Finite Group Databasebeta

← #81674

Subgroups — #81674

Order 384.

384 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5727C211yesmaximal, characteristic, complemented
2#5681C211yesmaximal, characteristic, complemented
3C3×#489C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5057C211yesmaximal, characteristic, complemented
6#4819C211yesmaximal, characteristic, complemented
7#4794C211yesmaximal, characteristic, complemented
8#4791C211yesmaximal, characteristic, complemented
9#4788C211yesmaximal, characteristic, complemented
10#4786C211yesmaximal, characteristic
11#4785C211yesmaximal, characteristic, complemented
12#4761C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4711C211yesmaximal, characteristic, complemented
15#4701C211yesmaximal, characteristic, complemented
16#3373—31nomaximal, Sylow
17C2×D3×Q8—21no—
18C3×C4⋊Q8C2211yescharacteristic
19C3×#124C2211yescharacteristic
20#947C2211yescharacteristic
21#946—21no—
22Q8×Dic3C2211yescharacteristic
23C2×#215C2211yescharacteristic
24#940C2211yescharacteristic
25D4×Dic3C2211yescharacteristic
26C2×#213C2211yescharacteristic
27C2×D24C2211yescharacteristic
28#896—21no—
29#896—21no—
30#893C2211yescharacteristic
31#893C2211yescharacteristic
32#888C2211yescharacteristic
33#888C2211yescharacteristic
34#886—21no—
35#885—21no—
36#883—21no—
37#67⋊C4—21no—
38C4×D12C2211yescharacteristic
39C4×D12C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×#103C2211yescharacteristic
42C3×#103C2211yescharacteristic
43C3×#102C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#97C2211yescharacteristic
46#835C2211yescharacteristic
47#832C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#817C2211yescharacteristic
51#815C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#559—31no—
61D4×Q8—31no—
62#489—31no—
63#489D311yescharacteristic, complemented
64#486—31no—
65#476—31no—
66#476—31no—
67#475—31no—
68#463—31no—
69#463—31no—
70#459—31no—
71#446—31no—
72#442—31no—
73#442—31no—
74C4×SD16—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81D3×Q8—44no—
82C2×D12C2311yescharacteristic
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×Dic6—21no—
89C2×Dic6—21no—
90C2×Dic6D411yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97C3×C22⋊C4—21no—
98C3×C42C2311yescharacteristic
99#217—21no—
100#215—22no—
101#213—22no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
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⋊C4D411yescharacteristic
115Dic3⋊C4D411yescharacteristic
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121D24—22no—
122C22×Q8—61no—
123C2×SD16—31no—
124C2×D8—31no—
125C2×D8—31no—
126C4⋊Q8—31no—
127C4⋊Q8—31no—
128C4⋊Q8C2×D311yescharacteristic
129#127—31no—
130#124—31no—
131#124—61no—
132#124C2×D311yescharacteristic
133#123—61no—
134#123—61no—
135C22⋊Q8—61no—
136C22⋊Q8—61no—
137C22⋊Q8—61no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—31no—
144C4×D4—31no—
145C4×D4—61no—
146#106—31no—
147#106—31no—
148#106—31no—
149C4⋊C8—31no—
150C4⋊C8—31no—
151C4⋊C8—31no—
152#103—31no—
153#103C2×D311yescharacteristic
154#103C2×D311yescharacteristic
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—
166C22×D3—21no—
167C22×D3—21no—
168C3×Q8—21no—
169C3×Q8—21no—
170C3×Q8—42no—
171C3×D4—21no—
172C3×D4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2411yescharacteristic
179#67—42no—
180#67—81no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—81no—
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188D12—21no—
189D12—21no—
190D12—21no—
191D12#4112yes—
192C4×D3—42no—
193C4×D3—42no—
194C4×D3—42no—
195C4×D3—81no—
196Dic6—22no—
197Dic6—42no—
198Dic6—42no—
199C3×C8—21no—
200C3×C8—21no—
201C3⋊C8—21no—
202C3⋊C8—21no—
203C2×Q8—21no—
204C2×Q8—31no—
205C2×Q8—61no—
206C2×Q8—61no—
207C2×Q8—124no—
208C2×Q8C22×D311yescharacteristic
209C2×D4—31no—
210C2×D4—31no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4C22×D311yescharacteristic
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219SD16—62no—
220D8—62no—
221D8—62no—
222C2×C8—31no—
223C2×C8—31no—
224C2×C8C22×D311yescharacteristic
225C2×C8C22×D311yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4C22×D311yescharacteristic
236C4⋊C4C22×D311yescharacteristic
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C42—31no—
247C42—31no—
248C42—61no—
249C42C22×D311yescharacteristic
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3C22×D411yescharacteristic
254C2×D3—22no—
255C2×D3—41no—
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4C2×#4111yescharacteristic
266C3×C42−1+411yescharacteristic
267Dic3—21no—
268Dic3—21no—
269Dic3—21no—
270Dic3—41no—
271Dic3—41no—
272Dic3AGL(1,ℤ/8)12yes—
273C23—21no—
274C23—61no—
275C23—61no—
276Q8—21no—
277Q8—21no—
278Q8—42no—
279Q8—62no—
280Q8—122no—
281Q8—122no—
282D4—21no—
283D4—21no—
284D4—32no—
285D4—61no—
286D4—61no—
287D4—61no—
288D4—122no—
289D4—241no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4—241no—
303C2×C4—241no—
304C2×C4D3×D411yescharacteristic
305C2×C4D3×D411yescharacteristic
306C2×C4C23×D311yescharacteristic
307C8—21no—
308C8—21no—
309C8—61no—
310C8—61no—
311C2×C3—81no—
312C2×C3#54811yescharacteristic
313C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
314C2×C3#57711yescharacteristic
315D3—42no—
316D3—81no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—62no—
321C22—121no—
322C22—121no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22C2×D3×D411yescharacteristic
327C4—21no—
328C4—21no—
329C4—32no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—61no—
334C4—61no—
335C4—61no—
336C4—121no—
337C4—121no—
338C4D3×#4111yescharacteristic
339C4#101011yescharacteristic
340C3#337311yescharacteristic, complemented, Sylow
341C2—81no—
342C2—122no—
343C2—241no—
344C2#567711yescharacteristic
345C2D3×AGL(1,ℤ/8)11yescharacteristic
346C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#489C2
socleC22×C3C22×D4
prime radical#4786C2
hypercenter#489D3
hypocenterC3#3373
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #5727C2semidirect
2. #5681C2semidirect
3. C3×#489C2semidirect
4. #5084C2semidirect
5. #5057C2semidirect
6. #4819C2semidirect
7. #4794C2semidirect
8. #4791C2semidirect
9. #4788C2semidirect
11. #4785C2semidirect
12. #4761C2semidirect
13. #4760C2semidirect
14. #4711C2semidirect
15. #4701C2semidirect
63. #489D3semidirect
340. C3#3373semidirect

Maximal subgroups

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

subgroupindex
#57272
#56812
C3×#4892
#50842
#50572
#48192
#47942
#47912
#47882
#47862
#47852
#47612
#47602
#47112
#47012
#33733