Finite Group Databasebeta

← #27186904

Subgroups — #27186904

Order 12928.

324 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 100 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×#15692318C212yesmaximal
2#20266136C211yesmaximal, characteristic
3C4×C4⋊#87938C211yesmaximal, characteristic
4#20266050C212yesmaximal
5C4×Q8×D101C211yesmaximal, characteristic
6C2×D101⋊C8C2212yes—
7C2×C4⋊#87938C2212yes—
8C2×C4×#87938—21no—
9#15692318C418yescomplemented
10#15692310—21no—
11#15692310—21no—
12(C2×#87938)⋊C4—21no—
13C2×Q8×D101C2211yescharacteristic
14Q8×Dic101C411yescharacteristic, complemented
15C4×Dic202—21no—
16C4×Dic202C411yescharacteristic, complemented
17#15692220—21no—
18#15692220C411yescharacteristic, complemented
19C4⋊C4×D101—21no—
20C4⋊C4×D101C2211yescharacteristic
21C42×D101—21no—
22C42×D101C2211yescharacteristic
23C4×Q8×C101C411yescharacteristic, complemented
24C22×#87938—22no—
25D101⋊C8—22no—
26D101⋊C8C2×C414yes—
27C2×#11717581—22no—
28C4⋊#87938—22no—
29C4⋊#87938C2×C414yes—
30C4×#87938—42no—
31C2×Dic202—21no—
32C2×Dic202C2×C411yescharacteristic
33Q8×D101—22no—
34Q8×D101C2×C414yes—
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×C4×D101C2311yescharacteristic
40C4⋊Dic101—21no—
41C4⋊Dic101C2×C411yescharacteristic
42#13108053—21no—
43#13108053—21no—
44#13108053—21no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C4×Dic101—21no—
48C4×Dic101C2×C411yescharacteristic
49C4×Dic101D411yescharacteristic
50C4×Dic101D411yescharacteristic
51C2×Q8×C101C2×C411yescharacteristic
52C4⋊C4×C101—21no—
53C4⋊C4×C101C2×C411yescharacteristic
54C42×C101—21no—
55C42×C101C2×C411yescharacteristic
56C22×D101C2×D411yescharacteristic
57C2×#87938—24no—
58C2×#87938—42no—
59C2×#87938—42no—
60Q8×C101—21no—
61Q8×C101C4212yescomplemented
62C2×C4×C101—21no—
63C2×C4×C101—21no—
64C2×C4×C101C22⋊C411yescharacteristic
65C2×C4×C101C22⋊C411yescharacteristic
66C2×C4×C101C22×C411yescharacteristic
67C2×Dic101—21no—
68C2×Dic101—21no—
69C2×Dic101C22⋊C411yescharacteristic
70C2×Dic101C22⋊C411yescharacteristic
71C2×Dic101C2×D411yescharacteristic
72C4×D101—24no—
73C4×D101—21no—
74C4×D101—21no—
75C4×D101—22no—
76C4×D101—41no—
77C4×D101—41no—
78C4×D101C22×C412yes—
79C4×D101#4112yes—
80Dic202—21no—
81Dic202—24no—
82Dic202C4212yescomplemented
83#11717581—24no—
84C22×C101C2×C22⋊C411yescharacteristic
85C2×D101C4×D412yes—
86C2×D101AGL(1,ℤ/8)11yescharacteristic
87C2×D101#13712yes—
88C2×D101#13711yescharacteristic
89#87938—44no—
90C4×C101—22no—
91C4×C101—21no—
92C4×C101—21no—
93C4×C101—41no—
94C4×C101C2×C4211yescharacteristic
95C4×C101#11711yescharacteristic
96Dic101—22no—
97Dic101—21no—
98Dic101—21no—
99Dic101—41no—
100Dic101C4×D412yes—
101C2×C101C4×C22⋊C411yescharacteristic
102C2×C101#41711yescharacteristic
103C2×C101#41911yescharacteristic
104D101#44012yes—
105D101#44112yes—
106#1763—1011nomaximal, Sylow
107C101#176311yescharacteristic, complemented, Sylow
108C2×C4×Q8—1011no—
109C2×#103—1012no—
110C2×#97—1011no—
111C4×C4⋊C4—1011no—
112#340—1012no—
113C22×Q8—1011no—
114C22×C8—1012no—
115C4×Q8—1011no—
116C4×Q8—1011no—
117C4×Q8—1011no—
118C4×Q8—2021no—
119C4×Q8—2021no—
120C4×Q8#8793811yescharacteristic, complemented
121C2×C4⋊C4—1012no—
122C2×C4⋊C4—1011no—
123C2×C4⋊C4—2021no—
124C2×C42—1011no—
125C2×C42—2021no—
126C2×C42—2021no—
127#103—1018no—
128#97—2021no—
129#97—2021no—
130#95—2021no—
131C2×Q8—1014no—
132C2×Q8—1011no—
133C2×Q8—2021no—
134C2×Q8—2022no—
135C2×Q8C2×#8793811yescharacteristic
136C22×C4—1011no—
137C22×C4—1011no—
138C22×C4—1011no—
139C22×C4—2021no—
140C22×C4—2021no—
141C22×C4—2022no—
142C2×C8—1014no—
143C2×C8—2022no—
144C2×C8—2022no—
145C4⋊C4—21no—
146C4⋊C4—1014no—
147C4⋊C4—1011no—
148C4⋊C4—2021no—
149C4⋊C4—2021no—
150C4⋊C4—2021no—
151C4⋊C4—2022no—
152C4⋊C4—2021no—
153C4⋊C4C2×#8793811yescharacteristic
154C42—21no—
155C42—1011no—
156C42—1011no—
157C42—1011no—
158C42—4042no—
159C42—2021no—
160C42—2021no—
161C42—2021no—
162C42C2×#8793811yescharacteristic
163C23—1011no—
164Q8—21no—
165Q8—1012no—
166Q8—2024no—
167Q8—2021no—
168Q8C4×#8793812yescomplemented
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—1011no—
172C2×C4—1012no—
173C2×C4—1012no—
174C2×C4—1011no—
175C2×C4—1011no—
176C2×C4—4041no—
177C2×C4—4041no—
178C2×C4—4042no—
179C2×C4—4042no—
180C2×C4—2021no—
181C2×C4—2021no—
182C2×C4—2022no—
183C2×C4—2024no—
184C2×C4—2024no—
185C2×C4—2021no—
186C2×C4—2021no—
187C2×C4C22⋊#8793811yescharacteristic
188C2×C4C22⋊#8793811yescharacteristic
189C2×C4C22×#8793811yescharacteristic
190C8—2024no—
191C22—1012no—
192C22—1011no—
193C22—1011no—
194C22—1012no—
195C22C2×C22⋊#8793811yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—1012no—
201C4—4044no—
202C4—4041no—
203C4—2022no—
204C4—2021no—
205C4—2021no—
206C4C2×C4×#8793811yescharacteristic
207C4#1569233411yescharacteristic
208C2—1012no—
209C2—1012no—
210C2C4×C22⋊#8793811yescharacteristic
211C2#2026616811yescharacteristic
212C2#2026617011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC4×C101C2×C42
FrattiniC2×C4C22×#87938
FittingC4×Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C2×D4
cosocleC2×C4×D101C23
hypercenterC4×Q8#87938
hypocenterC101#1763
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
9. #15692318C4semidirect
14. Q8×Dic101C4semidirect
16. C4×Dic202C4semidirect
18. #15692220C4semidirect
23. C4×Q8×C101C4semidirect
61. Q8×C101C42semidirect
82. Dic202C42semidirect
107. C101#1763semidirect
120. C4×Q8#87938semidirect
168. Q8C4×#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923182
#202661362
C4×C4⋊#879382
#202660502
C4×Q8×D1012
#1763101