Finite Group Databasebeta

← #79738

Subgroups — #79738

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#5581C211yesmaximal, characteristic, complemented
2#5576C211yesmaximal, characteristic
3C3×C4×Q16C211yesmaximal, characteristic, complemented
4#5187C211yesmaximal, characteristic, complemented
5#5186C211yesmaximal, characteristic, complemented
6#5028C211yesmaximal, characteristic, complemented
7#5022C211yesmaximal, characteristic, complemented
8#4885C211yesmaximal, characteristic, complemented
9#4884C211yesmaximal, characteristic, complemented
10#4816C211yesmaximal, characteristic, complemented
11#4812C211yesmaximal, characteristic, complemented
12#4803C211yesmaximal, characteristic, complemented
13#4802C211yesmaximal, characteristic, complemented
14#4697C211yesmaximal, characteristic, complemented
15#4688C211yesmaximal, characteristic, complemented
16#3327—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×Q8C2211yescharacteristic
22#946C2211yescharacteristic
23#946C2211yescharacteristic
24C2×#216C2211yescharacteristic
25C2×#215C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×ΓL(1,𝔽25)C2211yescharacteristic
28#897—21no—
29#897C2211yescharacteristic
30#895—21no—
31#895C2211yescharacteristic
32#894—21no—
33#892C2211yescharacteristic
34D3×C4⋊C4C2211yescharacteristic
35#889—21no—
36#876—21no—
37#875—21no—
38#875C2211yescharacteristic
39C4×D12—21no—
40C4×D12C2211yescharacteristic
41#872—21no—
42#870C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×C4×C8C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#552—31no—
61#548—31no—
62#488—31no—
63#484—31no—
64#482—31no—
65#479—31no—
66#470—31no—
67#468—31no—
68#464—31no—
69#452—31no—
70#451—31no—
71C4×Q16D311yescharacteristic, complemented
72#436—31no—
73#419—31no—
74#417—31no—
75C2×C3×Q8C2311yescharacteristic
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×D3—21no—
88C2×C4×D3C2311yescharacteristic
89C2×C4×D3C2311yescharacteristic
90C2×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C3×Q16—22no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C42—21no—
100C3×C42—21no—
101C3×C42C2311yescharacteristic
102#216—41no—
103#215—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)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3—21no—
114C4⋊Dic3D411yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125ΓL(1,𝔽25)—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16C2×D311yescharacteristic
131C2×SD16—31no—
132C2×M16—31no—
133C2×M16—31no—
134#126—61no—
135#125—31no—
136#125—61no—
137#124—31no—
138#124—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144#121—31no—
145#121—61no—
146C4×Q8C2×D311yescharacteristic
147C4×Q8C2×D311yescharacteristic
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151#117—61no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107C2×D311yescharacteristic
155#106—31no—
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⋊C8—31no—
169C4×C8C2×D311yescharacteristic
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8#4112yes—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189D12—21no—
190D12—21no—
191D12—41no—
192D12—42no—
193D12—42no—
194C4×D3—21no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—42no—
199C4×D3—42no—
200C4×D3—42no—
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—31no—
211C2×Q8—61no—
212C2×Q8—62no—
213C2×Q8—121no—
214C2×Q8C22×D311yescharacteristic
215C2×Q8C22×D311yescharacteristic
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—61no—
219C2×D4—61no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225Q16—22no—
226Q16—121no—
227SD16—121no—
228M16—121no—
229M16—121no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C22⋊C4—31no—
248C22⋊C4—31no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C42—21no—
257C42—21no—
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—21no—
288Q8—21no—
289Q8—61no—
290Q8—61no—
291Q8—122no—
292Q8#23512yes—
293D4—61no—
294D4—61no—
295D4—121no—
296D4—122no—
297D4—122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—122no—
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×#23511yescharacteristic
349C4#100911yescharacteristic
350C3#332711yescharacteristic, complemented, Sylow
351C2—121no—
352C2—121no—
353C2—241no—
354C2#555011yescharacteristic
355C2#576611yescharacteristic
356C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×Q16C2
socleC22×C3C22×D4
prime radical#5576C2
hypercenterC4×Q16D3
hypocenterC3#3327
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #5581C2semidirect
3. C3×C4×Q16C2semidirect
4. #5187C2semidirect
5. #5186C2semidirect
6. #5028C2semidirect
7. #5022C2semidirect
8. #4885C2semidirect
9. #4884C2semidirect
10. #4816C2semidirect
11. #4812C2semidirect
12. #4803C2semidirect
13. #4802C2semidirect
14. #4697C2semidirect
15. #4688C2semidirect
71. C4×Q16D3semidirect
350. C3#3327semidirect

Maximal subgroups

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

subgroupindex
#55812
#55762
C3×C4×Q162
#51872
#51862
#50282
#50222
#48852
#48842
#48162
#48122
#48032
#48022
#46972
#46882
#33273