Finite Group Databasebeta

← #68303

Subgroups — #68303

Order 384.

346 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 65 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×#121C211yesmaximal, characteristic
2#4726C211yesmaximal, characteristic, complemented
3#4666C211yesmaximal, characteristic, complemented
4C3×#327C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1507—31nomaximal, Sylow
9C2×D3×D4—21no—
10C2×D3×D4C411yescharacteristic, complemented
11C22×C4×D3C2211yescharacteristic
12C3×#121C2211yescharacteristic
13#939—21no—
14#938C411yescharacteristic, complemented
15#930C2211yescharacteristic
16C2×(C2×D3)⋊C4—21no—
17C2×C4×Dic3—21no—
18C2×ΓL(1,𝔽25)—21no—
19#895C411yescharacteristic, complemented
20D3×C4⋊C4C411yescharacteristic, complemented
21#884—21no—
22D3×C22⋊C4—21no—
23C3×C22⋊C8C2211yescharacteristic, complemented
24C3×#95C2211yescharacteristic, complemented
25C3⋊#95C2211yescharacteristic, complemented
26#830C2211yescharacteristic, complemented
27(C2×D3)⋊C8—21no—
28C2×#121—31no—
29#407—31no—
30C4×C22⋊C4—31no—
31#327—31no—
32#327—31no—
33#327—31no—
34#327D311yescharacteristic, complemented
35C23×D3—21no—
36C23×D3D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4C2×C411yescharacteristic
39C22×C3×C4—21no—
40C22×C3×C4C2311yescharacteristic
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—21no—
44C22×Dic3—21no—
45C22×Dic3D411yescharacteristic
46D3×D4—42no—
47D3×D4—42no—
48D3×D4—42no—
49C2×D12—21no—
50C2×D12C2×C411yescharacteristic
51C2×C4×D3—21no—
52C2×C4×D3—42no—
53C2×C4×D3C2×C411yescharacteristic
54C2×C4×D3C2×C411yescharacteristic
55C2×C4×D3D411yescharacteristic
56C2×C4×D3D411yescharacteristic
57C2×C3×C8—21no—
58C3×C4⋊C4C2×C411yescharacteristic
59C3×C22⋊C4—21no—
60#217—21no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64C4⋊Dic3C2×C411yescharacteristic
65Dic3⋊C4—21no—
66C4×Dic3—42no—
67C2×C3⋊C8—21no—
68ΓL(1,𝔽25)—42no—
69C22×D4—31no—
70C22×D4—61no—
71C23×C4—31no—
72C2×M16—61no—
73#121—31no—
74#121—31no—
75#121—31no—
76#121—61no—
77#121—61no—
78#121C2×D311yescharacteristic
79C2×C4⋊C4—31no—
80C2×C22⋊C4—61no—
81C2×C22⋊C4—61no—
82C2×C42—61no—
83C22⋊C8—31no—
84C22⋊C8—61no—
85C22⋊C8C2×D311yescharacteristic, complemented
86#95—31no—
87#95C2×D311yescharacteristic, complemented
88C23×C3—21no—
89C23×C3C2×D411yescharacteristic
90C22×D3—21no—
91C22×D3—21no—
92C22×D3—22no—
93C22×D3—21no—
94C22×D3—41no—
95C22×D3—42no—
96C22×D3—42no—
97C22×D3—42no—
98C22×D3C22⋊C411yescharacteristic
99C22×D3C22⋊C411yescharacteristic
100C3×D4—41no—
101C3×D4—41no—
102C3×D4—41no—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—41no—
106C2×C3×C4—41no—
107C2×C3×C4—41no—
108C2×C3×C4C22×C411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110#67—42no—
111#67—41no—
112#67—41no—
113#67—42no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—21no—
117C2×Dic3—41no—
118C2×Dic3—41no—
119C2×Dic3C22⋊C411yescharacteristic
120C2×Dic3C22⋊C411yescharacteristic
121D12—41no—
122D12—42no—
123C4×D3—22no—
124C4×D3—22no—
125C4×D3—41no—
126C4×D3—41no—
127C4×D3—42no—
128C3×C8—41no—
129C3⋊C8—41no—
130C24—31no—
131C24—61no—
132C2×D4—21no—
133C2×D4—31no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4—122no—
139C2×D4—122no—
140C2×D4—122no—
141C2×D4C4×D311yescharacteristic
142C22×C4—21no—
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×C4C22×D311yescharacteristic
152M16—122no—
153C2×C8—21no—
154C2×C8—61no—
155C4⋊C4—31no—
156C4⋊C4—61no—
157C4⋊C4C4×D311yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—61no—
160C22⋊C4—64no—
161C22⋊C4—61no—
162C22⋊C4—61no—
163C42—122no—
164C22×C3—21no—
165C22×C3—21no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3C2×C22⋊C411yescharacteristic
170C22×C3AGL(1,ℤ/8)11yescharacteristic
171C22×C3AGL(1,ℤ/8)11yescharacteristic
172C2×D3—21no—
173C2×D3—21no—
174C2×D3—21no—
175C2×D3—21no—
176C2×D3—21no—
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—41no—
180C2×D3—41no—
181C2×D3—41no—
182C2×D3—84no—
183C2×D3#10412yes—
184C3×C4—21no—
185C3×C4—21no—
186C3×C4—42no—
187C3×C4—41no—
188Dic3—21no—
189Dic3—42no—
190Dic3—41no—
191Dic3#9912yes—
192C23—21no—
193C23—31no—
194C23—31no—
195C23—62no—
196C23—61no—
197C23—61no—
198C23—61no—
199C23—121no—
200C23—122no—
201C23—122no—
202C23—122no—
203C23D3×D411yescharacteristic
204D4—41no—
205D4—41no—
206D4—41no—
207D4—121no—
208D4—122no—
209D4—122no—
210D4—122no—
211D4—121no—
212D4—121no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—61no—
225C2×C4—121no—
226C2×C4—121no—
227C2×C4—122no—
228C2×C4—121no—
229C2×C4—121no—
230C2×C4C2×C4×D311yescharacteristic
231C2×C4D3×D411yescharacteristic
232C8—41no—
233C8—121no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3—81no—
237C2×C3C2×#9911yescharacteristic
238C2×C3#41811yescharacteristic
239C2×C3C2×#10411yescharacteristic
240D3—22no—
241D3—41no—
242D3—81no—
243C22—21no—
244C22—21no—
245C22—32no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—61no—
250C22—61no—
251C22—61no—
252C22—61no—
253C22—61no—
254C22—122no—
255C22—122no—
256C22—121no—
257C22—121no—
258C22—121no—
259C22—244no—
260C22D3×C22⋊C411yescharacteristic
261C22#91111yescharacteristic
262C22#91411yescharacteristic
263C4—21no—
264C4—21no—
265C4—32no—
266C4—42no—
267C4—41no—
268C4—61no—
269C4—122no—
270C4—121no—
271C3#150711yescharacteristic, complemented, Sylow
272C2—21no—
273C2—21no—
274C2—62no—
275C2—81no—
276C2—121no—
277C2—241no—
278C2D3×#9911yescharacteristic
279C2#477011yescharacteristic
280C2D3×#10411yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3C2×C22⋊C4
prime radicalD3×#121C2
cosocleC22×C3×C4C23
hypercenter#327D3
hypocenterC3#1507
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #4726C2semidirect
3. #4666C2semidirect
4. C3×#327C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4451C2semidirect
10. C2×D3×D4C4semidirect
14. #938C4semidirect
19. #895C4semidirect
20. D3×C4⋊C4C4semidirect
23. C3×C22⋊C8C22semidirect
24. C3×#95C22semidirect
25. C3⋊#95C22semidirect
26. #830C22semidirect
34. #327D3semidirect
85. C22⋊C8C2×D3semidirect
87. #95C2×D3semidirect
271. C3#1507semidirect

Maximal subgroups

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

subgroupindex
D3×#1212
#47262
#46662
C3×#3272
#45372
#44702
#44512
#15073