Finite Group Databasebeta

← #69595

Subgroups — #69595

Order 384.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#5486C211yesmaximal, characteristic
2C3×#385C211yesmaximal, characteristic, complemented
3#5213C211yesmaximal, characteristic
4#5210C211yesmaximal, characteristic, complemented
5C2×#806C211yesmaximal, characteristic
6#4527C211yesmaximal, characteristic, complemented
7#4526C211yesmaximal, characteristic, complemented
8#1867—31nomaximal, Sylow
9C23×C3×C4C2211yescharacteristic
10C22×#67C2211yescharacteristic
11C2×C3×C4⋊C4—21no—
12C2×C3×C22⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×#217C2211yescharacteristic
15C2×#217C2211yescharacteristic
16C2×#217C2211yescharacteristic
17C4×#67—21no—
18C4×#67—21no—
19C2×(C2×D3)⋊C4—21no—
20C2×Dic3⋊C4—21no—
21#67⋊C4—21no—
22#67⋊C4—21no—
23D3×C22⋊C4—21no—
24#877—21no—
25C3×#95—21no—
26C3⋊#95—21no—
27C3⋊#95—21no—
28#806C414yescomplemented
29#518—31no—
30C2×#99—31no—
31#386—31no—
32#385—31no—
33#385D311yescharacteristic, complemented
34#342—31no—
35#342—31no—
36C24×C3C2311yescharacteristic
37C23×D3D411yescharacteristic
38C22×C3×C4—21no—
39C22×C3×C4—21no—
40C22×C3×C4—21no—
41C22×C3×C4—21no—
42C22×C3×C4—42no—
43C22×C3×C4D411yescharacteristic
44C22×C3×C4D411yescharacteristic
45C2×#67—21no—
46C2×#67—21no—
47C2×#67—21no—
48C2×#67—21no—
49C2×#67C2×C412yes—
50C2×#67Q812yes—
51C22×Dic3—21no—
52C22×Dic3—21no—
53C22×Dic3D411yescharacteristic
54C22×Dic3D411yescharacteristic
55C22×Dic3D411yescharacteristic
56C2×C4×D3—21no—
57C2×C4×D3—21no—
58C3×C4⋊C4—42no—
59C3×C22⋊C4—21no—
60C3×C22⋊C4—21no—
61C3×C22⋊C4—21no—
62C3×C22⋊C4C2×C412yescomplemented
63#217—21no—
64#217—21no—
65#217—21no—
66#217—21no—
67#217—21no—
68#217C2×C412yescomplemented
69(C2×D3)⋊C4—21no—
70(C2×D3)⋊C4—21no—
71(C2×D3)⋊C4—42no—
72Dic3⋊C4—21no—
73Dic3⋊C4—21no—
74Dic3⋊C4—42no—
75C4×Dic3—21no—
76C4×Dic3—21no—
77C22×D4—31no—
78C23×C4C2×D311yescharacteristic
79C4×D4—61no—
80C4×D4—61no—
81C4×D4—61no—
82C4×D4—61no—
83#117—61no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—61no—
86C2×C22⋊C4—31no—
87C2×C22⋊C4—31no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—61no—
90C2×C22⋊C4—61no—
91C2×C22⋊C4C2×D311yescharacteristic
92C2×C22⋊C4C2×D311yescharacteristic
93#99—34no—
94#95—21no—
95#95—61no—
96#95—61no—
97C23×C3—41no—
98C23×C3—41no—
99C23×C3C22×C411yescharacteristic
100C23×C3C2×D411yescharacteristic
101C23×C3C2×D411yescharacteristic
102C23×C3C2×D411yescharacteristic
103C23×C3C2×Q811yescharacteristic
104C23×C3#4111yescharacteristic
105C23×C3#4111yescharacteristic
106C22×D3—21no—
107C22×D3—21no—
108C22×D3—41no—
109C22×D3—41no—
110C22×D3C4⋊C412yescomplemented
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—41no—
118C2×C3×C4—41no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4—41no—
122C2×C3×C4—41no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
125#67—22no—
126#67—22no—
127#67—22no—
128#67—22no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—41no—
135C2×Dic3—41no—
136C2×Dic3—41no—
137C2×Dic3—41no—
138C2×Dic3—41no—
139C2×Dic3—41no—
140C2×Dic3—41no—
141C2×Dic3C4⋊C412yescomplemented
142C4×D3—81no—
143C4×D3—81no—
144C24—31no—
145C24C22×D311yescharacteristic
146C2×D4—32no—
147C2×D4—32no—
148C2×D4—61no—
149C2×D4—61no—
150C2×D4—61no—
151C2×D4—61no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—42no—
160C22×C4—61no—
161C22×C4—61no—
162C22×C4—61no—
163C22×C4—61no—
164C22×C4#6711yescharacteristic
165C22×C4#6711yescharacteristic
166C4⋊C4—42no—
167C4⋊C4—61no—
168C4⋊C4—61no—
169C4⋊C4—122no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—32no—
174C22⋊C4—61no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C22⋊C4—122no—
182C22⋊C4C4×D312yescomplemented
183C42—61no—
184C42—61no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3C2×C4⋊C411yescharacteristic
196C22×C3C4×D411yescharacteristic
197C22×C3C4×D411yescharacteristic
198C22×C3#12011yescharacteristic
199C22×C3C22⋊Q811yescharacteristic
200C22×C3C22⋊Q811yescharacteristic
201C22×C3#12311yescharacteristic
202C2×D3—22no—
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—42no—
206C2×D3—81no—
207C3×C4—42no—
208C3×C4—41no—
209C3×C4—42no—
210C3×C4—41no—
211Dic3—22no—
212Dic3—42no—
213Dic3—41no—
214Dic3—41no—
215Dic3—81no—
216C23—32no—
217C23—41no—
218C23—41no—
219C23—61no—
220C23—61no—
221C23—121no—
222C23—121no—
223C23C2×C4×D311yescharacteristic
224C23#23511yescharacteristic
225C23D3×D411yescharacteristic
226C23D3×D411yescharacteristic
227C23D3×Q811yescharacteristic
228C23#23911yescharacteristic
229C23C2×#6711yescharacteristic
230D4—62no—
231D4—62no—
232D4—62no—
233D4—62no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—32no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—121no—
259C2×C4—121no—
260C2×C4—121no—
261C2×C4—241no—
262C2×C4—241no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3#38511yescharacteristic
270C2×C3B(4,𝔽2)11yescharacteristic
271C2×C3#45811yescharacteristic
272D3—42no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—62no—
284C22—122no—
285C22—121no—
286C22—121no—
287C22—241no—
288C22D3×C4⋊C411yescharacteristic
289C22#89311yescharacteristic
290C22#89411yescharacteristic
291C22#89611yescharacteristic
292C22C4×#6711yescharacteristic
293C22#93711yescharacteristic
294C22#94611yescharacteristic
295C4—42no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—62no—
300C4—122no—
301C4—121no—
302C4—121no—
303C4—241no—
304C3#186711yescharacteristic, complemented, Sylow
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—122no—
312C2#498911yescharacteristic
313C2#503211yescharacteristic
314C2#505311yescharacteristic
subgroupG/H
centerC22#937
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#385C2
socleC22×C3#120
prime radicalC22×#67C22
cosocleC24×C3C23
hypercenter#385D3
hypocenterC3#1867
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C3×#385C2semidirect
4. #5210C2semidirect
6. #4527C2semidirect
7. #4526C2semidirect
28. #806C4semidirect
33. #385D3semidirect
62. C3×C22⋊C4C2×C4semidirect
68. #217C2×C4semidirect
110. C22×D3C4⋊C4semidirect
141. C2×Dic3C4⋊C4semidirect
182. C22⋊C4C4×D3semidirect
304. C3#1867semidirect

Maximal subgroups

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

subgroupindex
#54862
C3×#3852
#52132
#52102
C2×#8062
#45272
#45262
#18673