Finite Group Databasebeta

← #78972

Subgroups — #78972

Order 384.

374 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic
2#5533C211yesmaximal, characteristic
3C3×#419C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5137C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic
7#4996C211yesmaximal, characteristic, complemented
8#4816C211yesmaximal, characteristic, complemented
9#4815C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11#4802C211yesmaximal, characteristic, complemented
12#4795C211yesmaximal, characteristic, complemented
13#4792C211yesmaximal, characteristic, complemented
14#4791C211yesmaximal, characteristic, complemented
15#4790C211yesmaximal, characteristic, complemented
16#3328—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#946—21no—
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#944—21no—
25#944C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C4×#67—21no—
29C4×#67C2211yescharacteristic
30#924—21no—
31#924C2211yescharacteristic
32#924C2211yescharacteristic
33C2×Dic12C2211yescharacteristic
34C2×ΓL(1,𝔽25)C2211yescharacteristic
35#897C2211yescharacteristic
36#892C2211yescharacteristic
37#888C2211yescharacteristic
38#887C2211yescharacteristic
39#885—21no—
40#879—21no—
41#876—21no—
42C4×Dic6—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#830C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#817C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#556—31no—
61D4×Q8—31no—
62#487—31no—
63#485—31no—
64#485—31no—
65#475—31no—
66#470—31no—
67#470—31no—
68#462—31no—
69#451—31no—
70#451—31no—
71C4×Q16—31no—
72#436—31no—
73#419—31no—
74#419D311yescharacteristic, 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×C4C2311yescharacteristic
82C2×#67D411yescharacteristic
83D3×Q8—22no—
84D3×Q8—41no—
85C2×C4×D3—21no—
86C2×C4×D3C2311yescharacteristic
87C2×Dic6—21no—
88C2×Dic6—21no—
89C2×Dic6C2311yescharacteristic
90C2×Dic6C2311yescharacteristic
91C3×M16—41no—
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97C3×C42—21no—
98#217—21no—
99#217—21no—
100#217D411yescharacteristic
101#216—22no—
102#216—41no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4D411yescharacteristic
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4D411yescharacteristic
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125Dic12—41no—
126ΓL(1,𝔽25)—41no—
127C22×Q8—31no—
128C22×Q8C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16—31no—
131C2×Q16—31no—
132C2×M16—31no—
133C2×M16C2×D311yescharacteristic
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—61no—
137#126—61no—
138#126—61no—
139#124—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8—61no—
145C22⋊Q8—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#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—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
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3C2×D411yescharacteristic
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—42no—
176C3×Q8#4112yes—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—42no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—22no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C4×D3—21no—
195C4×D3—21no—
196C4×D3—42no—
197Dic6—21no—
198Dic6—21no—
199Dic6—21no—
200Dic6—21no—
201Dic6—42no—
202Dic6—42no—
203C3×C8—21no—
204C3×C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×Q8—31no—
210C2×Q8—31no—
211C2×Q8—41no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—121no—
216C2×Q8C22×D311yescharacteristic
217C2×Q8C22×D311yescharacteristic
218C2×D4—31no—
219C22×C4—21no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4C22×D311yescharacteristic
223Q16—62no—
224Q16—121no—
225Q16—121no—
226M16—41no—
227M16—121no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8C22×D311yescharacteristic
231C2×C8C22×D311yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—31no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4C22×D311yescharacteristic
245C4⋊C4C22×D311yescharacteristic
246C22⋊C4—21no—
247C22⋊C4—31no—
248C22⋊C4—31no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C42—21no—
255C42—31no—
256C42—61no—
257C42—61no—
258C42—61no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3C22×D411yescharacteristic
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—21no—
266C3×C4—22no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4C2×#4111yescharacteristic
272C3×C4H(5,𝔽2)11yescharacteristic
273Dic3—21no—
274Dic3—21no—
275Dic3—41no—
276Dic3—41no—
277Dic3—41no—
278Dic3—41no—
279Dic3#13712yes—
280C23—31no—
281C23D3×D411yescharacteristic
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—42no—
286Q8—61no—
287Q8—61no—
288Q8—61no—
289Q8—61no—
290Q8—122no—
291Q8—122no—
292Q8#23512yes—
293D4—62no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—42no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—122no—
311C2×C4D3×D411yescharacteristic
312C2×C4C23×D311yescharacteristic
313C8—21no—
314C8—21no—
315C8—61no—
316C8—61no—
317C2×C3—41no—
318C2×C3#54611yescharacteristic
319C2×C3C2×#13711yescharacteristic
320C2×C3#57811yescharacteristic
321D3—41no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—61no—
326C22—61no—
327C22—61no—
328C22C2×D3×D411yescharacteristic
329C4—21no—
330C4—22no—
331C4—32no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—61no—
336C4—61no—
337C4—121no—
338C4—121no—
339C4—121no—
340C4—121no—
341C4C2×#23511yescharacteristic
342C4#100411yescharacteristic
343C3#332811yescharacteristic, complemented, Sylow
344C2—41no—
345C2—121no—
346C2#549011yescharacteristic
347C2D3×#13711yescharacteristic
348C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#419C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#419D3
hypocenterC3#3328
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. C3×#419C2semidirect
4. #5228C2semidirect
5. #5137C2semidirect
7. #4996C2semidirect
8. #4816C2semidirect
9. #4815C2semidirect
10. #4809C2semidirect
11. #4802C2semidirect
12. #4795C2semidirect
13. #4792C2semidirect
14. #4791C2semidirect
15. #4790C2semidirect
74. #419D3semidirect
343. C3#3328semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
#55332
C3×#4192
#52282
#51372
#51282
#49962
#48162
#48152
#48092
#48022
#47952
#47922
#47912
#47902
#33283