Finite Group Databasebeta

← #80440

Subgroups — #80440

Order 384.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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#5728C211yesmaximal, characteristic, complemented
2#5557C211yesmaximal, characteristic
3C3×#461C211yesmaximal, characteristic, complemented
4#5174C212yesmaximal
5#5166C212yesmaximal, complemented
6#5090C211yesmaximal, characteristic, complemented
7#5008C211yesmaximal, characteristic, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4816C212yesmaximal, complemented
11#4796C212yesmaximal, complemented
12#3252—31nomaximal, Sylow
13C2×C3×SD16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#946C2212yes—
17Q8×Dic3C2212yes—
18C2×#216C2212yes—
19#938C2211yescharacteristic
20#936C2211yescharacteristic
21C4×#67—21no—
22#926—21no—
23C2×ΓL(1,𝔽25)C2212yes—
24#897—21no—
25#897C2211yescharacteristic
26#892—21no—
27#890—21no—
28#890C2211yescharacteristic
29#887—21no—
30#883—21no—
31#879—21no—
32C42×D3C2211yescharacteristic
33#869C2211yescharacteristic
34C4×Dic6C2211yescharacteristic
35C3×C4⋊C8C2211yescharacteristic
36C3×#103C2212yes—
37#835C2212yes—
38#832C2212yes—
39#818C2212yes—
40#816C2212yes—
41Dic3⋊C8C2212yes—
42#808C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#533—31no—
45#529—31no—
46#479—32no—
47#477—31no—
48#476—31no—
49#470—32no—
50#468—32no—
51#464—32no—
52#462—31no—
53#461D311yescharacteristic, complemented
54#423—31no—
55C2×C3×Q8C2312yes—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—21no—
59C2×C4×D3D411yescharacteristic
60C2×C4×D3D411yescharacteristic
61C2×C4×D3C2311yescharacteristic
62C2×Dic6—21no—
63C2×Dic6C2311yescharacteristic
64C3×SD16—42no—
65C2×C3×C8C2312yes—
66C3×C4⋊C4—21no—
67C3×C4⋊C4—21no—
68C3×C4⋊C4C2311yescharacteristic
69C3×C22⋊C4—21no—
70C3×C42C2311yescharacteristic
71#217—21no—
72#217—21no—
73#217—21no—
74#216—42no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—22no—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3—22no—
81C4⋊Dic3C2312yes—
82C4⋊Dic3C2311yescharacteristic
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—22no—
86C4×Dic3—21no—
87C4×Dic3—22no—
88C4×Dic3D411yescharacteristic
89C4×Dic3D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91C2×C3⋊C8C2312yes—
92ΓL(1,𝔽25)—42no—
93C2×Q16—32no—
94C2×SD16C2×D312yes—
95C2×M16—32no—
96C4⋊Q8—31no—
97C4⋊Q8C2×D311yescharacteristic
98#126—61no—
99#125—31no—
100#125—61no—
101#124—31no—
102#123—61no—
103C22⋊Q8—32no—
104C22⋊Q8—31no—
105C22⋊Q8—61no—
106#121—31no—
107C4×Q8—32no—
108C4×Q8—31no—
109C4×Q8—61no—
110C4×D4—61no—
111C4×D4C2×D311yescharacteristic
112#117—61no—
113#117—61no—
114C2×C42—31no—
115#107—32no—
116#106—32no—
117C4⋊C8—32no—
118C4⋊C8—31no—
119C4⋊C8C2×D311yescharacteristic
120#103—32no—
121#103—32no—
122#103C2×D312yes—
123#102—32no—
124C23×C3—21no—
125C22×D3C2×D411yescharacteristic
126C3×Q8—22no—
127C3×Q8—22no—
128C3×D4—21no—
129C3×D4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—42no—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—42no—
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3C2×D411yescharacteristic
141C2×Dic3C2×D411yescharacteristic
142C2×Dic3C2×D411yescharacteristic
143C4×D3—22no—
144C4×D3—21no—
145C4×D3—21no—
146C4×D3—41no—
147Dic6—21no—
148Dic6—21no—
149Dic6—42no—
150C3×C8—22no—
151C3⋊C8—22no—
152C2×Q8—31no—
153C2×Q8—61no—
154C2×Q8C22×D312yes—
155C2×D4—61no—
156C2×D4C22×D311yescharacteristic
157C22×C4—21no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161Q16—122no—
162SD16—42no—
163M16—122no—
164C2×C8—32no—
165C2×C8C22×D312yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—32no—
169C4⋊C4—31no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—62no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—62no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C22⋊C4—61no—
183C22⋊C4—61no—
184C42—31no—
185C42—31no—
186C42—31no—
187C42—62no—
188C42—61no—
189C42C22×D311yescharacteristic
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3C22×D411yescharacteristic
194C2×D3—21no—
195C2×D3—21no—
196C2×D3—21no—
197C3×C4—21no—
198C3×C4—42no—
199C3×C4—41no—
200C3×C4#13712yes—
201C3×C4H(5,𝔽2)11yescharacteristic
202C3×C42−1+411yescharacteristic
203Dic3—21no—
204Dic3—21no—
205Dic3—21no—
206Dic3—41no—
207Dic3—42no—
208Dic3—41no—
209C23—21no—
210C23—31no—
211Q8—22no—
212Q8—22no—
213Q8—61no—
214Q8—61no—
215Q8—122no—
216D4—21no—
217D4—21no—
218D4—122no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—42no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—62no—
231C2×C4—121no—
232C2×C4D3×D411yescharacteristic
233C2×C4D3×D411yescharacteristic
234C2×C4C23×D311yescharacteristic
235C8—22no—
236C8—62no—
237C2×C3—81no—
238C2×C3#53711yescharacteristic
239C2×C3C2×#13711yescharacteristic
240C2×C3#57511yescharacteristic
241D3—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—61no—
246C22—61no—
247C22—61no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—42no—
252C4—61no—
253C4—61no—
254C4—61no—
255C4—121no—
256C4—122no—
257C4—121no—
258C4#91512yes—
259C4#100411yescharacteristic
260C4#101011yescharacteristic
261C3#325211yescharacteristic, complemented, Sylow
262C2—81no—
263C2—121no—
264C2#560811yescharacteristic
265C2C2×#91511yescharacteristic
266C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radical#938C22
hypercenter#461D3
hypocenterC3#3252
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#461C2semidirect
5. #5166C2semidirect
6. #5090C2semidirect
7. #5008C2semidirect
8. #4850C2semidirect
9. #4837C2semidirect
10. #4816C2semidirect
11. #4796C2semidirect
53. #461D3semidirect
261. C3#3252semidirect

Maximal subgroups

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

subgroupindex
#57282
#55572
C3×#4612
#51742
#51662
#50902
#50082
#48502
#48372
#48162
#47962
#32523