Finite Group Databasebeta

← #69145

Subgroups — #69145

Order 384.

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

normal subgroups only

This group has 234 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#5648C211yesmaximal, characteristic
2#4745C211yesmaximal, characteristic
3#4740C211yesmaximal, characteristic
4C3×#352C211yesmaximal, characteristic, complemented
5#4539C211yesmaximal, characteristic, complemented
6#4476C211yesmaximal, characteristic, complemented
7#4472C211yesmaximal, characteristic, complemented
8#2123—31nomaximal, Sylow
9C2×#237C2211yescharacteristic
10C22×Dic6C411yescharacteristic, complemented
11C3×#123C2211yescharacteristic
12#935C2211yescharacteristic
13#929C411yescharacteristic, complemented
14D3×M16—21no—
15#896—21no—
16#888—21no—
17#886C411yescharacteristic, complemented
18#885—21no—
19#878—21no—
20#877—21no—
21#877C411yescharacteristic, complemented
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#833C2211yescharacteristic
26#824—21no—
27#806—21no—
28#536—31no—
29#413—31no—
30#410—31no—
31#352—31no—
32#352—31no—
33#352—31no—
34#352D311yescharacteristic, complemented
35C2×C3×D4C2311yescharacteristic
36C22×C3×C4C2×C411yescharacteristic
37C2×#67C2×C411yescharacteristic
38C2×#67D411yescharacteristic
39C22×Dic3C2×C411yescharacteristic
40C22×Dic3C2×C411yescharacteristic
41C22×Dic3D411yescharacteristic
42#237—41no—
43#237—41no—
44C2×C4×D3D411yescharacteristic
45C2×Dic6—21no—
46C2×Dic6—21no—
47C2×Dic6—41no—
48C2×Dic6—41no—
49C2×Dic6D411yescharacteristic
50C3×M16—21no—
51C3×C4⋊C4—21no—
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4C2×C411yescharacteristic
55#217—21no—
56#217—21no—
57#217C2×C411yescharacteristic
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—21no—
60C4⋊Dic3—21no—
61Dic3⋊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—
71C22×Q8—31no—
72C2×M16—61no—
73C4⋊Q8—61no—
74#124—61no—
75#123—31no—
76#123—31no—
77#123—31no—
78#123C2×D311yescharacteristic
79C22⋊Q8—61no—
80C22⋊Q8—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×D3C22⋊C411yescharacteristic
92C3×D4—41no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4C2×D411yescharacteristic
98#67—41no—
99#67—41no—
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3C22⋊C411yescharacteristic
109C2×Dic3C22⋊C411yescharacteristic
110C2×Dic3C22⋊C411yescharacteristic
111C4×D3—21no—
112C4×D3—21no—
113Dic6—41no—
114Dic6—42no—
115Dic6—41no—
116Dic6—41no—
117C3×C8—41no—
118C3⋊C8—41no—
119#41—121no—
120#41—121no—
121C2×Q8—31no—
122C2×Q8—61no—
123C2×Q8—61no—
124C2×Q8—121no—
125C2×Q8—121no—
126C2×D4—31no—
127C2×D4—31no—
128C2×D4C22×D311yescharacteristic
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—31no—
132C22×C4—31no—
133C22×C4C4×D311yescharacteristic
134M16—21no—
135M16—61no—
136M16—121no—
137C2×C8—121no—
138C4⋊C4—21no—
139C4⋊C4—61no—
140C4⋊C4—61no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—121no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—31no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4C4×D311yescharacteristic
152C42—61no—
153C42—121no—
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3C2×C22⋊C411yescharacteristic
159C2×D3—21no—
160C2×D3—41no—
161C3×C4—21no—
162C3×C4—41no—
163C3×C4—41no—
164C3×C4—81no—
165Dic3—21no—
166Dic3—21no—
167Dic3—41no—
168Dic3—41no—
169Dic3—81no—
170Dic3#9911yescharacteristic
171Dic3#9911yescharacteristic
172C23—31no—
173C23C2×C4×D311yescharacteristic
174C23D3×D411yescharacteristic
175Q8—121no—
176Q8—121no—
177Q8—121no—
178Q8—122no—
179D4—41no—
180D4—121no—
181D4—121no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4D3×D411yescharacteristic
200C8—41no—
201C8—121no—
202C2×C3—21no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3C2×#9911yescharacteristic
206D3—41no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—61no—
212C22—121no—
213C22D3×C22⋊C411yescharacteristic
214C4—21no—
215C4—31no—
216C4—31no—
217C4—41no—
218C4—41no—
219C4—61no—
220C4—61no—
221C4—81no—
222C4—121no—
223C4—121no—
224C4—241no—
225C3#212311yescharacteristic, complemented, Sylow
226C2—21no—
227C2—41no—
228C2—41no—
229C2—121no—
230C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC3×#352C2
socleC2×C3C2×#99
prime radicalC2×#237C22
cosocleC2×C3×D4C23
hypercenter#352D3
hypocenterC3#2123
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
4. C3×#352C2semidirect
5. #4539C2semidirect
6. #4476C2semidirect
7. #4472C2semidirect
10. C22×Dic6C4semidirect
13. #929C4semidirect
17. #886C4semidirect
21. #877C4semidirect
34. #352D3semidirect
225. C3#2123semidirect

Maximal subgroups

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

subgroupindex
#56482
#47452
#47402
C3×#3522
#45392
#44762
#44722
#21233