Finite Group Databasebeta

← #79504

Subgroups — #79504

Order 384.

382 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#5551C211yesmaximal, characteristic
2#5541C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5137C211yesmaximal, characteristic, complemented
5#5133C211yesmaximal, characteristic, complemented
6#5116C211yesmaximal, characteristic, complemented
7#5113C211yesmaximal, characteristic, complemented
8#4849C211yesmaximal, characteristic, complemented
9#4847C211yesmaximal, characteristic, complemented
10#4733C211yesmaximal, characteristic, complemented
11#4733C211yesmaximal, characteristic, complemented
12#4723C211yesmaximal, characteristic, complemented
13#4722C211yesmaximal, characteristic, complemented
14#4710C211yesmaximal, characteristic, complemented
15#4703C211yesmaximal, characteristic, complemented
16#3315—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C22×C3×C8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#938—21no—
23#935—21no—
24#926C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26#924C2211yescharacteristic
27#924C2211yescharacteristic
28#924C2211yescharacteristic
29#924C2211yescharacteristic
30C2×Dic12C2211yescharacteristic
31C2×Dic12C2211yescharacteristic
32C2×#204C2211yescharacteristic
33D3×C4⋊C4—21no—
34#889—21no—
35#883—21no—
36#883—21no—
37#879—21no—
38#877—21no—
39C4×D12C2211yescharacteristic
40#870C2211yescharacteristic
41#869C2211yescharacteristic
42C4×Dic6C2211yescharacteristic
43C3×C4⋊C8C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46C3×#97C2211yescharacteristic
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#818C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#816C2211yescharacteristic
57#816C2211yescharacteristic
58#816C2211yescharacteristic
59#816C2211yescharacteristic
60#552—31no—
61#547—31no—
62#491—31no—
63#485—31no—
64#484—31no—
65#478—31no—
66#470—31no—
67#467—31no—
68#462—31no—
69#452—31no—
70#452—31no—
71#440—31no—
72#435D311yescharacteristic, complemented
73#417—31no—
74#416—31no—
75C2×C3×D4D411yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81#235—22no—
82#235—22no—
83#235—41no—
84#235—41no—
85C2×D12C2311yescharacteristic
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×Dic6C2311yescharacteristic
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C3×M16—41no—
93C2×C3×C8—41no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4D411yescharacteristic
99C3×C22⋊C4D411yescharacteristic
100C3×C22⋊C4D411yescharacteristic
101C3×C42C2311yescharacteristic
102#217—21no—
103#217—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107(C2×D3)⋊C4—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C4×Dic3—21no—
124Dic12—41no—
125Dic12—41no—
126#204—22no—
127C2×#41—31no—
128C2×#41—31no—
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C2×M16C2×D311yescharacteristic
133C22×C8C2×D311yescharacteristic
134C4⋊Q8—31no—
135#126—61no—
136#125—31no—
137#125—61no—
138#123—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8—31no—
145#121—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4C2×D311yescharacteristic
149#117—31no—
150#117—61no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8C2×D311yescharacteristic
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8C2×D311yescharacteristic
168C22⋊C8C2×D311yescharacteristic
169#97C2×D311yescharacteristic
170C23×C3C2×D411yescharacteristic
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C3×D4—41no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2411yescharacteristic
183#67—42no—
184#67—42no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—42no—
193D12—21no—
194D12#4112yes—
195C4×D3—42no—
196C4×D3—42no—
197C4×D3—42no—
198Dic6—21no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—21no—
204C3×C8—21no—
205C3×C8—21no—
206C3×C8—21no—
207C3×C8—21no—
208#41—62no—
209#41—62no—
210#41—121no—
211#41—121no—
212C2×Q8—31no—
213C2×Q8—31no—
214C2×Q8—31no—
215C2×D4—31no—
216C2×D4—61no—
217C2×D4—61no—
218C2×D4D1211yescharacteristic
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4C22×D311yescharacteristic
224C22×C4C22×D311yescharacteristic
225Q16—121no—
226Q16—121no—
227SD16—62no—
228M16—41no—
229C2×C8—41no—
230C2×C8C22×D311yescharacteristic
231C2×C8C22×D311yescharacteristic
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4D1211yescharacteristic
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4D1211yescharacteristic
256C22⋊C4D1211yescharacteristic
257C42—61no—
258C42—61no—
259C42C22×D311yescharacteristic
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3C22×D411yescharacteristic
267C2×D3—22no—
268C2×D3—41no—
269C2×D3—41no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4C2×#4111yescharacteristic
275C3×C4H(5,𝔽2)11yescharacteristic
276Dic3—41no—
277Dic3—41no—
278Dic3—41no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283C23—61no—
284C23C2×D1211yescharacteristic
285C23C2×D1211yescharacteristic
286Q8—61no—
287Q8—61no—
288Q8—61no—
289Q8—61no—
290Q8—61no—
291Q8—61no—
292D4—32no—
293D4—41no—
294D4—61no—
295D4—122no—
296D4—122no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4C2×D1211yescharacteristic
313C2×C4C2×D1211yescharacteristic
314C2×C4C2×D1211yescharacteristic
315C2×C4C2×D1211yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—21no—
320C8—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3#54611yescharacteristic
324C2×C3#57511yescharacteristic
325C2×C3#57811yescharacteristic
326D3—42no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—62no—
334C22—121no—
335C22—121no—
336C22C22×D1211yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—121no—
342C4—121no—
343C4—121no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4#100411yescharacteristic
349C4D3×#4111yescharacteristic
350C3#331511yescharacteristic, complemented, Sylow
351C2—41no—
352C2—41no—
353C2—122no—
354C2#555411yescharacteristic
355C2#574811yescharacteristic
356C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
prime radical#5551C2
hypercenter#435D3
hypocenterC3#3315
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #5541C2semidirect
3. C3×#435C2semidirect
4. #5137C2semidirect
5. #5133C2semidirect
6. #5116C2semidirect
7. #5113C2semidirect
8. #4849C2semidirect
9. #4847C2semidirect
10. #4733C2semidirect
11. #4733C2semidirect
12. #4723C2semidirect
13. #4722C2semidirect
14. #4710C2semidirect
15. #4703C2semidirect
72. #435D3semidirect
350. C3#3315semidirect

Maximal subgroups

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

subgroupindex
#55512
#55412
C3×#4352
#51372
#51332
#51162
#51132
#48492
#48472
#47332
#47232
#47222
#47102
#47032
#33153