Finite Group Databasebeta

← #27186896

Subgroups — #27186896

Order 12928.

320 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 92 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×#15692320C212yesmaximal
2C4×#13108084C211yesmaximal, characteristic, complemented
3#20266113C211yesmaximal, characteristic, complemented
4#20266047C212yesmaximal, complemented
5C4×#13108075C211yesmaximal, characteristic
6C2×#13108084C2212yes—
7C2×C4×#87938C2212yes—
8#15692320C418yescomplemented
9#15692310—21no—
10C4×#11717581—21no—
11(C2×#87938)⋊C4—21no—
12(C2×#87938)⋊C4—21no—
13C2×#13108075C2211yescharacteristic
14Q8×Dic101C411yescharacteristic, complemented
15C4×D404—21no—
16C4×D404C411yescharacteristic, complemented
17#15692217—21no—
18#15692217C411yescharacteristic, complemented
19#15692213—21no—
20#15692213C2211yescharacteristic
21C42×D101—21no—
22C42×D101C2211yescharacteristic
23C4×Q8×C101C411yescharacteristic, complemented
24C22×#87938—22no—
25#13108084—22no—
26#13108084C2×C414yescomplemented
27C2×#11717581—22no—
28C4×#87938—22no—
29C4×#87938C2×C414yescomplemented
30#13108075—22no—
31#13108075C2×C414yes—
32C2×D404—21no—
33C2×D404C2×C411yescharacteristic
34C2×C4×D101—21no—
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×C4×D101C2311yescharacteristic
40C4⋊Dic101—21no—
41C4⋊Dic101C2×C411yescharacteristic
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C4×Dic101C2×C411yescharacteristic
48C4×Dic101D411yescharacteristic
49C4×Dic101D411yescharacteristic
50C2×Q8×C101C2×C411yescharacteristic
51C4⋊C4×C101—21no—
52C4⋊C4×C101C2×C411yescharacteristic
53C42×C101—21no—
54C42×C101C2×C411yescharacteristic
55C22×D101—21no—
56C22×D101C2×D411yescharacteristic
57C2×#87938—24no—
58C2×#87938—42no—
59C2×#87938—42no—
60Q8×C101—21no—
61Q8×C101C4212yes—
62C2×C4×C101—21no—
63C2×C4×C101—21no—
64C2×C4×C101C22⋊C411yescharacteristic
65C2×C4×C101C22⋊C411yescharacteristic
66C2×C4×C101C22×C411yescharacteristic
67C2×Dic101—21no—
68C2×Dic101C22⋊C411yescharacteristic
69C2×Dic101C22⋊C411yescharacteristic
70C2×Dic101C2×D411yescharacteristic
71D404—24no—
72D404—21no—
73D404C4212yes—
74C4×D101—24no—
75C4×D101—41no—
76C4×D101—41no—
77C4×D101—42no—
78C4×D101—42no—
79C4×D101C22×C412yes—
80C4×D101#4112yes—
81#11717581—24no—
82C22×C101C2×C22⋊C411yescharacteristic
83C2×D101—21no—
84C2×D101—21no—
85C2×D101—22no—
86C2×D101—41no—
87C2×D101—41no—
88C2×D101C4×D412yes—
89#87938—44no—
90C4×C101—22no—
91C4×C101—21no—
92C4×C101—21no—
93C4×C101—41no—
94C4×C101C2×C4211yescharacteristic
95C4×C101#11711yescharacteristic
96Dic101—21no—
97Dic101—21no—
98Dic101—41no—
99Dic101C4×D412yes—
100C2×C101C4×C22⋊C411yescharacteristic
101C2×C101#42111yescharacteristic
102C2×C101#42111yescharacteristic
103D101—22no—
104D101—42no—
105#1759—1011nomaximal, Sylow
106C101#175911yescharacteristic, complemented, Sylow
107C4×#41—1011no—
108C2×#104—1012no—
109C4×M16—1011no—
110#376—1011no—
111#339—1012no—
112C2×#41—1011no—
113C2×M16—1012no—
114C4×Q8—1011no—
115C4×Q8#8793811yescharacteristic, complemented
116C4×D4—1011no—
117C4×D4—1011no—
118C4×D4—2021no—
119C4×D4—2021no—
120#117—1011no—
121#117—2021no—
122C2×C42—1011no—
123C2×C42—1012no—
124C2×C42—2021no—
125#104—1018no—
126#97—2021no—
127C4×C8—2021no—
128#95—2021no—
129#95—2021no—
130#41—1014no—
131#41—2022no—
132C2×Q8C2×#8793811yescharacteristic
133C2×D4—1011no—
134C2×D4—2021no—
135C22×C4—1011no—
136C22×C4—1011no—
137C22×C4—1011no—
138C22×C4—2021no—
139C22×C4—2021no—
140C22×C4—2022no—
141C22×C4—2021no—
142M16—1014no—
143M16—2022no—
144C2×C8—2022no—
145C4⋊C4—21no—
146C4⋊C4—1011no—
147C4⋊C4—2021no—
148C4⋊C4C2×#8793811yescharacteristic
149C22⋊C4—2021no—
150C22⋊C4—2021no—
151C22⋊C4—2021no—
152C42—21no—
153C42—1011no—
154C42—1011no—
155C42—1011no—
156C42—1014no—
157C42—2021no—
158C42—2021no—
159C42—2022no—
160C42C2×#8793811yescharacteristic
161C23—1011no—
162C23—2021no—
163Q8—21no—
164Q8C4×#8793812yes—
165D4—1012no—
166D4—2024no—
167D4—2021no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—1011no—
171C2×C4—1012no—
172C2×C4—1012no—
173C2×C4—1011no—
174C2×C4—1011no—
175C2×C4—4042no—
176C2×C4—4041no—
177C2×C4—4041no—
178C2×C4—4042no—
179C2×C4—4042no—
180C2×C4—4042no—
181C2×C4—2024no—
182C2×C4—2024no—
183C2×C4—2021no—
184C2×C4C22⋊#8793811yescharacteristic
185C2×C4C22⋊#8793811yescharacteristic
186C2×C4C22×#8793811yescharacteristic
187C8—2024no—
188C22—1012no—
189C22—4041no—
190C22—4041no—
191C22—2022no—
192C22—2021no—
193C22—2021no—
194C22C2×C22⋊#8793811yescharacteristic
195C4—22no—
196C4—21no—
197C4—21no—
198C4—41no—
199C4—1012no—
200C4—4044no—
201C4—4041no—
202C4—2021no—
203C4—2021no—
204C4C2×C4×#8793811yescharacteristic
205C4#1569233411yescharacteristic
206C2—4042no—
207C2—2022no—
208C2C4×C22⋊#8793811yescharacteristic
209C2#2026617411yescharacteristic
210C2#2026617611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC4×C101C2×C42
FrattiniC2×C4C22×#87938
FittingC4×Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×#13108075C22
cosocleC2×C4×D101C23
hypercenterC4×Q8#87938
hypocenterC101#1759
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
2. C4×#13108084C2semidirect
3. #20266113C2semidirect
4. #20266047C2semidirect
8. #15692320C4semidirect
14. Q8×Dic101C4semidirect
16. C4×D404C4semidirect
18. #15692217C4semidirect
23. C4×Q8×C101C4semidirect
26. #13108084C2×C4semidirect
29. C4×#87938C2×C4semidirect
106. C101#1759semidirect
115. C4×Q8#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923202
C4×#131080842
#202661132
#202660472
C4×#131080752
#1759101