Finite Group Databasebeta

← #77107

Subgroups — #77107

Order 384.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#883C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#4979C211yesmaximal, characteristic, complemented
7#4956C212yesmaximal, complemented
8#4950C212yesmaximal, complemented
9#4946C212yesmaximal, complemented
10#4664C211yesmaximal, characteristic, complemented
11#4656C211yesmaximal, characteristic, complemented
12#2406—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C23×Dic3C2212yes—
16C2×#237—28no—
17C2×#237—24no—
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2212yes—
24C2×C3×C22⋊C4C2212yes—
25C2×#217C2212yes—
26C2×#217C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2212yes—
33D3×C4⋊C4—24no—
34#883—28no—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×#123—32no—
42C22×C4⋊C4—31no—
43#395—31no—
44#395—31no—
45#394—31no—
46#394D311yescharacteristic, complemented
47#386—32no—
48#385—32no—
49#385—32no—
50#381—31no—
51C24×C3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×D4—24no—
54C2×C3×D4—22no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67D418yes—
62C22×Dic3—24no—
63C22×Dic3—24no—
64C22×Dic3—24no—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70#237—232no—
71C2×C4×D3—22no—
72C2×C4×D3—24no—
73C2×C4×D3—22no—
74C2×C4×D3—22no—
75C2×C4×D3—24no—
76C2×C4×D3—22no—
77C2×C4×D3D414yes—
78C2×Dic6—22no—
79C2×Dic6—24no—
80C3×C4⋊C4—22no—
81C3×C22⋊C4—24no—
82C3×C22⋊C4—24no—
83#217—24no—
84#217—24no—
85(C2×D3)⋊C4—24no—
86C4⋊Dic3—24no—
87C4⋊Dic3—22no—
88Dic3⋊C4—22no—
89Dic3⋊C4—24no—
90Dic3⋊C4—22no—
91C2×#41—64no—
92C2×#41—68no—
93C22×Q8—31no—
94C22×D4—32no—
95C22×D4C2×D311yescharacteristic
96C23×C4—31no—
97C23×C4—32no—
98C23×C4—31no—
99C23×C4—31no—
100#123—68no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—64no—
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4C2×D312yes—
112C2×C22⋊C4C2×D312yes—
113#95—32no—
114#95—31no—
115#95—32no—
116#95—32no—
117#95C2×D311yescharacteristic
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3C2×D414yes—
125C23×C3C2411yescharacteristic
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—22no—
129C22×D3C2×D414yes—
130C22×D3C2×D412yes—
131C3×D4—28no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4#4112yescomplemented
142#67—216no—
143C2×Dic3—216no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3C2×D414yes—
155C2×Dic3C2×D412yes—
156C2×Dic3#4112yescomplemented
157C4×D3—28no—
158C4×D3—28no—
159C4×D3—44no—
160Dic6—28no—
161C24—31no—
162C24C22×D312yes—
163#41—632no—
164C2×Q8—64no—
165C2×Q8—62no—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—38no—
169C2×D4—64no—
170C2×D4—64no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—32no—
174C22×C4—34no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—64no—
180C22×C4—62no—
181C22×C4—64no—
182C22×C4—64no—
183C22×C4—64no—
184C22×C4—64no—
185C22×C4—62no—
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C4⋊C4—64no—
194C4⋊C4—64no—
195C4⋊C4—62no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—64no—
199C22⋊C4—64no—
200C22⋊C4—64no—
201C22×C3—24no—
202C22×C3—24no—
203C22×C3—24no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3C22×D411yescharacteristic
209C22×C3C22×D412yes—
210C22×C3C2×#4111yescharacteristic
211C22×C3C2×#4111yescharacteristic
212C22×C32−1+412yes—
213C2×D3—22no—
214C2×D3—22no—
215C2×D3—24no—
216C2×D3—24no—
217C2×D3#12014yes—
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—41no—
221C3×C4—42no—
222Dic3—24no—
223Dic3—22no—
224Dic3—22no—
225Dic3—41no—
226Dic3—42no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—34no—
234C23—32no—
235C23—61no—
236C23—61no—
237C23—62no—
238C23D3×D414yes—
239C23C23×D311yescharacteristic
240Q8—68no—
241D4—28no—
242D4—616no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—32no—
252C2×C4—34no—
253C2×C4—32no—
254C2×C4—68no—
255C2×C4—68no—
256C2×C4—616no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—61no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—124no—
268C2×C4D3×D412yes—
269C2×C4#23712yescomplemented
270C2×C3—44no—
271C2×C3C2×#12011yescharacteristic
272C2×C3#52511yescharacteristic
273C2×C3#53611yescharacteristic
274C2×C3#54712yes—
275C2×C3#54712yes—
276D3—24no—
277C22—24no—
278C22—24no—
279C22—24no—
280C22—34no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—64no—
286C22—62no—
287C22—62no—
288C22—64no—
289C22C2×D3×D412yes—
290C22C2×D3×D411yescharacteristic
291C22C2×#23711yescharacteristic
292C22D3×#4111yescharacteristic
293C22#101012yes—
294C4—22no—
295C4—22no—
296C4—41no—
297C4—42no—
298C4—62no—
299C4—64no—
300C4—62no—
301C4—122no—
302C4—121no—
303C3#240611yescharacteristic, complemented, Sylow
304C2—44no—
305C2—64no—
306C2D3×#12011yescharacteristic
307C2#560611yescharacteristic
308C2#561112yes—
309C2#565512yes—
310C2#567511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×#237C2
hypercenter#394D3
hypocenterC3#2406
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. #4979C2semidirect
7. #4956C2semidirect
8. #4950C2semidirect
9. #4946C2semidirect
10. #4664C2semidirect
11. #4656C2semidirect
46. #394D3semidirect
141. C2×C3×C4#41semidirect
156. C2×Dic3#41semidirect
269. C2×C4#237semidirect
303. C3#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×D3×C4⋊C42
C2×#8832
C3×#3942
#52222
#49792
#49562
#49502
#49462
#46642
#46562
#24063