Finite Group Databasebeta

← #81552

Subgroups — #81552

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#5633C211yesmaximal, characteristic
2#5533C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4#5051C211yesmaximal, characteristic, complemented
5#5002C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4875C211yesmaximal, characteristic, complemented
8#4866C211yesmaximal, characteristic, complemented
9#4853C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11#4802C211yesmaximal, characteristic, complemented
12#4798C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4730C211yesmaximal, characteristic, complemented
16#3313—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#947C2211yescharacteristic
22C2×#216C2211yescharacteristic
23C4×#67C2211yescharacteristic
24#924C2211yescharacteristic
25C22×C3⋊C8C2211yescharacteristic
26C2×Dic12C2211yescharacteristic
27C2×#204C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29#897C2211yescharacteristic
30#896—21no—
31#895C2211yescharacteristic
32#893—21no—
33#892C2211yescharacteristic
34#888C2211yescharacteristic
35#887C2211yescharacteristic
36#887C2211yescharacteristic
37#885—21no—
38#885—21no—
39#884—21no—
40#879—21no—
41#876—21no—
42C4×Dic6—21no—
43C3×#107C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47C3×C22⋊C8C2211yescharacteristic
48#835C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#816C2211yescharacteristic
52#815C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#809C2211yescharacteristic
59#807C2211yescharacteristic
60#556—31no—
61#548—31no—
62#489—31no—
63#485D311yescharacteristic, complemented
64#482—31no—
65#477—31no—
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×Q8C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C2×#67—21no—
78C2×#67D411yescharacteristic
79D3×Q8—22no—
80D3×Q8—41no—
81#235—22no—
82#235—41no—
83C2×D12—21no—
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3C2311yescharacteristic
88C2×Dic6—21no—
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99C3×C22⋊C4—21no—
100C3×C42—21no—
101#217—21no—
102#217D411yescharacteristic
103#216—41no—
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)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C2×C3⋊C8—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124Dic12—22no—
125#204—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#126—61no—
136#126—61no—
137#124—31no—
138#124—61no—
139#124—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8C2×D311yescharacteristic
144#121—31no—
145#121—61no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8—61no—
149C4×D4—31no—
150C4×D4—61no—
151#117—31no—
152#117C2×D311yescharacteristic
153#107—31no—
154#107C2×D311yescharacteristic
155#106C2×D311yescharacteristic
156C4⋊C8—31no—
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⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2411yescharacteristic
183#67—41no—
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—42no—
197C4×D3—42no—
198Dic6—21no—
199Dic6—21no—
200Dic6—21no—
201Dic6—42no—
202Dic6—42no—
203Dic6#4112yes—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—62no—
209#41—121no—
210C2×Q8—31no—
211C2×Q8—31no—
212C2×Q8—61no—
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—61no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
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—31no—
236C4⋊C4—31no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4C22×D311yescharacteristic
243C4⋊C4C22×D311yescharacteristic
244C4⋊C4C22×D311yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—31no—
248C22⋊C4—31no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—21no—
256C42—31no—
257C42—61no—
258C42—61no—
259C42—61no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3C22×D411yescharacteristic
264C2×D3—21no—
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4C2×#4111yescharacteristic
276C3×C4H(5,𝔽2)11yescharacteristic
277Dic3—22no—
278Dic3—21no—
279Dic3—21no—
280Dic3—21no—
281Dic3—41no—
282Dic3—41no—
283C23—31no—
284C23—61no—
285C23D3×D411yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—32no—
289Q8—61no—
290Q8—61no—
291Q8—61no—
292Q8—122no—
293Q8—122no—
294D4—61no—
295D4—61no—
296D4—122no—
297D4—121no—
298D4—122no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3#54611yescharacteristic
323C2×C3#57511yescharacteristic
324C2×C3#57811yescharacteristic
325D3—41no—
326D3—81no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—61no—
331C22—61no—
332C22—61no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—61no—
343C4—62no—
344C4—61no—
345C4—61no—
346C4—121no—
347C4—121no—
348C4D3×#4111yescharacteristic
349C4#100911yescharacteristic
350C3#331311yescharacteristic, complemented, Sylow
351C2—41no—
352C2—121no—
353C2—241no—
354C2#565411yescharacteristic
355C2#576811yescharacteristic
356C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
prime radical#5633C2
hypercenter#485D3
hypocenterC3#3313
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #5533C2semidirect
3. C3×#485C2semidirect
4. #5051C2semidirect
5. #5002C2semidirect
6. #4886C2semidirect
7. #4875C2semidirect
8. #4866C2semidirect
9. #4853C2semidirect
10. #4809C2semidirect
11. #4802C2semidirect
12. #4798C2semidirect
13. #4794C2semidirect
14. #4733C2semidirect
15. #4730C2semidirect
63. #485D3semidirect
350. C3#3313semidirect

Maximal subgroups

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

subgroupindex
#56332
#55332
C3×#4852
#50512
#50022
#48862
#48752
#48662
#48532
#48092
#48022
#47982
#47942
#47332
#47302
#33133