Finite Group Databasebeta

← #68304

Subgroups — #68304

Order 384.

326 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 63 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#4664C211yesmaximal, characteristic, complemented
4C3×#327C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1510—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×ΓL(1,𝔽25)—21no—
18#895C411yescharacteristic, complemented
19D3×C4⋊C4C411yescharacteristic, complemented
20#884—21no—
21D3×C22⋊C4—21no—
22C3×C22⋊C8C2211yescharacteristic, complemented
23C3×#95C2211yescharacteristic, complemented
24C3⋊#95—21no—
25C3⋊#95C2211yescharacteristic, complemented
26#830C2211yescharacteristic, complemented
27(C2×D3)⋊C8—21no—
28C2×#121—31no—
29#407—31no—
30#381—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—21no—
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—42no—
64C4⋊Dic3C2×C411yescharacteristic
65Dic3⋊C4—21no—
66C2×C3⋊C8—21no—
67ΓL(1,𝔽25)—42no—
68C22×D4—31no—
69C22×D4—61no—
70C23×C4—31no—
71C2×M16—61no—
72#121—31no—
73#121—31no—
74#121—31no—
75#121—61no—
76#121—61no—
77#121C2×D311yescharacteristic
78C2×C4⋊C4—31no—
79C2×C22⋊C4—61no—
80C2×C22⋊C4—61no—
81C22⋊C8—31no—
82C22⋊C8—61no—
83C22⋊C8C2×D311yescharacteristic, complemented
84#95—31no—
85#95—61no—
86#95C2×D311yescharacteristic, complemented
87C23×C3—21no—
88C23×C3C2×D411yescharacteristic
89C22×D3—21no—
90C22×D3—21no—
91C22×D3—22no—
92C22×D3—21no—
93C22×D3—41no—
94C22×D3—42no—
95C22×D3—42no—
96C22×D3—42no—
97C22×D3C22⋊C411yescharacteristic
98C22×D3C22⋊C411yescharacteristic
99C3×D4—41no—
100C3×D4—41no—
101C3×D4—41no—
102C2×C3×C4—21no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4—41no—
107C2×C3×C4C22×C411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109#67—42no—
110#67—42no—
111#67—41no—
112#67—41no—
113C2×Dic3—21no—
114C2×Dic3—41no—
115C2×Dic3—41no—
116C2×Dic3—41no—
117C2×Dic3—41no—
118C2×Dic3C22⋊C411yescharacteristic
119C2×Dic3C22⋊C411yescharacteristic
120D12—41no—
121D12—42no—
122C4×D3—22no—
123C4×D3—22no—
124C4×D3—41no—
125C4×D3—41no—
126C4×D3—42no—
127C3×C8—41no—
128C3⋊C8—41no—
129C24—31no—
130C24—61no—
131C2×D4—21no—
132C2×D4—31no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—122no—
138C2×D4—122no—
139C2×D4—122no—
140C2×D4C4×D311yescharacteristic
141C22×C4—21no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—61no—
148C22×C4—61no—
149C22×C4—122no—
150C22×C4C22×D311yescharacteristic
151M16—122no—
152C2×C8—21no—
153C2×C8—61no—
154C4⋊C4—31no—
155C4⋊C4—61no—
156C4⋊C4C4×D311yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—61no—
159C22⋊C4—61no—
160C22⋊C4—61no—
161C22⋊C4—122no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3C2×C22⋊C411yescharacteristic
168C22×C3AGL(1,ℤ/8)11yescharacteristic
169C22×C3AGL(1,ℤ/8)11yescharacteristic
170C2×D3—21no—
171C2×D3—21no—
172C2×D3—21no—
173C2×D3—21no—
174C2×D3—21no—
175C2×D3—42no—
176C2×D3—42no—
177C2×D3—41no—
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—84no—
181C2×D3#9912yes—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—41no—
185C3×C4—81no—
186Dic3—21no—
187Dic3—21no—
188Dic3—41no—
189Dic3—81no—
190C23—21no—
191C23—31no—
192C23—31no—
193C23—62no—
194C23—61no—
195C23—61no—
196C23—61no—
197C23—121no—
198C23—122no—
199C23—122no—
200C23—122no—
201C23D3×D411yescharacteristic
202D4—41no—
203D4—41no—
204D4—41no—
205D4—121no—
206D4—122no—
207D4—122no—
208D4—122no—
209D4—121no—
210D4—121no—
211C2×C4—21no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4—122no—
224C2×C4—121no—
225C2×C4—121no—
226C2×C4—121no—
227C2×C4—121no—
228C2×C4C2×C4×D311yescharacteristic
229C2×C4D3×D411yescharacteristic
230C8—41no—
231C8—121no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—81no—
235C2×C3C2×#9911yescharacteristic
236C2×C3#41811yescharacteristic
237C2×C3#42111yescharacteristic
238D3—22no—
239D3—41no—
240D3—81no—
241C22—21no—
242C22—21no—
243C22—32no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—61no—
248C22—61no—
249C22—61no—
250C22—61no—
251C22—61no—
252C22—122no—
253C22—122no—
254C22—121no—
255C22—121no—
256C22—121no—
257C22—244no—
258C22D3×C22⋊C411yescharacteristic
259C22#91111yescharacteristic
260C22#91411yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—61no—
265C4—61no—
266C4—81no—
267C4—121no—
268C4—241no—
269C3#151011yescharacteristic, complemented, Sylow
270C2—21no—
271C2—21no—
272C2—62no—
273C2—81no—
274C2—121no—
275C2—241no—
276C2D3×#9911yescharacteristic
277C2#477011yescharacteristic
278C2#482111yescharacteristic
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#1510
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. #4664C2semidirect
4. C3×#327C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4451C2semidirect
10. C2×D3×D4C4semidirect
14. #938C4semidirect
18. #895C4semidirect
19. D3×C4⋊C4C4semidirect
22. C3×C22⋊C8C22semidirect
23. C3×#95C22semidirect
25. C3⋊#95C22semidirect
26. #830C22semidirect
34. #327D3semidirect
83. C22⋊C8C2×D3semidirect
86. #95C2×D3semidirect
269. C3#1510semidirect

Maximal subgroups

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

subgroupindex
D3×#1212
#47262
#46642
C3×#3272
#45372
#44702
#44512
#15103