Finite Group Databasebeta

← #76701

Subgroups — #76701

Order 384.

628 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 131 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×#936C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#4956C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4947C211yesmaximal, characteristic, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2631—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2212yes—
17C23×Dic3C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26#936—24no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31#67⋊C4—24no—
32C3×#95C2211yescharacteristic
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#124—31no—
39C2×C4×D4—31no—
40#397—32no—
41#393—31no—
42#393—31no—
43#392—31no—
44#392—32no—
45#392D311yescharacteristic, complemented
46#386—32no—
47#386—32no—
48#383—31no—
49C24×C3C2312yes—
50C24×C3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C2×C3×D4—24no—
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C2×C3×D4—42no—
56C2×C3×D4—42no—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—22no—
61C2×#67—22no—
62C2×#67—24no—
63C2×#67—22no—
64C2×#67—42no—
65C2×#67—42no—
66C22×Dic3—22no—
67C22×Dic3—42no—
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2312yes—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C2×C4×D3—22no—
74C2×C4×D3—42no—
75C2×Dic6—22no—
76C2×Dic6—42no—
77C3×C22⋊C4—22no—
78C3×C22⋊C4—42no—
79#217—24no—
80#217—24no—
81#217—42no—
82#217—42no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—42no—
85Dic3⋊C4—22no—
86C4×Dic3D414yescomplemented
87C22×Q8—31no—
88C22×D4—31no—
89C22×D4—32no—
90C22×D4C2×D311yescharacteristic
91C22×D4C2×D312yes—
92C23×C4—31no—
93C23×C4—31no—
94#124—64no—
95C4×D4—64no—
96C2×C4⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D312yes—
105C2×C42—31no—
106#95—32no—
107#95—31no—
108#95—32no—
109#95—32no—
110#95—32no—
111#95C2×D311yescharacteristic
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3#4112yes—
124C23×C3C2411yescharacteristic
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3#4112yes—
130C3×D4—44no—
131C3×D4—48no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4C2×D412yes—
139#67—28no—
140#67—48no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—44no—
153C2×Dic3C2×D412yes—
154C2×Dic3C2×D412yes—
155C4×D3—44no—
156Dic6—44no—
157C24—31no—
158C24C22×D312yes—
159C24C22×D311yescharacteristic
160C2×Q8—62no—
161C2×Q8—122no—
162C2×D4—24no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—64no—
171C2×D4—122no—
172C2×D4—122no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—31no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—122no—
181C22×C4—122no—
182C22×C4C22×D312yes—
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D311yescharacteristic
185C4⋊C4—62no—
186C22⋊C4—22no—
187C22⋊C4—42no—
188C22⋊C4—64no—
189C22⋊C4—64no—
190C22⋊C4—62no—
191C22⋊C4—122no—
192C22⋊C4—122no—
193C22⋊C4—122no—
194C42—34no—
195C22×C3—22no—
196C22×C3—24no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3C22×D411yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3H(5,𝔽2)12yes—
211C22×C3H(5,𝔽2)12yes—
212C2×D3—22no—
213C2×D3—24no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—42no—
217C3×C4—22no—
218C3×C4—41no—
219C3×C4—42no—
220Dic3—22no—
221Dic3—22no—
222Dic3—42no—
223Dic3—41no—
224Dic3—42no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—32no—
237C23—61no—
238C23—61no—
239C23—62no—
240C23—62no—
241C23#23712yes—
242C23C23×D311yescharacteristic
243Q8—124no—
244D4—44no—
245D4—48no—
246D4—68no—
247D4—128no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4—61no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—124no—
268C2×C4—124no—
269C2×C4D3×D412yes—
270C2×C3—42no—
271C2×C3—82no—
272C2×C3#52511yescharacteristic
273C2×C3#53511yescharacteristic
274C2×C3#53511yescharacteristic
275C2×C3#53812yes—
276C2×C3#53812yes—
277D3—42no—
278C22—24no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—62no—
291C22—64no—
292C22—121no—
293C22—121no—
294C22—122no—
295C22C2×D3×D411yescharacteristic
296C22C2×#23711yescharacteristic
297C22#100412yes—
298C22#100412yes—
299C22D3×#4111yescharacteristic
300C4—22no—
301C4—41no—
302C4—42no—
303C4—62no—
304C4—62no—
305C4—121no—
306C4—122no—
307C4—122no—
308C3#263111yescharacteristic, complemented, Sylow
309C2—42no—
310C2—82no—
311C2—122no—
312C2#559211yescharacteristic
313C2#559312yes—
314C2#560611yescharacteristic
315C2#561312yes—
316C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2631
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#936C2semidirect
2. C2×#67⋊C4C2semidirect
3. C3×#392C2semidirect
4. #5223C2semidirect
5. #5223C2semidirect
6. #5220C2semidirect
7. #4956C2semidirect
8. #4952C2semidirect
9. #4947C2semidirect
10. #4673C2semidirect
11. #4651C2semidirect
45. #392D3semidirect
86. C4×Dic3D4semidirect
308. C3#2631semidirect

Maximal subgroups

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

subgroupindex
C2×#9362
C2×#67⋊C42
C3×#3922
#52232
#52202
#49562
#49522
#49472
#46732
#46512
#26313