Finite Group Databasebeta

← #69144

Subgroups — #69144

Order 384.

264 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 55 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#5656C211yesmaximal, characteristic
2#4745C211yesmaximal, characteristic, complemented
3#4740C211yesmaximal, characteristic, complemented
4C3×#352C211yesmaximal, characteristic, complemented
5#4539C211yesmaximal, characteristic, complemented
6#4475C211yesmaximal, characteristic, complemented
7#4471C211yesmaximal, characteristic, complemented
8#2120—31nomaximal, Sylow
9C2×#237C2211yescharacteristic
10C22×C4×D3C411yescharacteristic, complemented
11C3×#123C2211yescharacteristic
12#937C2211yescharacteristic
13#929C411yescharacteristic, complemented
14D3×M16—21no—
15#896—21no—
16#893—21no—
17#884—21no—
18#882C411yescharacteristic, complemented
19#67⋊C4—21no—
20#877—21no—
21#877C411yescharacteristic, complemented
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#833C2211yescharacteristic
26#824—21no—
27#806—21no—
28#525—31no—
29#413—31no—
30#410—31no—
31#352—31no—
32#352D311yescharacteristic, complemented
33AΣL(1,𝔽16)—31no—
34AΣL(1,𝔽16)—31no—
35C23×D3C2×C411yescharacteristic
36C2×C3×D4C2311yescharacteristic
37C22×C3×C4C2×C411yescharacteristic
38C2×#67—21no—
39C2×#67C2×C411yescharacteristic
40C2×#67D411yescharacteristic
41C22×Dic3C2×C411yescharacteristic
42C22×Dic3D411yescharacteristic
43#237—41no—
44#237—41no—
45C2×D12—21no—
46C2×C4×D3—21no—
47C2×C4×D3—21no—
48C2×C4×D3—41no—
49C2×C4×D3—41no—
50C2×C4×D3D411yescharacteristic
51C2×Dic6D411yescharacteristic
52C3×M16—21no—
53C3×C4⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4C2×C411yescharacteristic
57#217—21no—
58#217C2×C411yescharacteristic
59(C2×D3)⋊C4—21no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—41no—
65C4×Dic3—21no—
66C4×Dic3—41no—
67#208—21no—
68ΓL(1,𝔽25)—41no—
69D3×C8—41no—
70C2×#41—31no—
71C23×C4—31no—
72C2×M16—61no—
73#123—31no—
74#123C2×D311yescharacteristic
75C22⋊Q8—61no—
76#121—61no—
77#120—31no—
78#120—31no—
79C4×D4—61no—
80C4×D4—61no—
81#117—31no—
82#117—61no—
83#101—61no—
84#100—31no—
85#100C2×D311yescharacteristic
86#99—31no—
87#99—61no—
88#99C2×D311yescharacteristic
89C23×C3C22×C411yescharacteristic
90C23×C3C2×D411yescharacteristic
91C22×D3—21no—
92C22×D3—42no—
93C22×D3—41no—
94C22×D3C22⋊C411yescharacteristic
95C3×D4—41no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4C2×D411yescharacteristic
101#67—41no—
102#67—41no—
103#67—42no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—41no—
110C2×Dic3C22⋊C411yescharacteristic
111C2×Dic3C22⋊C411yescharacteristic
112C2×Dic3C22⋊C411yescharacteristic
113D12—42no—
114C4×D3—21no—
115C4×D3—21no—
116C4×D3—42no—
117C4×D3—41no—
118C4×D3—41no—
119Dic6—41no—
120C3×C8—41no—
121C3⋊C8—41no—
122C24—31no—
123#41—121no—
124#41—121no—
125C2×Q8—31no—
126C2×D4—31no—
127C2×D4—31no—
128C2×D4—61no—
129C2×D4—61no—
130C2×D4C22×D311yescharacteristic
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—61no—
135C22×C4—61no—
136C22×C4—121no—
137C22×C4—121no—
138C22×C4C4×D311yescharacteristic
139M16—21no—
140M16—61no—
141M16—121no—
142C2×C8—121no—
143C4⋊C4—21no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—121no—
147C22⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—31no—
150C22⋊C4—61no—
151C22⋊C4—61no—
152C22⋊C4—61no—
153C22⋊C4—61no—
154C22⋊C4C4×D311yescharacteristic
155C42—61no—
156C42—121no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3C2×C22⋊C411yescharacteristic
162C2×D3—22no—
163C2×D3—21no—
164C2×D3—41no—
165C2×D3—42no—
166C2×D3—41no—
167C2×D3—41no—
168C2×D3—42no—
169C3×C4—21no—
170C3×C4—41no—
171C3×C4—41no—
172C3×C4—81no—
173Dic3—21no—
174Dic3—21no—
175Dic3—41no—
176Dic3—81no—
177Dic3#9911yescharacteristic
178Dic3#9911yescharacteristic
179C23—31no—
180C23—61no—
181C23—121no—
182C23—122no—
183C23C2×C4×D311yescharacteristic
184C23D3×D411yescharacteristic
185Q8—121no—
186D4—41no—
187D4—122no—
188D4—121no—
189D4—121no—
190D4—122no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—121no—
206C2×C4—121no—
207C2×C4—122no—
208C2×C4—121no—
209C2×C4D3×D411yescharacteristic
210C8—41no—
211C8—121no—
212C2×C3—21no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3C2×#9911yescharacteristic
216D3—41no—
217D3—42no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—62no—
223C22—61no—
224C22—121no—
225C22—121no—
226C22—122no—
227C22—122no—
228C22—121no—
229C22D3×C22⋊C411yescharacteristic
230C4—21no—
231C4—31no—
232C4—31no—
233C4—41no—
234C4—41no—
235C4—61no—
236C4—61no—
237C4—81no—
238C4—121no—
239C4—241no—
240C3#212011yescharacteristic, complemented, Sylow
241C2—21no—
242C2—41no—
243C2—41no—
244C2—122no—
245C2—121no—
246C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC3×#352C2
socleC2×C3C2×#99
prime radical#5656C2
cosocleC2×C3×D4C23
hypercenter#352D3
hypocenterC3#2120
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #4745C2semidirect
3. #4740C2semidirect
4. C3×#352C2semidirect
5. #4539C2semidirect
6. #4475C2semidirect
7. #4471C2semidirect
10. C22×C4×D3C4semidirect
13. #929C4semidirect
18. #882C4semidirect
21. #877C4semidirect
32. #352D3semidirect
240. C3#2120semidirect

Maximal subgroups

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

subgroupindex
#56562
#47452
#47402
C3×#3522
#45392
#44752
#44712
#21203