Finite Group Databasebeta

← #73376

Subgroups — #73376

Order 384.

914 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 173 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×#235C211yesmaximal, characteristic, complemented
2C2×#930C212yesmaximal, complemented
3C2×#929C211yesmaximal, characteristic, complemented
4C2×C3×#95C211yesmaximal, characteristic, complemented
5#5213C212yesmaximal, complemented
6#4673C211yesmaximal, characteristic, complemented
7#4672C211yesmaximal, characteristic, complemented
8#4671C211yesmaximal, characteristic, complemented
9#4668C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11#4656C211yesmaximal, characteristic, complemented
12#2430—31nomaximal, Sylow
13C23×C3×C4C2212yes—
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2212yes—
17C2×#235—24no—
18C2×#235—28no—
19C22×D12C2211yescharacteristic
20C22×D12C2212yes—
21C22×C4×D3C2212yes—
22C22×Dic6C2211yescharacteristic
23C2×#217C2211yescharacteristic
24#930—28no—
25#929—24no—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2212yes—
32C2×C4⋊Dic3C2212yes—
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C3×#95C2211yescharacteristic
36C3×#95C2212yes—
37C3×#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×#123—31no—
42C2×#121—32no—
43#395—31no—
44#394—31no—
45#393—31no—
46#392—31no—
47#386—32no—
48#386—32no—
49#385—32no—
50C2×#95D311yescharacteristic, complemented
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C23×D3C2312yes—
54C22×C3×C4—24no—
55C22×C3×C4—24no—
56C22×C3×C4—24no—
57C22×C3×C4—42no—
58C22×C3×C4D414yescomplemented
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2312yes—
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—22no—
66C2×#67—24no—
67C2×#67D418yescomplemented
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2312yes—
72#235—232no—
73C2×D12—24no—
74C2×D12—22no—
75C2×D12—24no—
76C2×D12—24no—
77C2×D12—42no—
78C2×C4×D3—24no—
79C2×C4×D3—24no—
80C2×C4×D3—24no—
81C2×Dic6—22no—
82C2×Dic6—24no—
83#217—22no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—24no—
89(C2×D3)⋊C4—42no—
90C4⋊Dic3—24no—
91Dic3⋊C4—22no—
92Dic3⋊C4—22no—
93C2×#41—68no—
94C2×#41—64no—
95C22×Q8—31no—
96C22×D4—32no—
97C22×D4—31no—
98C22×D4—31no—
99C22×D4—32no—
100C23×C4—32no—
101C23×C4C2×D312yes—
102C23×C4C2×D311yescharacteristic
103#123—64no—
104#121—68no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115#95—32no—
116#95—32no—
117#95C2×D312yes—
118#95C2×D311yescharacteristic
119#95C2×D311yescharacteristic
120C23×C3—22no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3C2×D412yes—
124C23×C3C2×D414yes—
125C23×C3C2411yescharacteristic
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—21no—
133C22×D3—22no—
134C22×D3—22no—
135C22×D3—21no—
136C22×D3—21no—
137C22×D3C2×D414yes—
138C2×C3×C4—28no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—48no—
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2×D412yes—
149C2×C3×C4#4114yescomplemented
150#67—216no—
151#67—44no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3C2×D414yes—
162D12—28no—
163D12—48no—
164C4×D3—216no—
165Dic6—28no—
166C24—32no—
167C24—31no—
168C24C22×D311yescharacteristic
169#41—632no—
170C2×Q8—64no—
171C2×Q8—62no—
172C2×D4—38no—
173C2×D4—64no—
174C2×D4—64no—
175C2×D4—64no—
176C2×D4—64no—
177C2×D4—62no—
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—64no—
181C2×D4—122no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—32no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—42no—
190C22×C4—64no—
191C22×C4—64no—
192C22×C4—64no—
193C22×C4D1214yescomplemented
194C22×C4C22×D312yes—
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D311yescharacteristic
197C22×C4C22×D312yes—
198C4⋊C4—64no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C22⋊C4—64no—
202C22⋊C4—64no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C22⋊C4—64no—
207C22⋊C4—122no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3#12014yes—
213C22×C3C22×D411yescharacteristic
214C22×C3C22×D412yes—
215C22×C3C2×#4112yes—
216C22×C3H(5,𝔽2)11yescharacteristic
217C22×C32−1+411yescharacteristic
218C2×D3—24no—
219C2×D3—24no—
220C2×D3—24no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—41no—
226C2×D3—41no—
227C2×D3—41no—
228C2×D3—42no—
229C2×D3—42no—
230C3×C4—22no—
231C3×C4—22no—
232C3×C4—24no—
233C3×C4—42no—
234Dic3—24no—
235Dic3—41no—
236Dic3—41no—
237Dic3—41no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—34no—
242C23—62no—
243C23—62no—
244C23—62no—
245C23—62no—
246C23—62no—
247C23—62no—
248C23—61no—
249C23—62no—
250C23—62no—
251C23—61no—
252C23—61no—
253C23C2×D1212yes—
254C23D3×D414yes—
255C23C23×D311yescharacteristic
256Q8—68no—
257D4—68no—
258D4—616no—
259D4—128no—
260D4—124no—
261C2×C4—28no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—34no—
270C2×C4—48no—
271C2×C4—616no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4C2×D1212yes—
282C2×C4C2×D1212yes—
283C2×C4#23514yescomplemented
284C2×C3—24no—
285C2×C3C2×#12011yescharacteristic
286C2×C3#53211yescharacteristic
287C2×C3#53711yescharacteristic
288C2×C3#54612yes—
289C2×C3#54712yes—
290D3—44no—
291D3—81no—
292C22—22no—
293C22—24no—
294C22—24no—
295C22—22no—
296C22—64no—
297C22—64no—
298C22—64no—
299C22—122no—
300C22—122no—
301C22—122no—
302C22—122no—
303C22—121no—
304C22—121no—
305C22—122no—
306C22—122no—
307C22—121no—
308C22#88214yes—
309C22C22×D1211yescharacteristic
310C22C2×#23512yes—
311C22C2×D3×D412yes—
312C22#100411yescharacteristic
313C22#100711yescharacteristic
314C4—22no—
315C4—22no—
316C4—24no—
317C4—42no—
318C4—64no—
319C4—121no—
320C4—121no—
321C4—121no—
322C3#243011yescharacteristic, complemented, Sylow
323C2—24no—
324C2—124no—
325C2—241no—
326C2#547711yescharacteristic
327C2C2×#88211yescharacteristic
328C2#549012yes—
329C2#550712yes—
330C2#551011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#95C2
hypercenterC2×#95D3
hypocenterC3#2430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C2×#930C2semidirect
3. C2×#929C2semidirect
4. C2×C3×#95C2semidirect
5. #5213C2semidirect
6. #4673C2semidirect
7. #4672C2semidirect
8. #4671C2semidirect
9. #4668C2semidirect
10. #4665C2semidirect
11. #4656C2semidirect
50. C2×#95D3semidirect
58. C22×C3×C4D4semidirect
67. C2×#67D4semidirect
149. C2×C3×C4#41semidirect
193. C22×C4D12semidirect
283. C2×C4#235semidirect
322. C3#2430semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×#9302
C2×#9292
C2×C3×#952
#52132
#46732
#46722
#46712
#46682
#46652
#46562
#24303