Finite Group Databasebeta

← #77124

Subgroups — #77124

Order 384.

632 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 143 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×#938C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#883C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#4981C211yesmaximal, characteristic, complemented
7#4956C212yesmaximal, complemented
8#4948C212yesmaximal, complemented
9#4948C212yesmaximal, complemented
10#4667C211yesmaximal, characteristic, complemented
11#4661C211yesmaximal, characteristic, complemented
12#2618—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24#938—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33D3×C4⋊C4—24no—
34#883—28no—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×#123—32no—
40C2×#121—31no—
41C22×C4⋊C4—31no—
42#398—31no—
43#398—31no—
44#396—32no—
45#396—32no—
46#394—31no—
47#394D311yescharacteristic, complemented
48#386—32no—
49#380—31no—
50C24×C3C2312yes—
51C23×D3C2311yescharacteristic
52C2×C3×D4—22no—
53C2×C3×D4—42no—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C2×#67—24no—
59C2×#67—24no—
60C2×#67—24no—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—42no—
70C2×C4×D3—42no—
71C2×C4×D3D414yes—
72C2×C4×D3D414yescomplemented
73C3×C4⋊C4—22no—
74C3×C22⋊C4—24no—
75C3×C22⋊C4—42no—
76#217—24no—
77#217—24no—
78#217—42no—
79(C2×D3)⋊C4—24no—
80C4⋊Dic3—22no—
81C4⋊Dic3—24no—
82C4⋊Dic3—22no—
83Dic3⋊C4—24no—
84Dic3⋊C4—22no—
85Dic3⋊C4—22no—
86Dic3⋊C4—42no—
87C22×D4—32no—
88C22×D4C2×D311yescharacteristic
89C23×C4—31no—
90C23×C4—31no—
91C23×C4—31no—
92#123—68no—
93#121—64no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—64no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4C2×D312yes—
108C2×C22⋊C4C2×D312yes—
109#95—31no—
110#95—32no—
111#95—32no—
112#95C2×D311yescharacteristic
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3C2411yescharacteristic
121C22×D3—22no—
122C22×D3—22no—
123C22×D3C2×D412yes—
124C22×D3C2×D412yes—
125C22×D3#4112yes—
126C3×D4—44no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D412yes—
137#67—48no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3C2×D412yes—
150C2×Dic3C2×D412yes—
151C4×D3—28no—
152C4×D3—44no—
153C4×D3—44no—
154C24—31no—
155C24C22×D312yes—
156C2×D4—22no—
157C2×D4—42no—
158C2×D4—64no—
159C2×D4—64no—
160C2×D4—64no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—34no—
164C22×C4—34no—
165C22×C4—32no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4—122no—
172C22×C4—122no—
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D312yes—
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—64no—
181C4⋊C4—62no—
182C4⋊C4—62no—
183C4⋊C4—64no—
184C4⋊C4—122no—
185C22⋊C4—24no—
186C22⋊C4—42no—
187C22⋊C4—64no—
188C22⋊C4—64no—
189C22⋊C4—64no—
190C22⋊C4—122no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3C22×D411yescharacteristic
199C22×C3C22×D411yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C3H(5,𝔽2)12yes—
202C22×C32−1+412yes—
203C2×D3—22no—
204C2×D3—22no—
205C2×D3—24no—
206C2×D3—24no—
207C2×D3#12114yes—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—41no—
211C3×C4—42no—
212Dic3—22no—
213Dic3—22no—
214Dic3—42no—
215Dic3—41no—
216Dic3—42no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—32no—
225C23—32no—
226C23—32no—
227C23—62no—
228C23—62no—
229C23C23×D311yescharacteristic
230D4—44no—
231D4—128no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—68no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—61no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—124no—
255C2×C4—124no—
256C2×C4D3×D412yes—
257C2×C4D3×D412yes—
258C2×C3—82no—
259C2×C3C2×#12111yescharacteristic
260C2×C3#53411yescharacteristic
261C2×C3#53611yescharacteristic
262C2×C3#53712yes—
263C2×C3#53912yes—
264D3—24no—
265C22—34no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—64no—
274C22—62no—
275C22—62no—
276C22—64no—
277C22C2×D3×D411yescharacteristic
278C22C2×D3×D411yescharacteristic
279C22#100412yes—
280C22D3×#4111yescharacteristic
281C22#101012yes—
282C4—22no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—62no—
287C4—62no—
288C4—121no—
289C4—122no—
290C4—122no—
291C3#261811yescharacteristic, complemented, Sylow
292C2—64no—
293C2—82no—
294C2#558911yescharacteristic
295C2D3×#12111yescharacteristic
296C2#560812yes—
297C2#566012yes—
298C2#567511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#938C2
hypercenter#394D3
hypocenterC3#2618
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×D3×C4⋊C4C2semidirect
3. C2×#883C2semidirect
4. C3×#394C2semidirect
5. #5222C2semidirect
6. #4981C2semidirect
7. #4956C2semidirect
8. #4948C2semidirect
9. #4948C2semidirect
10. #4667C2semidirect
11. #4661C2semidirect
47. #394D3semidirect
72. C2×C4×D3D4semidirect
291. C3#2618semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×D3×C4⋊C42
C2×#8832
C3×#3942
#52222
#49812
#49562
#49482
#46672
#46612
#26183