Finite Group Databasebeta

← #80419

Subgroups — #80419

Order 384.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5732C211yesmaximal, characteristic, complemented
2#5539C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5165C211yesmaximal, characteristic
5#5165C211yesmaximal, characteristic
6#5164C211yesmaximal, characteristic, complemented
7Dic3×SD16C211yesmaximal, characteristic, complemented
8#5082C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic
10#4847C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4800C211yesmaximal, characteristic, complemented
13#4796C211yesmaximal, characteristic, complemented
14#4794C211yesmaximal, characteristic, complemented
15#4791C211yesmaximal, characteristic, complemented
16#3376—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#947C2211yescharacteristic
22#947C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25D4×Dic3C2211yescharacteristic
26D4×Dic3C2211yescharacteristic
27C2×#213C2211yescharacteristic
28#924—21no—
29C2×Dic3⋊C4—21no—
30#898—21no—
31#898—21no—
32#897—21no—
33#892—21no—
34#890C2211yescharacteristic
35#888C2211yescharacteristic
36#878—21no—
37#878—21no—
38C4×D12C2211yescharacteristic
39#870C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×C4⋊C8C2211yescharacteristic
42C3×#103C2211yescharacteristic
43C3×#103C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#803C2211yescharacteristic
60#556—31no—
61#554—31no—
62#491—31no—
63#489—31no—
64#488—31no—
65#479—31no—
66#478—31no—
67#475—31no—
68#463—31no—
69#463—31no—
70#461D311yescharacteristic, complemented
71#446—31no—
72#442—31no—
73#440—31no—
74C4×SD16—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C22×Dic3—21no—
80C22×Dic3—21no—
81C2×D12C2311yescharacteristic
82C2×C4×D3—21no—
83C2×Dic6—21no—
84C2×Dic6D411yescharacteristic
85C3×SD16—22no—
86C3×SD16—22no—
87C2×C3×C8C2311yescharacteristic
88C2×C3×C8C2311yescharacteristic
89C3×C4⋊C4—21no—
90C3×C4⋊C4—21no—
91C3×C4⋊C4C2311yescharacteristic
92C3×C22⋊C4—21no—
93C3×C42C2311yescharacteristic
94#217—21no—
95#217—21no—
96#213—22no—
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3D411yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4D411yescharacteristic
114Dic3⋊C4D411yescharacteristic
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×SD16C2×D311yescharacteristic
122C2×SD16C2×D311yescharacteristic
123C2×D8—31no—
124C4⋊Q8—31no—
125C4⋊Q8C2×D311yescharacteristic
126#126—61no—
127#126—61no—
128#125—31no—
129#125—31no—
130#124—31no—
131#124—31no—
132C22⋊Q8—61no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136C4×Q8—31no—
137C4×Q8—31no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4C2×D311yescharacteristic
143#117—61no—
144C2×C4⋊C4—61no—
145#107—31no—
146#106—31no—
147#106—31no—
148C4⋊C8—31no—
149C4⋊C8—31no—
150C4⋊C8C2×D311yescharacteristic
151#103—31no—
152#103—31no—
153#103—31no—
154#103—31no—
155#103C2×D311yescharacteristic
156#103C2×D311yescharacteristic
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#97—31no—
162#97—31no—
163C4×C8—31no—
164C23×C3—21no—
165C22×D3—21no—
166C3×Q8—21no—
167C3×Q8—21no—
168C3×Q8—21no—
169C3×Q8—21no—
170C3×D4—21no—
171C3×D4#4112yes—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—81no—
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2411yescharacteristic
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—42no—
183C2×Dic3—81no—
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188D12—21no—
189D12—21no—
190C4×D3—42no—
191Dic6—41no—
192Dic6—81no—
193C3×C8—21no—
194C3×C8—21no—
195C3⋊C8—21no—
196C3⋊C8—21no—
197C2×Q8—31no—
198C2×Q8—61no—
199C2×Q8C22×D311yescharacteristic
200C2×Q8C22×D311yescharacteristic
201C2×D4—31no—
202C2×D4C22×D311yescharacteristic
203C22×C4—21no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207SD16—22no—
208SD16—22no—
209D8—62no—
210C2×C8—31no—
211C2×C8—31no—
212C2×C8C22×D311yescharacteristic
213C2×C8C22×D311yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4C22×D311yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C42—31no—
239C42—31no—
240C42—61no—
241C42—61no—
242C42C22×D311yescharacteristic
243C22×C3—22no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3C22×D411yescharacteristic
247C2×D3—41no—
248C2×D3—41no—
249C2×D3—41no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4C2×#4111yescharacteristic
256C3×C42−1+411yescharacteristic
257Dic3—21no—
258Dic3—21no—
259Dic3—21no—
260Dic3—21no—
261Dic3—41no—
262Dic3—41no—
263Dic3—41no—
264C23—21no—
265C23—61no—
266Q8—21no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—121no—
271Q8—241no—
272D4—21no—
273D4—61no—
274D4—61no—
275D4#23712yes—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—81no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—241no—
290C2×C4D3×D411yescharacteristic
291C2×C4D3×D411yescharacteristic
292C2×C4C23×D311yescharacteristic
293C8—21no—
294C8—21no—
295C8—61no—
296C8—61no—
297C2×C3—42no—
298C2×C3#54811yescharacteristic
299C2×C3#57511yescharacteristic
300C2×C3#57711yescharacteristic
301D3—81no—
302C22—22no—
303C22—41no—
304C22—41no—
305C22—121no—
306C22—121no—
307C22—121no—
308C22C2×D3×D411yescharacteristic
309C4—21no—
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—61no—
315C4—61no—
316C4—61no—
317C4—61no—
318C4—121no—
319C4—121no—
320C4—121no—
321C4C2×#23711yescharacteristic
322C4#101011yescharacteristic
323C3#337611yescharacteristic, complemented, Sylow
324C2—42no—
325C2—241no—
326C2#559811yescharacteristic
327C2#576211yescharacteristic
328C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×#213C22
hypercenter#461D3
hypocenterC3#3376
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
2. #5539C2semidirect
3. C3×#461C2semidirect
6. #5164C2semidirect
7. Dic3×SD16C2semidirect
8. #5082C2semidirect
10. #4847C2semidirect
11. #4828C2semidirect
12. #4800C2semidirect
13. #4796C2semidirect
14. #4794C2semidirect
15. #4791C2semidirect
70. #461D3semidirect
323. C3#3376semidirect

Maximal subgroups

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

subgroupindex
#57322
#55392
C3×#4612
#51652
#51642
Dic3×SD162
#50822
#50142
#48472
#48282
#48002
#47962
#47942
#47912
#33763