3313
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
3. C3×#477C2semidirect
4. #5089C2semidirect
5. #5030C2semidirect
6. #4888C2semidirect
7. #4887C2semidirect
8. #4886C2semidirect
9. #4885C2semidirect
10. #4845C2semidirect
11. #4836C2semidirect
12. #4809C2semidirect
13. #4808C2semidirect
14. #4804C2semidirect
15. #4802C2semidirect
65. #477D3semidirect
350. C3#3313semidirect

Maximal subgroups

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

subgroupindex
#57322
#55762
C3×#4772
#50892
#50302
#48882
#48872
#48862
#48852
#48452
#48362
#48092
#48082
#48042
#48022
#33133