Finite Group Databasebeta

← #2424

Subgroups — #2424

Order 128.

461 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×#121C211yesmaximal, characteristic, complemented
2C2×#121C211yesmaximal, characteristic, complemented
3C2×#120C211yesmaximal, characteristic, complemented
4C2×#120C211yesmaximal, characteristic, complemented
5C2×C4×D4C211yesmaximal, characteristic, complemented
6C22×C22⋊C4C211yesmaximal, characteristic, complemented
7#396C211yesmaximal, characteristic, complemented
8#394C211yesmaximal, characteristic, complemented
9#394C211yesmaximal, characteristic, complemented
10#392C211yesmaximal, characteristic, complemented
11#390C211yesmaximal, characteristic, complemented
12#388C211yesmaximal, characteristic, complemented
13#386C211yesmaximal, characteristic, complemented
14#386C211yesmaximal, characteristic, complemented
15#385C211yesmaximal, characteristic, complemented
16C25C2211yescharacteristic
17C22×D4C2211yescharacteristic
18C22×D4C2211yescharacteristic
19C22×D4C2211yescharacteristic
20C22×D4C2211yescharacteristic
21C22×D4C2211yescharacteristic
22C22×D4C2211yescharacteristic
23C22×D4C2211yescharacteristic, complemented
24C23×C4C2211yescharacteristic
25C23×C4C2211yescharacteristic
26C23×C4C2211yescharacteristic, complemented
27#121—24no—
28#121—24no—
29#120—24no—
30#120—24no—
31C4×D4—24no—
32C2×C4⋊C4C2211yescharacteristic, complemented
33C2×C4⋊C4C2211yescharacteristic, complemented
34C2×C4⋊C4C2211yescharacteristic, complemented
35C2×C4⋊C4C2211yescharacteristic, complemented
36C2×C22⋊C4—24no—
37C2×C22⋊C4C2211yescharacteristic
38C2×C22⋊C4C2211yescharacteristic
39C2×C22⋊C4C2211yescharacteristic
40C2×C22⋊C4C2211yescharacteristic
41C2×C22⋊C4C2211yescharacteristic
42C2×C22⋊C4C2211yescharacteristic
43C2×C22⋊C4C2211yescharacteristic
44C2×C22⋊C4C2211yescharacteristic
45C2×C22⋊C4C2211yescharacteristic
46C2×C22⋊C4C2211yescharacteristic
47C2×C22⋊C4C2211yescharacteristic, complemented
48C2×C22⋊C4C2211yescharacteristic, complemented
49C2×C22⋊C4C2211yescharacteristic, complemented
50C2×C22⋊C4C2211yescharacteristic, complemented
51C2×C22⋊C4C2211yescharacteristic, complemented
52C2×C42C2211yescharacteristic, complemented
53#95C2211yescharacteristic, complemented
54#95C2211yescharacteristic, complemented
55#95C2211yescharacteristic, complemented
56#95C2211yescharacteristic, complemented
57C24—22no—
58C24—22no—
59C24—22no—
60C24—22no—
61C24—22no—
62C24—41no—
63C24—41no—
64C24C2311yescharacteristic
65C24C2311yescharacteristic
66C24C2311yescharacteristic
67C24C2311yescharacteristic
68C24C2311yescharacteristic
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—22no—
72C2×D4—22no—
73C2×D4—22no—
74C2×D4—22no—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4D414yescomplemented
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4C2311yescharacteristic
100C22×C4C2311yescharacteristic
101C22×C4C2311yescharacteristic
102C22×C4C2311yescharacteristic
103C22×C4C2311yescharacteristic
104C22×C4C2311yescharacteristic
105C22×C4C2311yescharacteristic
106C22×C4C2311yescharacteristic
107C22×C4C2311yescharacteristic
108C22×C4C2311yescharacteristic
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—41no—
113C22⋊C4—22no—
114C22⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—41no—
126C22⋊C4D414yescomplemented
127C22⋊C4D414yescomplemented
128C42—22no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—28no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—44no—
160C23—44no—
161C23—44no—
162C23—44no—
163C23—44no—
164C23—44no—
165C23C2×D412yes—
166C23C2×D412yes—
167C23C2×D412yes—
168C23C2×D412yes—
169C23#4112yes—
170C23#4112yes—
171C23C2411yescharacteristic
172D4—28no—
173D4—44no—
174D4—44no—
175D4—44no—
176D4—44no—
177D4—44no—
178D4—44no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4C2×D412yes—
208C2×C4C2×D412yes—
209C2×C4C2×D412yes—
210C2×C4C2×D412yes—
211C2×C4C2×D412yes—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—416no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22#12114yescomplemented
248C22C22×D411yescharacteristic
249C22C22×D411yescharacteristic
250C22C22×D411yescharacteristic
251C22C2×#4111yescharacteristic
252C22C2×#4111yescharacteristic
253C22H(5,𝔽2)11yescharacteristic
254C22H(5,𝔽2)11yescharacteristic
255C4—22no—
256C4—22no—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—41no—
265C2—24no—
266C2—42no—
267C2—42no—
268C2—42no—
269C2—81no—
270C2C2×#12111yescharacteristic
271C2#52511yescharacteristic
272C2#53811yescharacteristic
273C2D4211yescharacteristic
274C2D4211yescharacteristic
275C2#54611yescharacteristic
276C2#54611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#121C2semidirect
2. C2×#121C2semidirect
3. C2×#120C2semidirect
4. C2×#120C2semidirect
5. C2×C4×D4C2semidirect
6. C22×C22⋊C4C2semidirect
7. #396C2semidirect
8. #394C2semidirect
9. #394C2semidirect
10. #392C2semidirect
11. #390C2semidirect
12. #388C2semidirect
13. #386C2semidirect
14. #386C2semidirect
15. #385C2semidirect
23. C22×D4C22semidirect
26. C23×C4C22semidirect
32. C2×C4⋊C4C22semidirect
33. C2×C4⋊C4C22semidirect
34. C2×C4⋊C4C22semidirect
35. C2×C4⋊C4C22semidirect
47. C2×C22⋊C4C22semidirect
48. C2×C22⋊C4C22semidirect
49. C2×C22⋊C4C22semidirect
50. C2×C22⋊C4C22semidirect
51. C2×C22⋊C4C22semidirect
52. C2×C42C22semidirect
53. #95C22semidirect
54. #95C22semidirect
55. #95C22semidirect
56. #95C22semidirect
89. C2×D4D4semidirect
126. C22⋊C4D4semidirect
127. C22⋊C4D4semidirect
247. C22#121semidirect

Maximal subgroups

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

subgroupindex
C2×#1212
C2×#1202
C2×C4×D42
C22×C22⋊C42
#3962
#3942
#3922
#3902
#3882
#3862
#3852