Finite Group Databasebeta

← #68775

Subgroups — #68775

Order 384.

334 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 83 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×#909C211yesmaximal, characteristic
2#5523C211yesmaximal, characteristic
3#5004C211yesmaximal, characteristic, complemented
4#4929C211yesmaximal, characteristic, complemented
5C2×#805C211yesmaximal, characteristic
6C3×#339C211yesmaximal, characteristic, complemented
7#4554C211yesmaximal, characteristic, complemented
8#1882—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C2×C3×M16C2211yescharacteristic
12C3×#117C2211yescharacteristic
13C2×C3×C42C2211yescharacteristic
14C2×C4⋊Dic3C2211yescharacteristic
15C2×#208C2211yescharacteristic
16#909—22no—
17#909C414yescomplemented
18C2×Dic12—21no—
19C2×#204—21no—
20#892—21no—
21D3×C4⋊C4—21no—
22#67⋊C4—22no—
23C4×D12—21no—
24C4×Dic6—21no—
25C3⋊#95—22no—
26#810—21no—
27#810—21no—
28#805—22no—
29C2×#137—31no—
30#520—31no—
31C2×#104—31no—
32#419—31no—
33#391—31no—
34#343—31no—
35#339D311yescharacteristic, complemented
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C2×#67—21no—
39C22×Dic3—21no—
40C22×Dic3D412yes—
41#235—21no—
42#235—21no—
43#235C2×C412yes—
44#235D412yes—
45C2×D12D411yescharacteristic
46C2×C4×D3—21no—
47C2×C4×D3—22no—
48C2×Dic6—21no—
49C2×Dic6—21no—
50C2×Dic6—21no—
51C2×Dic6—21no—
52C2×Dic6—21no—
53C2×Dic6C2×C412yescomplemented
54C2×Dic6D411yescharacteristic
55C3×M16—21no—
56C3×M16C2×C412yescomplemented
57C2×C3×C8—21no—
58C3×C4⋊C4D411yescharacteristic
59C3×C4⋊C4D411yescharacteristic
60C3×C22⋊C4—21no—
61C3×C42—21no—
62C3×C42—21no—
63C3×C42—21no—
64(C2×D3)⋊C4—22no—
65C4⋊Dic3—21no—
66C4⋊Dic3—21no—
67Dic3⋊C4—22no—
68C4×Dic3—22no—
69#208—21no—
70#208—21no—
71C2×C3⋊C8—21no—
72Dic12—24no—
73#204—24no—
74C2×#41—31no—
75C22×Q8—31no—
76#137—34no—
77#137—62no—
78C2×Q16—61no—
79C2×SD16—61no—
80C2×M16—31no—
81C2×M16C2×D311yescharacteristic
82C4×Q8—61no—
83C4×D4—61no—
84C4×D4—62no—
85#117—61no—
86#117C2×D311yescharacteristic
87C2×C4⋊C4—31no—
88C2×C4⋊C4—61no—
89C2×C42C2×D311yescharacteristic
90#104—62no—
91#103—61no—
92#103—61no—
93#95—62no—
94C23×C3C2×D411yescharacteristic
95C22×D3—21no—
96C2×C3×C4—22no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4C22×C411yescharacteristic
102C2×C3×C4C2×D411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4#4111yescharacteristic
106C2×C3×C4#4111yescharacteristic
107#67—22no—
108#67—22no—
109C2×Dic3—21no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114C2×Dic3—42no—
115D12—21no—
116D12C22⋊C412yes—
117C4×D3—22no—
118C4×D3—22no—
119C4×D3—82no—
120Dic6—21no—
121Dic6—22no—
122Dic6—22no—
123Dic6—41no—
124Dic6—41no—
125Dic6C22⋊C412yes—
126C3×C8—22no—
127C3⋊C8—41no—
128#41—32no—
129#41—32no—
130#41—61no—
131#41—61no—
132C2×Q8—32no—
133C2×Q8—31no—
134C2×Q8—61no—
135C2×Q8—61no—
136C2×Q8—61no—
137C2×Q8—61no—
138C2×Q8—61no—
139C2×D4—31no—
140C2×D4—61no—
141C22×C4—21no—
142C22×C4—32no—
143C22×C4—62no—
144C22×C4—61no—
145C22×C4—61no—
146C22×C4C22×D311yescharacteristic
147Q16—64no—
148SD16—64no—
149M16—21no—
150M16—61no—
151M16—61no—
152M16C4×D312yescomplemented
153C2×C8—21no—
154C2×C8—61no—
155C4⋊C4—62no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4D1211yescharacteristic
159C4⋊C4D1211yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—62no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—62no—
166C22×C3—21no—
167C22×C3—21no—
168C22×C3C4×D411yescharacteristic
169C22×C3#12011yescharacteristic
170C22×C3#12111yescharacteristic
171C2×D3—22no—
172C2×D3—41no—
173C2×D3—41no—
174C3×C4—42no—
175C3×C4—41no—
176C3×C4—41no—
177C3×C4C2×C22⋊C411yescharacteristic
178C3×C4C4×D411yescharacteristic
179C3×C4#12111yescharacteristic
180C3×C4#12311yescharacteristic
181Dic3—22no—
182Dic3—42no—
183Dic3—42no—
184C23—61no—
185C23D3×D411yescharacteristic
186Q8—32no—
187Q8—61no—
188Q8—62no—
189Q8—62no—
190Q8—121no—
191Q8—121no—
192D4—32no—
193D4—62no—
194D4—62no—
195D4—61no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—61no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—121no—
207C2×C4—121no—
208C2×C4—122no—
209C2×C4—242no—
210C2×C4C2×C4×D311yescharacteristic
211C2×C4C2×D1211yescharacteristic
212C2×C4#23511yescharacteristic
213C2×C4D3×D411yescharacteristic
214C2×C4D3×D411yescharacteristic
215C2×C4#23911yescharacteristic
216C8—22no—
217C8—121no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3#38611yescharacteristic
221C2×C3#45511yescharacteristic
222C2×C3#45611yescharacteristic
223D3—42no—
224C22—21no—
225C22—21no—
226C22—62no—
227C22—121no—
228C22—121no—
229C22#88211yescharacteristic
230C22#88411yescharacteristic
231C22#89311yescharacteristic
232C4—42no—
233C4—41no—
234C4—41no—
235C4—62no—
236C4—122no—
237C4—122no—
238C4C4×D1211yescharacteristic
239C4D3×C22⋊C411yescharacteristic
240C4#89411yescharacteristic
241C4#89511yescharacteristic
242C3#188211yescharacteristic, complemented, Sylow
243C2—21no—
244C2—21no—
245C2—122no—
246C2#466511yescharacteristic
247C2#482511yescharacteristic
248C2#482611yescharacteristic
subgroupG/H
centerC22#882
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3#120
prime radicalC2×#235C22
cosocleC22×C3×C4C23
hypercenter#339D3
hypocenterC3#1882
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
3. #5004C2semidirect
4. #4929C2semidirect
6. C3×#339C2semidirect
7. #4554C2semidirect
17. #909C4semidirect
35. #339D3semidirect
53. C2×Dic6C2×C4semidirect
56. C3×M16C2×C4semidirect
152. M16C4×D3semidirect
242. C3#1882semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
#55232
#50042
#49292
C2×#8052
C3×#3392
#45542
#18823