Finite Group Databasebeta

← #80521

Subgroups — #80521

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#5681C211yesmaximal, characteristic, complemented
2D4×Dic6C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5165C211yesmaximal, characteristic, complemented
5Dic3×SD16C211yesmaximal, characteristic, complemented
6#5152C211yesmaximal, characteristic, complemented
7#5150C211yesmaximal, characteristic
8#5055C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4842C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4794C211yesmaximal, characteristic, complemented
13#4791C211yesmaximal, characteristic, complemented
14#4761C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3373—31nomaximal, Sylow
17C22×Dic6—21no—
18C2×C3×SD16C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×#124C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#947C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#940C2211yescharacteristic
25D4×Dic3C2211yescharacteristic
26D4×Dic3C2211yescharacteristic
27D4×Dic3C2211yescharacteristic
28C2×#213C2211yescharacteristic
29#924—21no—
30#888C2211yescharacteristic
31#888C2211yescharacteristic
32#886—21no—
33#885—21no—
34#883—21no—
35#67⋊C4—21no—
36#878—21no—
37#878—21no—
38C4×D12C2211yescharacteristic
39#869C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×C4⋊C8C2211yescharacteristic
42C3×#103C2211yescharacteristic
43C3×#102C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#817C2211yescharacteristic
52#817C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#803C2211yescharacteristic
60#559—31no—
61D4×Q8—31no—
62#489—31no—
63#489—31no—
64#486—31no—
65#476—31no—
66#476—31no—
67#475—31no—
68#463—31no—
69#463D311yescharacteristic, complemented
70#459—31no—
71#446—31no—
72#442—31no—
73#442—31no—
74C4×SD16—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C22×Dic3—21no—
82C22×Dic3—21no—
83C22×Dic3—21no—
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×Dic6—21no—
87C2×Dic6—21no—
88C2×Dic6—21no—
89C2×Dic6—44no—
90C2×Dic6D411yescharacteristic
91C3×SD16—22no—
92C3×D8—22no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97C3×C22⋊C4—21no—
98C3×C22⋊C4—21no—
99C3×C42C2311yescharacteristic
100#217—21no—
101#217—21no—
102#217—21no—
103#213—22no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3D411yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4D411yescharacteristic
116Dic3⋊C4D411yescharacteristic
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C22×Q8—61no—
123C2×SD16C2×D311yescharacteristic
124C2×D8—31no—
125C2×D8C2×D311yescharacteristic
126C4⋊Q8—31no—
127C4⋊Q8—31no—
128C4⋊Q8—31no—
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—61no—
145C4×D4C2×D311yescharacteristic
146#106—31no—
147#106—31no—
148#106—31no—
149C4⋊C8—31no—
150C4⋊C8—31no—
151C4⋊C8C2×D311yescharacteristic
152#103—31no—
153#103—31no—
154#103C2×D311yescharacteristic
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102C2×D311yescharacteristic
162#97—31no—
163#97—31no—
164C4×C8—31no—
165C23×C3—21no—
166C23×C3—21no—
167C22×D3—21no—
168C3×Q8—21no—
169C3×Q8—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4#4112yes—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—42no—
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2411yescharacteristic
180#67—42no—
181#67—81no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—81no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192D12—21no—
193D12—21no—
194C4×D3—81no—
195Dic6—22no—
196Dic6—42no—
197Dic6—42no—
198Dic6—42no—
199C3×C8—21no—
200C3×C8—21no—
201C3⋊C8—21no—
202C3⋊C8—21no—
203C2×Q8—31no—
204C2×Q8—61no—
205C2×Q8—61no—
206C2×Q8—61no—
207C2×Q8—124no—
208C2×Q8C22×D311yescharacteristic
209C2×D4—31no—
210C2×D4—61no—
211C2×D4—61no—
212C2×D4C22×D311yescharacteristic
213C2×D4C22×D311yescharacteristic
214C22×C4—21no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219SD16—22no—
220D8—22no—
221D8—62no—
222C2×C8—31no—
223C2×C8—31no—
224C2×C8C22×D311yescharacteristic
225C2×C8C22×D311yescharacteristic
226C4⋊C4—31no—
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⋊C4—61no—
236C4⋊C4C22×D311yescharacteristic
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—21no—
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—22no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C22×D411yescharacteristic
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4C2×#4111yescharacteristic
265C3×C42−1+411yescharacteristic
266Dic3—21no—
267Dic3—21no—
268Dic3—21no—
269Dic3—41no—
270Dic3—41no—
271Dic3—41no—
272Dic3AGL(1,ℤ/8)12yes—
273C23—21no—
274C23—21no—
275C23—61no—
276Q8—21no—
277Q8—21no—
278Q8—62no—
279Q8—122no—
280Q8—122no—
281Q8—122no—
282D4—21no—
283D4—21no—
284D4—21no—
285D4—61no—
286D4—61no—
287D4—122no—
288D4—241no—
289D4#23712yes—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—42no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
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—42no—
312C2×C3—81no—
313C2×C3#54811yescharacteristic
314C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
315C2×C3#57711yescharacteristic
316D3—81no—
317C22—22no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22C2×D3×D411yescharacteristic
327C4—21no—
328C4—21no—
329C4—32no—
330C4—41no—
331C4—41no—
332C4—61no—
333C4—61no—
334C4—61no—
335C4—121no—
336C4—121no—
337C4—121no—
338C4C2×#23711yescharacteristic
339C4#101011yescharacteristic
340C3#337311yescharacteristic, complemented, Sylow
341C2—42no—
342C2—81no—
343C2—241no—
344C2#559811yescharacteristic
345C2#577011yescharacteristic
346C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
prime radical#5150C2
hypercenter#463D3
hypocenterC3#3373
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #5681C2semidirect
2. D4×Dic6C2semidirect
3. C3×#463C2semidirect
4. #5165C2semidirect
5. Dic3×SD16C2semidirect
6. #5152C2semidirect
8. #5055C2semidirect
9. #5014C2semidirect
10. #4842C2semidirect
11. #4828C2semidirect
12. #4794C2semidirect
13. #4791C2semidirect
14. #4761C2semidirect
15. #4760C2semidirect
69. #463D3semidirect
340. C3#3373semidirect

Maximal subgroups

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

subgroupindex
#56812
D4×Dic62
C3×#4632
#51652
Dic3×SD162
#51522
#51502
#50552
#50142
#48422
#48282
#47942
#47912
#47612
#47602
#33733