Finite Group Databasebeta

← #83455

Subgroups — #83455

Order 384.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#542C211yesmaximal, characteristic
2#5519C211yesmaximal, characteristic, complemented
3#5081C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5#5056C211yesmaximal, characteristic, complemented
6#5055C211yesmaximal, characteristic, complemented
7#5045C211yesmaximal, characteristic, complemented
8#5043C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic
10#5034C211yesmaximal, characteristic
11#5022C211yesmaximal, characteristic, complemented
12#5020C211yesmaximal, characteristic, complemented
13#5009C211yesmaximal, characteristic, complemented
14#5008C211yesmaximal, characteristic, complemented
15#5000C211yesmaximal, characteristic, complemented
16#3244—31nomaximal, Sylow
17C3×C4⋊Q8C2211yescharacteristic
18C3×#126—21no—
19C3×#124—21no—
20C3×#124C2211yescharacteristic
21C3×#123—21no—
22C3×C22⋊Q8—21no—
23C3×C22⋊Q8C2211yescharacteristic
24C3×#121C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4C2211yescharacteristic
27C3×#117C2211yescharacteristic
28#926C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30#890—21no—
31#888—21no—
32#878—21no—
33#878—21no—
34#870C2211yescharacteristic
35#870C2211yescharacteristic
36#869C2211yescharacteristic
37#869C2211yescharacteristic
38#835C2211yescharacteristic
39#835C2211yescharacteristic
40#835C2211yescharacteristic
41#835C2211yescharacteristic
42#832C2211yescharacteristic
43#832C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#830C2211yescharacteristic
47#830C2211yescharacteristic
48#808C2211yescharacteristic
49#808C2211yescharacteristic
50#808C2211yescharacteristic
51#808C2211yescharacteristic
52#808C2211yescharacteristic
53#808C2211yescharacteristic
54#807C2211yescharacteristic
55#807C2211yescharacteristic
56C4⋊(C3⋊C8)C2211yescharacteristic
57C4⋊(C3⋊C8)C2211yescharacteristic
58#803C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#544—31no—
61#542D311yescharacteristic
62#501—31no—
63#498—31no—
64#489—31no—
65#488—31no—
66#485—31no—
67#483—31no—
68#482—31no—
69#481—31no—
70#479—31no—
71#478—31no—
72#476—31no—
73#475—31no—
74#433—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4C2311yescharacteristic
81C22×Dic3—21no—
82C2×Dic6—21no—
83C2×Dic6—21no—
84C3×C4⋊C4—21no—
85C3×C4⋊C4—21no—
86C3×C4⋊C4—21no—
87C3×C4⋊C4D411yescharacteristic
88C3×C4⋊C4D411yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C22⋊C4—21no—
92C3×C22⋊C4—21no—
93C3×C22⋊C4—21no—
94C3×C22⋊C4—21no—
95C3×C22⋊C4—21no—
96C3×C22⋊C4D411yescharacteristic
97C3×C22⋊C4D411yescharacteristic
98C3×C42—21no—
99C3×C42C2311yescharacteristic
100C3×C42C2311yescharacteristic
101#217—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3C2311yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113C4×Dic3—21no—
114C2×C3⋊C8C2311yescharacteristic
115C2×C3⋊C8C2311yescharacteristic
116C2×C3⋊C8C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C4⋊Q8—31no—
119C4⋊Q8—31no—
120C4⋊Q8—61no—
121C4⋊Q8C2×D311yescharacteristic
122#126—21no—
123#125—31no—
124#125—31no—
125#125—61no—
126#124—21no—
127#124C2×D311yescharacteristic
128#123—21no—
129C22⋊Q8—21no—
130C22⋊Q8—61no—
131C22⋊Q8—61no—
132C22⋊Q8C2×D311yescharacteristic
133#121C2×D311yescharacteristic
134C4×Q8C2×D311yescharacteristic
135C4×D4C2×D311yescharacteristic
136#117—31no—
137#117C2×D311yescharacteristic
138C2×C4⋊C4—31no—
139#107—31no—
140#107—31no—
141#106—31no—
142#106—31no—
143#106—31no—
144#106—31no—
145#106—31no—
146#106—31no—
147C4⋊C8—31no—
148C4⋊C8—31no—
149#103—31no—
150#103—31no—
151#103—31no—
152#103—31no—
153#102—31no—
154#102—31no—
155#102—31no—
156#102—31no—
157C22⋊C8—31no—
158C22⋊C8—31no—
159#97—31no—
160C4×C8—31no—
161C23×C3—21no—
162C23×C3C2×D411yescharacteristic
163C3×Q8—21no—
164C3×Q8—21no—
165C3×Q8—81no—
166C3×D4—21no—
167C3×D4—21no—
168C3×D4—81no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—81no—
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—81no—
186Dic6—81no—
187Dic6—81no—
188C3⋊C8—21no—
189C3⋊C8—21no—
190C3⋊C8—21no—
191C3⋊C8—21no—
192C2×Q8—21no—
193C2×Q8—61no—
194C2×Q8—61no—
195C2×Q8C22×D311yescharacteristic
196C2×D4—21no—
197C2×D4C22×D311yescharacteristic
198C22×C4—21no—
199C22×C4—61no—
200C22×C4C22×D311yescharacteristic
201C2×C8—31no—
202C2×C8—31no—
203C2×C8—31no—
204C2×C8—31no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—31no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—31no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4#6711yescharacteristic
220C4⋊C4#6711yescharacteristic
221C4⋊C4C22×D311yescharacteristic
222C4⋊C4C22×D311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—21no—
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—61no—
229C22⋊C4#6711yescharacteristic
230C22⋊C4#6711yescharacteristic
231C42—21no—
232C42—61no—
233C42C22×D311yescharacteristic
234C42C22×D311yescharacteristic
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3C22×D411yescharacteristic
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—41no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C42−1+411yescharacteristic
251C3×C42−1+411yescharacteristic
252Dic3—41no—
253Dic3—41no—
254Dic3—41no—
255Dic3—41no—
256C23—21no—
257C23C2×#6711yescharacteristic
258Q8—21no—
259Q8—21no—
260Q8—81no—
261Q8—241no—
262Q8—241no—
263D4—21no—
264D4—21no—
265D4—81no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—81no—
276C2×C4—241no—
277C2×C4C2×#6711yescharacteristic
278C2×C4C2×#6711yescharacteristic
279C2×C4C2×#6711yescharacteristic
280C2×C4C2×#6711yescharacteristic
281C2×C4C2×#6711yescharacteristic
282C2×C4C23×D311yescharacteristic
283C8—61no—
284C8—61no—
285C8—61no—
286C8—61no—
287C2×C3—41no—
288C2×C3—81no—
289C2×C3#53611yescharacteristic
290C2×C3#57711yescharacteristic
291C2×C3#57711yescharacteristic
292C22—21no—
293C22—21no—
294C22—21no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22C22×#6711yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—121no—
308C4—121no—
309C4—121no—
310C4—121no—
311C4#101011yescharacteristic
312C4#101011yescharacteristic
313C3#324411yescharacteristic, complemented, Sylow
314C2—41no—
315C2—81no—
316C2#582511yescharacteristic
317C2#583611yescharacteristic
318C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#542C2
socleC22×C3C22×D4
prime radicalC3×#121C22
hypercenter#542D3
hypocenterC3#3244
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
2. #5519C2semidirect
3. #5081C2semidirect
4. #5080C2semidirect
5. #5056C2semidirect
6. #5055C2semidirect
7. #5045C2semidirect
8. #5043C2semidirect
11. #5022C2semidirect
12. #5020C2semidirect
13. #5009C2semidirect
14. #5008C2semidirect
15. #5000C2semidirect
313. C3#3244semidirect

Maximal subgroups

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

subgroupindex
C3×#5422
#55192
#50812
#50802
#50562
#50552
#50452
#50432
#50352
#50342
#50222
#50202
#50092
#50082
#50002
#32443