Finite Group Databasebeta

← #71795

Subgroups — #71795

Order 384.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2#5816C211yesmaximal, characteristic
3C2×#942C211yesmaximal, characteristic
4C2×#836C211yesmaximal, characteristic, complemented
5#5225C211yesmaximal, characteristic, complemented
6C2×#805C211yesmaximal, characteristic
7#4523C211yesmaximal, characteristic, complemented
8#2005—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×C3×Q8C2211yescharacteristic
11C2×#235C2211yescharacteristic
12C3×C4⋊Q8—22no—
13C3×C4⋊Q8—22no—
14C2×C3×C4⋊C4—22no—
15C2×C3×C42C2211yescharacteristic
16#947—21no—
17#946—22no—
18#944—21no—
19C2×#216—21no—
20#942—22no—
21#942—22no—
22C2×#215—21no—
23#929—22no—
24#926C2211yescharacteristic
25C2×#208C2211yescharacteristic
26C2×#208C2211yescharacteristic
27#836—22no—
28#835—22no—
29#805—22no—
30C2×#137—31no—
31#536—31no—
32C2×C4⋊Q8D311yescharacteristic, complemented
33C2×#104—31no—
34#419—31no—
35C2×#101—31no—
36#339—31no—
37C2×C3×Q8—21no—
38C2×C3×Q8—21no—
39C2×C3×Q8—21no—
40C2×C3×Q8—21no—
41C2×C3×Q8—21no—
42C2×C3×Q8—21no—
43C2×C3×Q8—21no—
44C2×C3×Q8—21no—
45C2×C3×Q8—21no—
46C2×C3×Q8—21no—
47C2×C3×Q8—21no—
48C2×C3×Q8D412yes—
49C22×C3×C4—21no—
50C22×C3×C4—21no—
51C22×C3×C4D412yes—
52C22×C3×C4C2311yescharacteristic
53C2×#67—21no—
54#235—21no—
55#235—21no—
56#235—21no—
57#235—21no—
58C2×D12D411yescharacteristic
59C2×C4×D3—21no—
60C2×Dic6D411yescharacteristic
61C3×C4⋊C4—28no—
62C3×C42—21no—
63C3×C42D412yes—
64#217—21no—
65#216—24no—
66#215—24no—
67(C2×D3)⋊C4—24no—
68C4⋊Dic3D412yes—
69Dic3⋊C4—24no—
70C4×Dic3—21no—
71#208—21no—
72#208—21no—
73#208—21no—
74#208D412yes—
75C2×C3⋊C8—21no—
76C2×C3⋊C8—21no—
77C2×#41—31no—
78C22×Q8C2×D311yescharacteristic
79C22×Q8C2×D311yescharacteristic
80#137—62no—
81#137—62no—
82C2×Q16—61no—
83C2×SD16—61no—
84C2×M16—31no—
85C2×M16—31no—
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88C4⋊Q8—61no—
89#124—61no—
90#123—62no—
91C22⋊Q8—62no—
92#117—31no—
93C2×C4⋊C4—22no—
94C2×C42C2×D311yescharacteristic
95#104—62no—
96#103—62no—
97#101—62no—
98C23×C3C2×D411yescharacteristic
99C22×D3—21no—
100C3×Q8—22no—
101C3×Q8—22no—
102C3×Q8—41no—
103C3×Q8—41no—
104C3×Q8—41no—
105C3×Q8—41no—
106C3×Q8—41no—
107C3×Q8—41no—
108C2×C3×C4—24no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—41no—
115C2×C3×C4—41no—
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2×D411yescharacteristic
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4#4111yescharacteristic
122#67—41no—
123#67—41no—
124C2×Dic3—22no—
125C2×Dic3—21no—
126D12—21no—
127D12—21no—
128C4×D3—41no—
129C4×D3—41no—
130Dic6—21no—
131Dic6—21no—
132C3⋊C8—22no—
133C3⋊C8—41no—
134#41—61no—
135#41—61no—
136#41—61no—
137#41—61no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—31no—
150C2×Q8#6712yes—
151C2×D4—31no—
152C2×D4—61no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—61no—
156C22×C4#6712yes—
157C22×C4C22×D311yescharacteristic
158Q16—64no—
159SD16—64no—
160M16—32no—
161M16—61no—
162M16—61no—
163M16—61no—
164C2×C8—61no—
165C2×C8—61no—
166C4⋊C4—28no—
167C4⋊C4—32no—
168C4⋊C4—64no—
169C22⋊C4—64no—
170C22⋊C4—61no—
171C42—21no—
172C42—61no—
173C42#6712yes—
174C22×C3—22no—
175C22×C3#12011yescharacteristic
176C22×C3#12011yescharacteristic
177C22×C3#12111yescharacteristic
178C2×D3—41no—
179C2×D3—42no—
180C3×C4—24no—
181C3×C4—42no—
182C3×C4—42no—
183C3×C4#12011yescharacteristic
184C3×C4#12111yescharacteristic
185C3×C4#12111yescharacteristic
186C3×C4#12711yescharacteristic
187Dic3—41no—
188Dic3—42no—
189C23—61no—
190C23C2×#6711yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—41no—
194Q8—41no—
195Q8—41no—
196Q8—41no—
197Q8—41no—
198Q8—41no—
199Q8—61no—
200Q8—61no—
201D4—61no—
202D4—61no—
203D4—121no—
204D4—121no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4—121no—
216C2×C4—121no—
217C2×C4D3×D411yescharacteristic
218C2×C4D3×D411yescharacteristic
219C2×C4D3×D411yescharacteristic
220C2×C4#23711yescharacteristic
221C2×C4C2×#6711yescharacteristic
222C2×C4C2×#6711yescharacteristic
223C8—62no—
224C8—121no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3#39211yescharacteristic
228C2×C3#45612yes—
229D3—81no—
230C22—22no—
231C22—121no—
232C22—122no—
233C22#93711yescharacteristic
234C22#93911yescharacteristic
235C22#95311yescharacteristic
236C4—24no—
237C4—42no—
238C4—42no—
239C4—121no—
240C4—122no—
241C4#93711yescharacteristic
242C4#93811yescharacteristic
243C4#93911yescharacteristic
244C4#94011yescharacteristic
245C3#200511yescharacteristic, complemented, Sylow
246C2—21no—
247C2—21no—
248C2—241no—
249C2#509512yes—
250C2#522311yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC2×C3×C4⋊Q8C2
socleC22×C3#120
prime radicalC2×#235C22
hypercenterC2×C4⋊Q8D3
hypocenterC3#2005
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊Q8C2semidirect
4. C2×#836C2semidirect
5. #5225C2semidirect
7. #4523C2semidirect
32. C2×C4⋊Q8D3semidirect
245. C3#2005semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
#58162
C2×#9422
C2×#8362
#52252
C2×#8052
#45232
#20053