Finite Group Databasebeta

← #76946

Subgroups — #76946

Order 384.

676 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 151 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×#940C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3C2×#885C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4957C212yesmaximal, complemented
8#4957C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4672C211yesmaximal, characteristic, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2548—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2212yes—
16C22×D12C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23#940—24no—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35#887—24no—
36#885—28no—
37C3×#95C2211yescharacteristic
38C3⋊#95C2211yescharacteristic
39C2×#127—31no—
40C2×#124—32no—
41C2×C4×Q8—31no—
42#394—31no—
43#394—31no—
44#394—31no—
45#394D311yescharacteristic, complemented
46#390—32no—
47#390—32no—
48#390—32no—
49C4×C4⋊C4—31no—
50C24×C3C2312yes—
51C23×D3C2312yes—
52C2×C3×D4—22no—
53C2×C3×D4—42no—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C2×#67—24no—
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—24no—
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C2×D12—22no—
70C2×D12—42no—
71C2×Dic6—22no—
72C2×Dic6—22no—
73C2×Dic6D414yescomplemented
74C3×C4⋊C4—22no—
75C3×C22⋊C4—24no—
76C3×C22⋊C4—42no—
77#217—24no—
78#217—42no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—42no—
82(C2×D3)⋊C4—42no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85Dic3⋊C4—24no—
86C4×Dic3—24no—
87C4×Dic3—22no—
88C4×Dic3—22no—
89C4×Dic3D414yes—
90C22×Q8—31no—
91C22×D4—32no—
92C22×D4—32no—
93C22×D4—31no—
94C22×D4C2×D311yescharacteristic
95#127—64no—
96#124—68no—
97C4×Q8—64no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4C2×D311yescharacteristic
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D312yes—
109C2×C22⋊C4C2×D312yes—
110C2×C42—31no—
111C2×C42—31no—
112C2×C42—31no—
113C2×C42—32no—
114#95—31no—
115#95C2×D311yescharacteristic
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3C2411yescharacteristic
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C3×D4—44no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2×D412yes—
142#67—48no—
143#67—48no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3C2×D412yes—
152C2×Dic3C2×D412yes—
153C2×Dic3C2×D412yes—
154C2×Dic3C2×D412yes—
155C2×Dic3#4112yes—
156C2×Dic3#4114yes—
157D12—44no—
158Dic6—28no—
159C24—32no—
160C24C22×D312yes—
161C2×Q8—34no—
162C2×Q8—62no—
163C2×Q8—62no—
164C2×D4—22no—
165C2×D4—42no—
166C2×D4—64no—
167C2×D4—64no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—64no—
172C2×D4—62no—
173C2×D4—122no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—31no—
179C22×C4C22×D312yes—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—62no—
185C4⋊C4—64no—
186C4⋊C4—62no—
187C22⋊C4—24no—
188C22⋊C4—42no—
189C22⋊C4—64no—
190C22⋊C4—64no—
191C22⋊C4—64no—
192C22⋊C4—122no—
193C22⋊C4—122no—
194C22⋊C4—122no—
195C42—34no—
196C42—62no—
197C42—62no—
198C42—64no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3C22×D411yescharacteristic
207C22×C3C22×D411yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4112yes—
210C22×C3H(5,𝔽2)12yes—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—42no—
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—41no—
221C3×C4—42no—
222Dic3—22no—
223Dic3—41no—
224Dic3—42no—
225Dic3#12114yes—
226Dic3#12414yescomplemented
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23C23×D311yescharacteristic
242Q8—68no—
243D4—44no—
244D4—124no—
245D4—128no—
246D4—128no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—34no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4D3×D412yes—
269C2×C4D3×D412yes—
270C2×C3—82no—
271C2×C3C2×#12111yescharacteristic
272C2×C3C2×#12411yescharacteristic
273C2×C3#53511yescharacteristic
274C2×C3#55012yes—
275C2×C3#55912yes—
276D3—82no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22—122no—
290C22—122no—
291C22C2×D3×D411yescharacteristic
292C22C2×D3×D411yescharacteristic
293C22#100412yes—
294C22D3×#4111yescharacteristic
295C22D3×#4112yes—
296C4—22no—
297C4—22no—
298C4—34no—
299C4—34no—
300C4—41no—
301C4—42no—
302C4—62no—
303C4—121no—
304C4—122no—
305C3#254811yescharacteristic, complemented, Sylow
306C2—82no—
307C2—242no—
308C2#559211yescharacteristic
309C2#559912yes—
310C2D3×#12111yescharacteristic
311C2#566412yes—
312C2D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#940C2
hypercenter#394D3
hypocenterC3#2548
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
3. C2×#885C2semidirect
4. C3×#394C2semidirect
5. #5222C2semidirect
6. #4991C2semidirect
7. #4957C2semidirect
8. #4957C2semidirect
9. #4957C2semidirect
10. #4672C2semidirect
11. #4649C2semidirect
45. #394D3semidirect
73. C2×Dic6D4semidirect
226. Dic3#124semidirect
305. C3#2548semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
C2×#8872
C2×#8852
C3×#3942
#52222
#49912
#49572
#46722
#46492
#25483