Finite Group Databasebeta

← #68754

Subgroups — #68754

Order 384.

306 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 79 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
1D3×#117C211yesmaximal, characteristic
2#5126C211yesmaximal, characteristic
3C4×(C2×D3)⋊C4C211yesmaximal, characteristic
4C3×#339C211yesmaximal, characteristic, complemented
5#4555C211yesmaximal, characteristic, complemented
6#4532C211yesmaximal, characteristic, complemented
7#4523C211yesmaximal, characteristic, complemented
8#1826—31nomaximal, Sylow
9C22×C4×D3C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11C3×#117C2211yescharacteristic
12C2×C3×C42C2211yescharacteristic
13C2×(C2×D3)⋊C4—21no—
14#926C2211yescharacteristic
15C2×C4×Dic3C2211yescharacteristic
16C2×#208C2211yescharacteristic
17#892C411yescharacteristic, complemented
18#892C411yescharacteristic, complemented
19D3×C4⋊C4C411yescharacteristic, complemented
20D3×C4⋊C4C411yescharacteristic, complemented
21D3×C22⋊C4—21no—
22#877—21no—
23#872—21no—
24C42×D3—21no—
25C3⋊#95—21no—
26(C2×D3)⋊C8—21no—
27(C2×D3)⋊C8—21no—
28C2×#117—31no—
29#407—31no—
30C4×C22⋊C4—31no—
31#339—31no—
32#339—31no—
33#339—31no—
34#339D311yescharacteristic, complemented
35C23×D3D411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C22×Dic3—21no—
39C22×Dic3D411yescharacteristic
40C2×C4×D3—21no—
41C2×C4×D3—21no—
42C2×C4×D3—21no—
43C2×C4×D3—21no—
44C2×C4×D3—21no—
45C2×C4×D3—21no—
46C2×C4×D3C2×C411yescharacteristic
47C2×C4×D3C2×C411yescharacteristic
48C2×C4×D3D411yescharacteristic
49C2×C4×D3D411yescharacteristic
50C3×M16—21no—
51C3×M16—21no—
52C2×C3×C8—21no—
53C3×C4⋊C4C2×C411yescharacteristic
54C3×C4⋊C4C2×C411yescharacteristic
55C3×C22⋊C4—21no—
56C3×C42—21no—
57C3×C42—21no—
58C3×C42—21no—
59#217—21no—
60(C2×D3)⋊C4—24no—
61(C2×D3)⋊C4—21no—
62(C2×D3)⋊C4—21no—
63C4⋊Dic3C2×C411yescharacteristic
64C4⋊Dic3C2×C411yescharacteristic
65Dic3⋊C4—21no—
66Dic3⋊C4—21no—
67C4×Dic3—21no—
68C4×Dic3—21no—
69C4×Dic3—21no—
70C4×Dic3—21no—
71C4×Dic3—21no—
72#208—21no—
73#208—21no—
74C2×C3⋊C8—21no—
75C23×C4—31no—
76C2×M16—31no—
77C2×M16C2×D311yescharacteristic
78#117—31no—
79#117—31no—
80#117—31no—
81#117—61no—
82#117—61no—
83#117C2×D311yescharacteristic
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C22⋊C4—61no—
87C2×C22⋊C4—61no—
88C2×C42—31no—
89C2×C42—61no—
90C2×C42C2×D311yescharacteristic
91C22⋊C8—61no—
92C22⋊C8—61no—
93#95—61no—
94C23×C3C2×D411yescharacteristic
95C22×D3—21no—
96C22×D3—21no—
97C22×D3—22no—
98C22×D3—41no—
99C22×D3C22⋊C411yescharacteristic
100C22×D3C22⋊C411yescharacteristic
101C2×C3×C4—22no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4C22×C411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4#4111yescharacteristic
109C2×C3×C4#4111yescharacteristic
110C2×C3×C4#4111yescharacteristic
111C2×C3×C4#4111yescharacteristic
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—22no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—41no—
118C2×Dic3—41no—
119C2×Dic3C22⋊C411yescharacteristic
120C2×Dic3C22⋊C411yescharacteristic
121C4×D3—21no—
122C4×D3—21no—
123C4×D3—21no—
124C4×D3—21no—
125C4×D3—21no—
126C4×D3—21no—
127C4×D3—21no—
128C4×D3—21no—
129C4×D3—42no—
130C4×D3—42no—
131C3×C8—41no—
132C3⋊C8—41no—
133C24—31no—
134C22×C4—21no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—61no—
141C22×C4—61no—
142C22×C4—61no—
143C22×C4—61no—
144C22×C4—61no—
145C22×C4—61no—
146C22×C4—61no—
147C22×C4C22×D311yescharacteristic
148M16—21no—
149M16—21no—
150M16—61no—
151M16—61no—
152C2×C8—21no—
153C2×C8—61no—
154C4⋊C4—31no—
155C4⋊C4—31no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4C4×D311yescharacteristic
159C4⋊C4C4×D311yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—61no—
162C22⋊C4—61no—
163C22⋊C4—64no—
164C22⋊C4—61no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—61no—
169C42—61no—
170C42—61no—
171C42—61no—
172C42—61no—
173C22×C3—21no—
174C22×C3—21no—
175C22×C3C2×C22⋊C411yescharacteristic
176C22×C3#12311yescharacteristic
177C22×C3#12311yescharacteristic
178C2×D3—21no—
179C2×D3—21no—
180C2×D3—21no—
181C2×D3—21no—
182C2×D3—21no—
183C2×D3—42no—
184C2×D3—42no—
185C2×D3#10412yes—
186C3×C4—42no—
187C3×C4—41no—
188C3×C4—41no—
189C3×C4#11711yescharacteristic
190C3×C4#11711yescharacteristic
191C3×C4#12311yescharacteristic
192C3×C4#12311yescharacteristic
193Dic3—21no—
194Dic3—21no—
195Dic3—42no—
196Dic3—41no—
197Dic3—41no—
198C23—31no—
199C23—31no—
200C23—62no—
201C23—61no—
202C23—61no—
203C23—121no—
204C23D3×D411yescharacteristic
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—62no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—121no—
228C2×C4—121no—
229C2×C4C2×C4×D311yescharacteristic
230C2×C4#23511yescharacteristic
231C2×C4#23511yescharacteristic
232C2×C4D3×D411yescharacteristic
233C2×C4#23711yescharacteristic
234C2×C4#23911yescharacteristic
235C8—41no—
236C8—121no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3#38111yescharacteristic
240C2×C3C2×#10411yescharacteristic
241C2×C3#42111yescharacteristic
242D3—22no—
243D3—41no—
244C22—21no—
245C22—21no—
246C22—32no—
247C22—61no—
248C22—61no—
249C22—61no—
250C22—61no—
251C22—61no—
252C22—122no—
253C22—122no—
254C22D3×C22⋊C411yescharacteristic
255C22#88311yescharacteristic
256C22#89411yescharacteristic
257C4—42no—
258C4—41no—
259C4—41no—
260C4—61no—
261C4—61no—
262C4—122no—
263C4—121no—
264C4—121no—
265C4#87211yescharacteristic
266C4#88311yescharacteristic
267C4#89211yescharacteristic
268C4#89411yescharacteristic
269C3#182611yescharacteristic, complemented, Sylow
270C2—21no—
271C2—21no—
272C2—62no—
273C2—121no—
274C2#466411yescharacteristic
275C2D3×#10411yescharacteristic
276C2#482111yescharacteristic
subgroupG/H
centerC2×C4D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3C2×C22⋊C4
prime radicalC23×D3D4
cosocleC22×C3×C4C23
hypercenter#339D3
hypocenterC3#1826
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
4. C3×#339C2semidirect
5. #4555C2semidirect
6. #4532C2semidirect
7. #4523C2semidirect
17. #892C4semidirect
18. #892C4semidirect
19. D3×C4⋊C4C4semidirect
20. D3×C4⋊C4C4semidirect
34. #339D3semidirect
269. C3#1826semidirect

Maximal subgroups

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

subgroupindex
D3×#1172
#51262
C4×(C2×D3)⋊C42
C3×#3392
#45552
#45322
#45232
#18263