Finite Group Databasebeta

← #77173

Subgroups — #77173

Order 384.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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×D3×Q8C211yesmaximal, characteristic
2C2×#896C211yesmaximal, characteristic
3C2×#888C212yesmaximal, complemented
4C3×#395C211yesmaximal, characteristic, complemented
5#5230C211yesmaximal, characteristic, complemented
6#4989C212yesmaximal
7#4979C211yesmaximal, characteristic, complemented
8#4668C212yesmaximal
9#4666C211yesmaximal, characteristic
10#4656C211yesmaximal, characteristic, complemented
11#4646C212yesmaximal, complemented
12#2434—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×D3×Q8—24no—
15C2×D3×Q8—28no—
16C22×C4×D3C2212yes—
17C22×C4×D3C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C22×Dic6C2212yes—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2212yes—
32#896—24no—
33#888—28no—
34C3×#95C2212yes—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2211yescharacteristic
40C23×Q8—31no—
41C2×C4⋊Q8—32no—
42C2×C22⋊Q8—31no—
43#395—31no—
44#395—31no—
45#395—31no—
46#395D311yescharacteristic, complemented
47#391—32no—
48#385—32no—
49#385—32no—
50C4×C22⋊C4—31no—
51C23×D3C2311yescharacteristic
52C2×C3×Q8—22no—
53C2×C3×Q8—24no—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C22×C3×C4C2312yes—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64D3×Q8—232no—
65C2×C4×D3—24no—
66C2×C4×D3—24no—
67C2×C4×D3—24no—
68C2×C4×D3—24no—
69C2×C4×D3D414yes—
70C2×Dic6—22no—
71C2×Dic6—24no—
72C2×Dic6—24no—
73C2×Dic6—24no—
74C2×Dic6—24no—
75C2×Dic6—22no—
76C2×Dic6D418yes—
77C3×C4⋊C4—24no—
78C3×C4⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4Q818yescomplemented
82C4⋊Dic3—24no—
83Dic3⋊C4—24no—
84Dic3⋊C4—22no—
85Dic3⋊C4—24no—
86Dic3⋊C4—22no—
87C4×Dic3—24no—
88C22×Q8—32no—
89C22×Q8—31no—
90C22×Q8—31no—
91C22×Q8—64no—
92C22×Q8—68no—
93C22×Q8C2×D311yescharacteristic
94C23×C4—31no—
95C23×C4—32no—
96C4⋊Q8—68no—
97C22⋊Q8—64no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4C2×D311yescharacteristic
104C2×C4⋊C4C2×D312yes—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—31no—
108C2×C42—32no—
109#95—31no—
110#95—32no—
111#95—32no—
112#95—32no—
113#95C2×D311yescharacteristic
114#95C2×D312yes—
115C23×C3C2411yescharacteristic
116C22×D3—22no—
117C22×D3—21no—
118C22×D3—21no—
119C22×D3C2×D412yes—
120C22×D3C2×Q814yes—
121C3×Q8—28no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D414yes—
134C2×C3×C4C2×Q814yescomplemented
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3C2×D414yes—
145C2×Dic3C2×D414yes—
146C2×Dic3C2×D412yes—
147C2×Dic3C2×Q814yescomplemented
148C4×D3—216no—
149C4×D3—28no—
150Dic6—28no—
151Dic6—216no—
152Dic6—44no—
153C24—31no—
154C2×Q8—22no—
155C2×Q8—24no—
156C2×Q8—38no—
157C2×Q8—64no—
158C2×Q8—64no—
159C2×Q8—64no—
160C2×Q8—62no—
161C2×Q8—64no—
162C2×Q8—632no—
163C2×Q8—62no—
164C22×C4—31no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—34no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—64no—
171C22×C4—64no—
172C22×C4—64no—
173C22×C4—64no—
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177C22×C4C22×D312yes—
178C22×C4C22×D312yes—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—62no—
182C4⋊C4—64no—
183C4⋊C4—64no—
184C4⋊C4—62no—
185C4⋊C4—64no—
186C22⋊C4—38no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C42—64no—
190C22×C3C22×D411yescharacteristic
191C22×C3C22×D412yes—
192C22×C3C22×Q812yes—
193C22×C32−1+411yescharacteristic
194C22×C32−1+411yescharacteristic
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—24no—
198C2×D3—24no—
199C2×D3C4⋊Q814yescomplemented
200C3×C4—22no—
201C3×C4—24no—
202C3×C4—24no—
203C3×C4—41no—
204C3×C4—41no—
205Dic3—24no—
206Dic3—24no—
207Dic3—41no—
208Dic3—41no—
209Dic3#12014yes—
210C23—32no—
211C23—34no—
212C23—62no—
213C23—61no—
214C23—61no—
215C23C23×D311yescharacteristic
216Q8—28no—
217Q8—616no—
218Q8—68no—
219Q8—124no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—34no—
231C2×C4—32no—
232C2×C4—34no—
233C2×C4—34no—
234C2×C4—68no—
235C2×C4—616no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4D3×D412yes—
246C2×C4D3×D414yes—
247C2×C4D3×Q814yescomplemented
248C2×C3C2×#12011yescharacteristic
249C2×C3C2×C4⋊Q811yescharacteristic
250C2×C3#53611yescharacteristic
251C2×C3D4×Q812yes—
252C2×C3D4×Q812yes—
253D3—24no—
254C22—34no—
255C22—64no—
256C22—64no—
257C22—62no—
258C22—62no—
259C22C2×D3×D412yes—
260C22C2×D3×D411yescharacteristic
261C22C2×D3×Q812yes—
262C22#100711yescharacteristic
263C22#101011yescharacteristic
264C4—24no—
265C4—22no—
266C4—24no—
267C4—34no—
268C4—41no—
269C4—41no—
270C4—64no—
271C4—64no—
272C4—121no—
273C4—121no—
274C3#243411yescharacteristic, complemented, Sylow
275C2—64no—
276C2D3×#12011yescharacteristic
277C2#562811yescharacteristic
278C2#563212yes—
279C2D3×C4⋊Q811yescharacteristic
280C2#572712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC23×D3C23
hypercenter#395D3
hypocenterC3#2434
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. C2×#888C2semidirect
4. C3×#395C2semidirect
5. #5230C2semidirect
7. #4979C2semidirect
10. #4656C2semidirect
11. #4646C2semidirect
46. #395D3semidirect
81. (C2×D3)⋊C4Q8semidirect
134. C2×C3×C4C2×Q8semidirect
147. C2×Dic3C2×Q8semidirect
199. C2×D3C4⋊Q8semidirect
247. C2×C4D3×Q8semidirect
274. C3#2434semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×#8962
C2×#8882
C3×#3952
#52302
#49892
#49792
#46682
#46662
#46562
#46462
#24343