Finite Group Databasebeta

← #79851

Subgroups — #79851

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5553C211yesmaximal, characteristic, complemented
2#5539C211yesmaximal, characteristic, complemented
3C3×#442C211yesmaximal, characteristic, complemented
4#5150C211yesmaximal, characteristic, complemented
5#5150C211yesmaximal, characteristic, complemented
6#5015C211yesmaximal, characteristic, complemented
7#5009C211yesmaximal, characteristic, complemented
8#4871C211yesmaximal, characteristic, complemented
9#4870C211yesmaximal, characteristic, complemented
10#4785C211yesmaximal, characteristic, complemented
11#4764C211yesmaximal, characteristic, complemented
12#4760C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4713C211yesmaximal, characteristic, complemented
15#4701C211yesmaximal, characteristic, complemented
16#3366—31nomaximal, Sylow
17C2×#235—21no—
18C2×C3×D8C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#940C2211yescharacteristic
22#940C2211yescharacteristic
23D4×Dic3C2211yescharacteristic
24D4×Dic3C2211yescharacteristic
25C2×#213C2211yescharacteristic
26C2×#213C2211yescharacteristic
27#930—21no—
28#924—21no—
29C2×Dic3⋊C4—21no—
30#893C2211yescharacteristic
31#893C2211yescharacteristic
32#890C2211yescharacteristic
33#888C2211yescharacteristic
34#884—21no—
35#884—21no—
36#878—21no—
37#878—21no—
38#874C2211yescharacteristic
39#870C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×#106C2211yescharacteristic
42C3×#102C2211yescharacteristic
43C3×#102C2211yescharacteristic
44C3×#97C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#815C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#554—31no—
61#550—31no—
62#490—31no—
63#490—31no—
64#486—31no—
65#478—31no—
66#478—31no—
67#476—31no—
68#459—31no—
69#459—31no—
70#459—31no—
71#446—31no—
72#442—31no—
73#442D311yescharacteristic, complemented
74C4×D8—31no—
75C2×C3×D4C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85#235—44no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12C2311yescharacteristic
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×Dic6—21no—
93C2×Dic6D411yescharacteristic
94C3×D8—22no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C42C2311yescharacteristic
102#217—21no—
103#217—21no—
104#213—22no—
105#213—22no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3D411yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×#41—61no—
123C2×D8—31no—
124C2×D8—31no—
125C2×D8C2×D311yescharacteristic
126C4⋊Q8—31no—
127#127—31no—
128#127—31no—
129#127—31no—
130#125—31no—
131#125—31no—
132C22⋊Q8—61no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135#121—61no—
136#121—61no—
137#121—61no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4C2×D311yescharacteristic
144C4×D4C2×D311yescharacteristic
145C2×C4⋊C4—61no—
146#107—31no—
147#106—31no—
148#106C2×D311yescharacteristic
149C4⋊C8—31no—
150C4⋊C8—31no—
151C4⋊C8—31no—
152#102—31no—
153#102—31no—
154#102—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102C2×D311yescharacteristic
161#102C2×D311yescharacteristic
162#97—31no—
163#97C2×D311yescharacteristic
164C4×C8—31no—
165C23×C3—21no—
166C23×C3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4#4112yescomplemented
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—42no—
176C2×C3×C4—81no—
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2411yescharacteristic
180#67—42no—
181#67—42no—
182#67—81no—
183#67—81no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—42no—
187C2×Dic3—81no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192D12—21no—
193D12—21no—
194D12—21no—
195D12—21no—
196D12—42no—
197D12—81no—
198C4×D3—42no—
199C4×D3—42no—
200Dic6—22no—
201Dic6—81no—
202C3×C8—21no—
203C3×C8—21no—
204C3⋊C8—21no—
205C3⋊C8—21no—
206#41—124no—
207C2×Q8—31no—
208C2×Q8—61no—
209C2×D4—31no—
210C2×D4—31no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4C22×D311yescharacteristic
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225D8—22no—
226D8—62no—
227D8—62no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8C22×D311yescharacteristic
231C2×C8C22×D311yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—31no—
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
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C42—31no—
251C42—31no—
252C42C22×D311yescharacteristic
253C22×C3—22no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3C22×D411yescharacteristic
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—41no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4C2×#4111yescharacteristic
271C3×C42−1+411yescharacteristic
272Dic3—21no—
273Dic3—21no—
274Dic3—21no—
275Dic3—41no—
276Dic3—41no—
277Dic3AGL(1,ℤ/8)12yes—
278C23—21no—
279C23—21no—
280C23—61no—
281C23—61no—
282Q8—62no—
283Q8—241no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—61no—
288D4—61no—
289D4—61no—
290D4—61no—
291D4—122no—
292D4—122no—
293D4—122no—
294D4—241no—
295D4—241no—
296D4—241no—
297D4#23512yescomplemented
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—42no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—81no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—241no—
312C2×C4D3×D411yescharacteristic
313C2×C4D3×D411yescharacteristic
314C2×C4C23×D311yescharacteristic
315C8—21no—
316C8—21no—
317C8—61no—
318C8—61no—
319C2×C3—42no—
320C2×C3—81no—
321C2×C3#54811yescharacteristic
322C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
323C2×C3#57611yescharacteristic
324D3—81no—
325D3—81no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—121no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22C2×D3×D411yescharacteristic
339C4—21no—
340C4—21no—
341C4—32no—
342C4—41no—
343C4—41no—
344C4—61no—
345C4—61no—
346C4—61no—
347C4—121no—
348C4—121no—
349C4C2×#23511yescharacteristic
350C4#101011yescharacteristic
351C3#336611yescharacteristic, complemented, Sylow
352C2—42no—
353C2—81no—
354C2—241no—
355C2—241no—
356C2#555211yescharacteristic
357C2D3×AGL(1,ℤ/8)11yescharacteristic
358C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
hypercenter#442D3
hypocenterC3#3366
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #5553C2semidirect
2. #5539C2semidirect
3. C3×#442C2semidirect
4. #5150C2semidirect
5. #5150C2semidirect
6. #5015C2semidirect
7. #5009C2semidirect
8. #4871C2semidirect
9. #4870C2semidirect
10. #4785C2semidirect
11. #4764C2semidirect
12. #4760C2semidirect
13. #4756C2semidirect
14. #4713C2semidirect
15. #4701C2semidirect
73. #442D3semidirect
172. C3×D4#41semidirect
297. D4#235semidirect
351. C3#3366semidirect

Maximal subgroups

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

subgroupindex
#55532
#55392
C3×#4422
#51502
#50152
#50092
#48712
#48702
#47852
#47642
#47602
#47562
#47132
#47012
#33663