Finite Group Databasebeta

← #76970

Subgroups — #76970

Order 384.

816 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 153 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×#237C211yesmaximal, characteristic
2C2×#896C211yesmaximal, characteristic, complemented
3C2×#878C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#4979C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4959C212yesmaximal, complemented
9#4942C212yesmaximal, complemented
10#4667C211yesmaximal, characteristic, complemented
11#4656C211yesmaximal, characteristic, complemented
12#2616—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C23×Dic3C2212yes—
16C2×#237—28no—
17C2×#237—24no—
18C22×C4×D3C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2212yes—
23C2×C3×C22⋊C4C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2212yes—
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34#896—24no—
35#878—28no—
36C3×#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×C22⋊Q8—31no—
42C2×C22⋊Q8—32no—
43#395—31no—
44#395—31no—
45#394—31no—
46#394—32no—
47#394—32no—
48#394D311yescharacteristic, complemented
49#380—32no—
50#380—31no—
51C24×C3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×D4—24no—
54C2×C3×D4—22no—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—24no—
62C22×Dic3—24no—
63C22×Dic3—24no—
64C22×Dic3D418yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70#237—232no—
71C2×C4×D3—24no—
72C2×C4×D3D414yes—
73C2×Dic6—22no—
74C2×Dic6—24no—
75C2×Dic6—24no—
76C2×Dic6—22no—
77C3×C4⋊C4—22no—
78C3×C22⋊C4—24no—
79C3×C22⋊C4—42no—
80#217—24no—
81#217—42no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—42no—
85C4⋊Dic3—24no—
86Dic3⋊C4—22no—
87Dic3⋊C4—24no—
88Dic3⋊C4—24no—
89Dic3⋊C4—22no—
90C2×#41—68no—
91C2×#41—64no—
92C22×Q8—31no—
93C22×Q8—31no—
94C22×D4—32no—
95C22×D4C2×D311yescharacteristic
96C23×C4—32no—
97C23×C4—31no—
98C22⋊Q8—68no—
99C22⋊Q8—64no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4C2×D311yescharacteristic
106C2×C22⋊C4—32no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4C2×D312yes—
112C2×C22⋊C4C2×D312yes—
113#95—32no—
114#95—32no—
115#95—31no—
116#95C2×D311yescharacteristic
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3C2×D414yes—
124C23×C3C2411yescharacteristic
125C22×D3—22no—
126C22×D3—21no—
127C22×D3—22no—
128C22×D3—21no—
129C22×D3C2×D412yes—
130C3×D4—28no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4C2×D412yes—
141#67—216no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—216no—
145C2×Dic3—21no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3C2×D414yes—
154C2×Dic3C2×D414yes—
155C2×Dic3C2×D412yes—
156C4×D3—28no—
157Dic6—28no—
158Dic6—44no—
159C24—31no—
160C24C22×D312yes—
161#41—632no—
162C2×Q8—64no—
163C2×Q8—64no—
164C2×Q8—62no—
165C2×Q8—62no—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—64no—
171C22×C4—31no—
172C22×C4—38no—
173C22×C4—32no—
174C22×C4—34no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—64no—
179C22×C4—64no—
180C22×C4—64no—
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D312yes—
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—64no—
187C4⋊C4—64no—
188C4⋊C4—62no—
189C4⋊C4—64no—
190C4⋊C4—62no—
191C22⋊C4—24no—
192C22⋊C4—42no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—122no—
197C22⋊C4—122no—
198C22×C3—24no—
199C22×C3—24no—
200C22×C3—24no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3C22×D411yescharacteristic
206C22×C3C22×D412yes—
207C22×C3H(5,𝔽2)11yescharacteristic
208C22×C32−1+411yescharacteristic
209C22×C32−1+412yes—
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—42no—
214C2×D3—42no—
215C3×C4—22no—
216C3×C4—41no—
217C3×C4—42no—
218C3×C4—41no—
219Dic3—24no—
220Dic3—41no—
221Dic3—42no—
222Dic3—41no—
223Dic3#12014yes—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—32no—
231C23—62no—
232C23—61no—
233C23—62no—
234C23—61no—
235C23D3×D414yes—
236C23C23×D311yescharacteristic
237Q8—68no—
238Q8—124no—
239D4—28no—
240D4—616no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—34no—
251C2×C4—32no—
252C2×C4—34no—
253C2×C4—68no—
254C2×C4—616no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4D3×D412yes—
266C2×C3—44no—
267C2×C3C2×#12011yescharacteristic
268C2×C3#53612yes—
269C2×C3#53611yescharacteristic
270C2×C3#53712yes—
271C2×C3#53711yescharacteristic
272D3—42no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—62no—
281C22—62no—
282C22—62no—
283C22—122no—
284C22—122no—
285C22C2×D3×D412yes—
286C22C2×D3×D411yescharacteristic
287C22#100411yescharacteristic
288C22#101011yescharacteristic
289C22#101012yes—
290C4—22no—
291C4—34no—
292C4—41no—
293C4—42no—
294C4—41no—
295C4—64no—
296C4—121no—
297C4—122no—
298C4—121no—
299C3#261611yescharacteristic, complemented, Sylow
300C2—44no—
301C2—122no—
302C2D3×#12011yescharacteristic
303C2#559712yes—
304C2#560811yescharacteristic
305C2#564812yes—
306C2#567511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×#237C2
hypercenter#394D3
hypocenterC3#2616
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#896C2semidirect
3. C2×#878C2semidirect
4. C3×#394C2semidirect
5. #5222C2semidirect
6. #4979C2semidirect
7. #4961C2semidirect
8. #4959C2semidirect
9. #4942C2semidirect
10. #4667C2semidirect
11. #4656C2semidirect
48. #394D3semidirect
299. C3#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#8962
C2×#8782
C3×#3942
#52222
#49792
#49612
#49592
#49422
#46672
#46562
#26163