Finite Group Databasebeta

← #77249

Subgroups — #77249

Order 384.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#239C211yesmaximal, characteristic
2C2×#894C212yesmaximal, complemented
3C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#395C211yesmaximal, characteristic, complemented
5#5230C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4989C212yesmaximal, complemented
8#4672C211yesmaximal, characteristic, complemented
9#4668C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11#4664C211yesmaximal, characteristic, complemented
12#2406—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#239—28no—
15C2×#239—24no—
16C22×D12C2211yescharacteristic
17C22×D12C2212yes—
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2212yes—
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C2×C3×C4⋊C4C2212yes—
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33#894—28no—
34D3×C4⋊C4—24no—
35C3×#95C2211yescharacteristic
36C3×#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×#123—32no—
42C22×C4⋊C4—31no—
43#395—31no—
44#395D311yescharacteristic, complemented
45#394—31no—
46#394—31no—
47#386—32no—
48#385—32no—
49#385—32no—
50#381—31no—
51C23×D3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×Q8—24no—
54C2×C3×Q8—22no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64#239—232no—
65C2×D12—24no—
66C2×D12—24no—
67C2×D12—24no—
68C2×D12—22no—
69C2×D12D418yes—
70C2×C4×D3—22no—
71C2×C4×D3—24no—
72C2×C4×D3—24no—
73C2×C4×D3—24no—
74C2×C4×D3—22no—
75C2×C4×D3—24no—
76C2×C4×D3—22no—
77C2×C4×D3—22no—
78C2×C4×D3—24no—
79C2×C4×D3D414yes—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—24no—
87C4⋊Dic3—22no—
88Dic3⋊C4—22no—
89Dic3⋊C4—24no—
90Dic3⋊C4—22no—
91C2×#41—64no—
92C2×#41—68no—
93C22×Q8C2×D311yescharacteristic
94C22×D4—32no—
95C22×D4—31no—
96C23×C4—31no—
97C23×C4—31no—
98C23×C4—31no—
99C23×C4—32no—
100#123—68no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—64no—
106C2×C4⋊C4C2×D311yescharacteristic
107C2×C4⋊C4C2×D312yes—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113#95—32no—
114#95—31no—
115#95—32no—
116#95C2×D311yescharacteristic
117#95C2×D312yes—
118C23×C3C2411yescharacteristic
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3C2×D414yes—
129C22×D3C2×D414yes—
130C22×D3C2×D412yes—
131C3×Q8—28no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4C2×D414yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4#4112yescomplemented
145C2×Dic3—21no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3C2×D412yes—
154C2×Dic3#4112yescomplemented
155D12—28no—
156D12—216no—
157C4×D3—28no—
158C4×D3—216no—
159C4×D3—28no—
160C4×D3—44no—
161C24—31no—
162C24—32no—
163#41—632no—
164C2×Q8—22no—
165C2×Q8—24no—
166C2×D4—38no—
167C2×D4—64no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—62no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—34no—
174C22×C4—32no—
175C22×C4—31no—
176C22×C4—64no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—64no—
180C22×C4—64no—
181C22×C4—62no—
182C22×C4—64no—
183C22×C4—64no—
184C22×C4—62no—
185C22×C4C22×D312yes—
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D312yes—
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4—62no—
193C4⋊C4—62no—
194C4⋊C4—62no—
195C4⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—64no—
200C22⋊C4—64no—
201C22×C3C22×D411yescharacteristic
202C22×C3C22×D412yes—
203C22×C3C2×#4111yescharacteristic
204C22×C3C2×#4111yescharacteristic
205C22×C32−1+412yes—
206C2×D3—22no—
207C2×D3—22no—
208C2×D3—24no—
209C2×D3—24no—
210C2×D3—24no—
211C2×D3—24no—
212C2×D3—24no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3#12014yes—
218C3×C4—24no—
219C3×C4—22no—
220C3×C4—22no—
221C3×C4—41no—
222C3×C4—42no—
223Dic3—22no—
224Dic3—22no—
225Dic3—41no—
226Dic3—42no—
227C23—34no—
228C23—32no—
229C23—34no—
230C23—61no—
231C23—62no—
232C23—61no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23C23×D311yescharacteristic
240Q8—28no—
241D4—616no—
242D4—68no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—68no—
256C2×C4—616no—
257C2×C4—68no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—124no—
267C2×C4D3×D412yes—
268C2×C4D3×D414yes—
269C2×C4#23912yescomplemented
270C2×C3C2×#12011yescharacteristic
271C2×C3#52511yescharacteristic
272C2×C3#53611yescharacteristic
273C2×C3#54712yes—
274C2×C3#54712yes—
275D3—24no—
276D3—44no—
277C22—34no—
278C22—62no—
279C22—64no—
280C22—62no—
281C22—64no—
282C22—64no—
283C22—64no—
284C22—64no—
285C22—122no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22C2×D3×D411yescharacteristic
290C22C2×D3×D412yes—
291C22C2×#23911yescharacteristic
292C22#100712yes—
293C22D3×#4111yescharacteristic
294C4—22no—
295C4—24no—
296C4—22no—
297C4—41no—
298C4—42no—
299C4—62no—
300C4—62no—
301C4—122no—
302C4—121no—
303C3#240611yescharacteristic, complemented, Sylow
304C2—64no—
305C2—124no—
306C2D3×#12011yescharacteristic
307C2#562711yescharacteristic
308C2#563112yes—
309C2#572411yescharacteristic
310C2#572612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×#239C2
hypercenter#395D3
hypocenterC3#2406
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#894C2semidirect
3. C2×D3×C4⋊C4C2semidirect
4. C3×#395C2semidirect
5. #5230C2semidirect
6. #4991C2semidirect
7. #4989C2semidirect
8. #4672C2semidirect
9. #4668C2semidirect
10. #4665C2semidirect
11. #4664C2semidirect
44. #395D3semidirect
144. C2×C3×C4#41semidirect
154. C2×Dic3#41semidirect
269. C2×C4#239semidirect
303. C3#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#2392
C2×#8942
C2×D3×C4⋊C42
C3×#3952
#52302
#49912
#49892
#46722
#46682
#46652
#46642
#24063