Finite Group Databasebeta

← #3306

Subgroups — #3306

Order 128.

221 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 88 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#547C211yesmaximal, characteristic, complemented
2#546C211yesmaximal, characteristic, complemented
3#489C211yesmaximal, characteristic, complemented
4#483C211yesmaximal, characteristic, complemented
5#481C211yesmaximal, characteristic, complemented
6#476C211yesmaximal, characteristic, complemented
7#469C211yesmaximal, characteristic, complemented
8#465C211yesmaximal, characteristic, complemented
9#463C211yesmaximal, characteristic, complemented
10#449C211yesmaximal, characteristic, complemented
11#447C211yesmaximal, characteristic, complemented
12#442C211yesmaximal, characteristic, complemented
13#435C211yesmaximal, characteristic, complemented
14#417C211yesmaximal, characteristic, complemented
15C2×#102C211yesmaximal, characteristic, complemented
16C2×#41—21no—
17C2×#41C2211yescharacteristic
18C22×D4C2211yescharacteristic
19C2×SD16C2211yescharacteristic, complemented
20C2×D8C2211yescharacteristic
21C2×D8C2211yescharacteristic
22C2×M16C2211yescharacteristic, complemented
23C22×C8C2211yescharacteristic
24C4⋊Q8C2211yescharacteristic, complemented
25#124C2211yescharacteristic
26#123—21no—
27#123—21no—
28#123—21no—
29C22⋊Q8—21no—
30C22⋊Q8C2211yescharacteristic
31#121—21no—
32#121C2211yescharacteristic
33#121C2211yescharacteristic
34#121C2211yescharacteristic, complemented
35#120—21no—
36C4×D4C2211yescharacteristic
37C4×D4C2211yescharacteristic
38C4×D4C2211yescharacteristic, complemented
39C2×C4⋊C4C2211yescharacteristic
40C2×C4⋊C4C2211yescharacteristic, complemented
41C2×C22⋊C4—21no—
42#106C2211yescharacteristic, complemented
43#106C2211yescharacteristic, complemented
44#106C2211yescharacteristic, complemented
45C4⋊C8C2211yescharacteristic, complemented
46#103C2211yescharacteristic, complemented
47#103C2211yescharacteristic, complemented
48#103C2211yescharacteristic, complemented
49#102C2211yescharacteristic
50#102C2211yescharacteristic
51#102C2211yescharacteristic
52#102C2211yescharacteristic
53#102C2211yescharacteristic
54#102C2211yescharacteristic
55#102C2211yescharacteristic, complemented
56C22⋊C8C2211yescharacteristic
57C22⋊C8C2211yescharacteristic, complemented
58#97C2211yescharacteristic, complemented
59C24—21no—
60#41—22no—
61#41—41no—
62#41—44no—
63C2×Q8—21no—
64C2×Q8C2311yescharacteristic
65C2×D4—22no—
66C2×D4—21no—
67C2×D4—21no—
68C2×D4—21no—
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—41no—
72C2×D4D411yescharacteristic
73C2×D4C2311yescharacteristic
74C2×D4C2311yescharacteristic
75C2×D4C2311yescharacteristic
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4C2311yescharacteristic
82C22×C4C2311yescharacteristic
83SD16—41no—
84D8—22no—
85D8—41no—
86M16—41no—
87C2×C8—41no—
88C2×C8C2311yescharacteristic
89C2×C8C2311yescharacteristic
90C2×C8C2311yescharacteristic
91C2×C8C2311yescharacteristic
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4D411yescharacteristic
97C4⋊C4C2311yescharacteristic
98C4⋊C4C2311yescharacteristic
99C4⋊C4C2311yescharacteristic
100C4⋊C4C2311yescharacteristic
101C22⋊C4—21no—
102C22⋊C4—21no—
103C22⋊C4—21no—
104C22⋊C4—21no—
105C22⋊C4—21no—
106C22⋊C4—21no—
107C22⋊C4—21no—
108C22⋊C4—21no—
109C22⋊C4D411yescharacteristic
110C22⋊C4D411yescharacteristic
111C42C2311yescharacteristic
112C23—21no—
113C23—21no—
114C23—21no—
115C23—42no—
116C23—42no—
117C23—81no—
118C23C2×D411yescharacteristic
119C23C2×D411yescharacteristic
120Q8—21no—
121Q8—21no—
122Q8—42no—
123D4—22no—
124D4—21no—
125D4—21no—
126D4—21no—
127D4—21no—
128D4—21no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—42no—
133D4—81no—
134D4#4112yescomplemented
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—81no—
149C2×C4C2×D411yescharacteristic
150C2×C4C2×D411yescharacteristic
151C2×C4C2×D411yescharacteristic
152C2×C4C2×D411yescharacteristic
153C2×C4C2411yescharacteristic
154C8—21no—
155C8—21no—
156C8—21no—
157C8—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—22no—
162C22—21no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—84no—
173C22AGL(1,ℤ/8)12yes—
174C22C22×D411yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—41no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4C2×#4111yescharacteristic
185C4H(5,𝔽2)11yescharacteristic
186C2—22no—
187C2—41no—
188C2—42no—
189C2—81no—
190C2—81no—
191C2#54611yescharacteristic
192C2C2×AGL(1,ℤ/8)11yescharacteristic
193C2#57711yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #547C2semidirect
2. #546C2semidirect
3. #489C2semidirect
4. #483C2semidirect
5. #481C2semidirect
6. #476C2semidirect
7. #469C2semidirect
8. #465C2semidirect
9. #463C2semidirect
10. #449C2semidirect
11. #447C2semidirect
12. #442C2semidirect
13. #435C2semidirect
14. #417C2semidirect
15. C2×#102C2semidirect
19. C2×SD16C22semidirect
22. C2×M16C22semidirect
24. C4⋊Q8C22semidirect
34. #121C22semidirect
38. C4×D4C22semidirect
40. C2×C4⋊C4C22semidirect
42. #106C22semidirect
43. #106C22semidirect
44. #106C22semidirect
45. C4⋊C8C22semidirect
46. #103C22semidirect
47. #103C22semidirect
48. #103C22semidirect
55. #102C22semidirect
57. C22⋊C8C22semidirect
58. #97C22semidirect
134. D4#41semidirect

Maximal subgroups

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

subgroupindex
#5472
#5462
#4892
#4832
#4812
#4762
#4692
#4652
#4632
#4492
#4472
#4422
#4352
#4172
C2×#1022