Finite Group Databasebeta

← #68955

Subgroups — #68955

Order 384.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#878C211yesmaximal, characteristic
2#4942C211yesmaximal, characteristic
3Dic3×C22⋊C4C211yesmaximal, characteristic
4C3×#342C211yesmaximal, characteristic
5#4527C211yesmaximal, characteristic
6#4527C211yesmaximal, characteristic
7#4526C211yesmaximal, characteristic
8#1996—31nomaximal, Sylow
9C23×Dic3C2211yescharacteristic
10C22×Dic6C411yescharacteristic, complemented
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×#217C2211yescharacteristic
15C2×#217C2211yescharacteristic
16C2×#217C2211yescharacteristic
17C2×C4⋊Dic3C411yescharacteristic, complemented
18C2×Dic3⋊C4—21no—
19C2×Dic3⋊C4C411yescharacteristic, complemented
20C2×Dic3⋊C4C411yescharacteristic, complemented
21C2×C4×Dic3—21no—
22#878—21no—
23#878—21no—
24#878—21no—
25#878—21no—
26C3⋊#95—21no—
27C3⋊#95—21no—
28C2×C22⋊Q8—31no—
29#380—31no—
30C4×C22⋊C4—31no—
31#342—31no—
32#342—31no—
33#342—31no—
34#342D311yescharacteristic
35C24×C3C2311yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4C2×C411yescharacteristic
39C22×C3×C4C2×C411yescharacteristic
40C22×Dic3—21no—
41C22×Dic3—21no—
42C22×Dic3—42no—
43C22×Dic3C2×C411yescharacteristic
44C22×Dic3C2×C411yescharacteristic
45C22×Dic3C2×C411yescharacteristic
46C22×Dic3C2×C411yescharacteristic
47C22×Dic3D411yescharacteristic
48C22×Dic3D411yescharacteristic
49C22×Dic3D411yescharacteristic
50C22×Dic3D411yescharacteristic
51C2×Dic6—21no—
52C2×Dic6—21no—
53C2×Dic6—42no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4—21no—
58C3×C22⋊C4—21no—
59C3×C22⋊C4Q812yescomplemented
60#217—21no—
61#217—21no—
62#217—21no—
63#217—21no—
64#217—21no—
65#217Q812yescomplemented
66C4⋊Dic3—21no—
67C4⋊Dic3—21no—
68Dic3⋊C4—21no—
69Dic3⋊C4—21no—
70Dic3⋊C4—21no—
71Dic3⋊C4—21no—
72Dic3⋊C4—42no—
73C4×Dic3—42no—
74C22×Q8—31no—
75C23×C4—31no—
76C22⋊Q8—61no—
77C22⋊Q8—61no—
78C22⋊Q8—61no—
79C22⋊Q8—61no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—61no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—31no—
87C2×C22⋊C4C2×D311yescharacteristic
88C2×C22⋊C4C2×D311yescharacteristic
89C2×C22⋊C4C2×D311yescharacteristic
90C2×C42—61no—
91#95—61no—
92#95—61no—
93C23×C3—41no—
94C23×C3—41no—
95C23×C3C22×C411yescharacteristic
96C23×C3C2×D411yescharacteristic
97C23×C3C2×D411yescharacteristic
98C23×C3C2×Q811yescharacteristic
99C23×C3C2×Q811yescharacteristic
100C23×C3#4111yescharacteristic
101C23×C3#4111yescharacteristic
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—41no—
106C2×C3×C4—41no—
107C2×C3×C4—41no—
108C2×C3×C4—41no—
109C2×C3×C4—41no—
110C2×C3×C4—41no—
111C2×C3×C4—41no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—22no—
118C2×Dic3—41no—
119C2×Dic3—41no—
120C2×Dic3—41no—
121C2×Dic3—41no—
122C2×Dic3—41no—
123C2×Dic3—41no—
124C2×Dic3—41no—
125C2×Dic3—41no—
126C2×Dic3—41no—
127C2×Dic3—41no—
128C2×Dic3C22⋊C411yescharacteristic
129C2×Dic3C22⋊C411yescharacteristic
130C2×Dic3C22⋊C411yescharacteristic
131C2×Dic3C22⋊C411yescharacteristic
132Dic6—42no—
133Dic6—42no—
134C24C22×D311yescharacteristic
135C2×Q8—61no—
136C2×Q8—61no—
137C2×Q8—122no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—61no—
149C22×C4—61no—
150C22×C4—122no—
151C22×C4C4×D311yescharacteristic
152C22×C4C4×D311yescharacteristic
153C4⋊C4—61no—
154C4⋊C4—61no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4—122no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—32no—
166C22⋊C4—61no—
167C22⋊C4—61no—
168C22⋊C4—61no—
169C22⋊C4—61no—
170C22⋊C4—61no—
171C22⋊C4Dic612yescomplemented
172C42—122no—
173C22×C3—21no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—21no—
177C22×C3—21no—
178C22×C3—21no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3C2×C22⋊C411yescharacteristic
184C22×C3C4×Q811yescharacteristic
185C22×C3C4×Q811yescharacteristic
186C22×C3C22⋊Q811yescharacteristic
187C22×C3C22⋊Q811yescharacteristic
188C22×C3#12411yescharacteristic
189C22×C3C4⋊Q811yescharacteristic
190C3×C4—42no—
191C3×C4—41no—
192C3×C4—41no—
193C3×C4—81no—
194Dic3—21no—
195Dic3—21no—
196Dic3—21no—
197Dic3—42no—
198Dic3—41no—
199Dic3—41no—
200Dic3—81no—
201Dic3#9912yes—
202C23—41no—
203C23—41no—
204C23C2×Dic611yescharacteristic
205C23C2×C4×D311yescharacteristic
206C23#23511yescharacteristic
207C23D3×D411yescharacteristic
208C23D3×D411yescharacteristic
209C23#23711yescharacteristic
210C23D3×Q811yescharacteristic
211Q8—122no—
212Q8—122no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—62no—
233C2×C4—121no—
234C2×C4—121no—
235C2×C4—121no—
236C2×C4—121no—
237C2×C4—121no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—121no—
241C2×C4—121no—
242C2×C4—121no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—21no—
249C2×C3#39111yescharacteristic
250C2×C3C2×#9911yescharacteristic
251C2×C3#41011yescharacteristic
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22C4×Dic611yescharacteristic
263C22#87811yescharacteristic
264C22D3×C22⋊C411yescharacteristic
265C22#88511yescharacteristic
266C22#88711yescharacteristic
267C22#88811yescharacteristic
268C22#89611yescharacteristic
269C4—32no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—61no—
274C4—61no—
275C4—61no—
276C4—81no—
277C4—122no—
278C4—121no—
279C4—121no—
280C4—241no—
281C3#199611yescharacteristic, complemented, Sylow
282C2—21no—
283C2—21no—
284C2—21no—
285C2—21no—
286C2—21no—
287C2—21no—
288C2#464611yescharacteristic
289C2#474011yescharacteristic
290C2D3×#9911yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#342C2
socleC22×C3C2×C22⋊C4
prime radical, cosocleC24×C3C23
hypercenter#342D3
hypocenterC3#1996
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
10. C22×Dic6C4semidirect
17. C2×C4⋊Dic3C4semidirect
19. C2×Dic3⋊C4C4semidirect
20. C2×Dic3⋊C4C4semidirect
59. C3×C22⋊C4Q8semidirect
65. #217Q8semidirect
171. C22⋊C4Dic6semidirect
281. C3#1996semidirect

Maximal subgroups

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

subgroupindex
C2×#8782
#49422
Dic3×C22⋊C42
C3×#3422
#45272
#45262
#19963