Finite Group Databasebeta

← #83413

Subgroups — #83413

Order 384.

378 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C3×#536C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic
3#5234C211yesmaximal, characteristic
4#5228C212yesmaximal, complemented
5#5052C212yesmaximal, complemented
6#5051C212yesmaximal, complemented
7#5050C212yesmaximal, complemented
8#5045C212yesmaximal, complemented
9#4996C212yesmaximal, complemented
10#3202—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C3×C4⋊Q8—21no—
15C3×#124—21no—
16C3×#123—21no—
17C3×#123—21no—
18C3×C22⋊Q8C2212yes—
19C3×C22⋊Q8C2212yes—
20C3×#117C2211yescharacteristic
21#944—21no—
22C2×#216C2212yes—
23C2×#216C2212yes—
24#936—21no—
25#935—21no—
26#935—21no—
27#926C2211yescharacteristic
28#924C2212yes—
29#924C2212yes—
30C2×#208C2211yescharacteristic
31C22×C3⋊C8C2211yescharacteristic
32#835C2212yes—
33#835C2212yes—
34#830C2212yes—
35#830C2212yes—
36#810C2212yes—
37#810C2212yes—
38#808C2212yes—
39#807C2212yes—
40#536—31no—
41#536D311yescharacteristic
42#485—32no—
43#485—32no—
44#470—32no—
45#467—32no—
46#451—32no—
47#451—32no—
48#408—31no—
49C3×#41—41no—
50C3×#41—41no—
51C2×C3×Q8—21no—
52C2×C3×Q8—41no—
53C2×C3×Q8—41no—
54C2×C3×Q8D411yescharacteristic
55C2×C3×Q8C2312yes—
56C2×C3×D4D411yescharacteristic
57C2×C3×D4D411yescharacteristic
58C2×C3×D4D411yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×Dic3—21no—
64C2×Dic6—21no—
65C2×Dic6—41no—
66C2×Dic6—41no—
67C2×Dic6C2312yes—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4C2312yes—
71C3×C22⋊C4—21no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—22no—
74C3×C42—21no—
75#217—22no—
76#217—22no—
77#217—21no—
78#216—42no—
79#216—42no—
80C4⋊Dic3C2312yes—
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83C4×Dic3—21no—
84#208—41no—
85C2×C3⋊C8—41no—
86C2×C3⋊C8C2312yes—
87C2×C3⋊C8C2312yes—
88C2×#41C2×D311yescharacteristic
89C22×Q8—31no—
90C22×Q8C2×D311yescharacteristic
91C2×Q16—32no—
92C2×Q16—32no—
93C2×M16—31no—
94C22×C8—31no—
95C4⋊Q8—21no—
96C4⋊Q8—61no—
97#124—21no—
98#124—61no—
99#123—21no—
100#123—21no—
101#123—61no—
102#123—61no—
103C22⋊Q8—32no—
104C22⋊Q8—32no—
105C22⋊Q8C2×D312yes—
106C22⋊Q8C2×D312yes—
107#117—31no—
108#117C2×D311yescharacteristic
109#107—32no—
110#106—32no—
111#103—32no—
112#103—32no—
113#103—32no—
114#103—32no—
115C22⋊C8—32no—
116C22⋊C8—32no—
117C23×C3C2×D411yescharacteristic
118C23×C3C2×D411yescharacteristic
119C23×C3C2×D411yescharacteristic
120C3×Q8—22no—
121C3×Q8—22no—
122C3×Q8—41no—
123C3×Q8—42no—
124C3×D4—41no—
125C3×D4—41no—
126C3×D4—41no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—42no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2411yescharacteristic
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—42no—
143Dic6—22no—
144Dic6—22no—
145Dic6—42no—
146C3⋊C8—22no—
147C3⋊C8—22no—
148#41—41no—
149#41—41no—
150C2×Q8—21no—
151C2×Q8—32no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—61no—
155C2×Q8—121no—
156C2×Q8—121no—
157C2×Q8#6711yescharacteristic
158C2×Q8C22×D312yes—
159C2×D4#6711yescharacteristic
160C2×D4#6711yescharacteristic
161C2×D4#6711yescharacteristic
162C22×C4—21no—
163C22×C4—61no—
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D311yescharacteristic
166C22×C4C22×D311yescharacteristic
167Q16—122no—
168Q16—122no—
169M16—121no—
170C2×C8—32no—
171C2×C8—32no—
172C2×C8—121no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—32no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4C22×D312yes—
179C22⋊C4—21no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—61no—
183C22⋊C4—62no—
184C22⋊C4—62no—
185C42—21no—
186C42—61no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3C22×D411yescharacteristic
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—42no—
198C3×C4—42no—
199C3×C4H(5,𝔽2)11yescharacteristic
200C3×C4H(5,𝔽2)11yescharacteristic
201Dic3—42no—
202Dic3—42no—
203C23C2×#6711yescharacteristic
204C23C2×#6711yescharacteristic
205C23C2×#6711yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—41no—
209Q8—42no—
210Q8—62no—
211Q8—62no—
212Q8—122no—
213D4—41no—
214D4—41no—
215D4—41no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—122no—
228C2×C4C2×#6711yescharacteristic
229C2×C4C2×#6711yescharacteristic
230C2×C4C2×#6711yescharacteristic
231C2×C4C23×D311yescharacteristic
232C8—62no—
233C8—62no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3#53411yescharacteristic
238C2×C3#57812yes—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22C22×#6711yescharacteristic
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—122no—
252C4—122no—
253C4#100411yescharacteristic
254C4#100411yescharacteristic
255C3#320211yescharacteristic, complemented, Sylow
256C2—41no—
257C2—41no—
258C2—41no—
259C2#580411yescharacteristic
260C2#583812yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#536D3
hypocenterC3#3202
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
4. #5228C2semidirect
5. #5052C2semidirect
6. #5051C2semidirect
7. #5050C2semidirect
8. #5045C2semidirect
9. #4996C2semidirect
255. C3#3202semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#58252
#52342
#52282
#50522
#50512
#50502
#50452
#49962
#32023