Finite Group Databasebeta

← #27171151

Subgroups — #27171151

Order 12928.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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#20265645C211yesmaximal, characteristic
2#20265200C211yesmaximal, characteristic, complemented
3C2×#15692183C211yesmaximal, characteristic
4C2×#15692181C211yesmaximal, characteristic
5C4×#13108053C211yesmaximal, characteristic, complemented
6#20264936C211yesmaximal, characteristic, complemented
7#339×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9C2×#13108058C2211yescharacteristic
10#15692220C411yescharacteristic, complemented
11#15692220C411yescharacteristic, complemented
12#15692217C411yescharacteristic, complemented
13#15692217C411yescharacteristic, complemented
14#15692215—21no—
15#15692215—21no—
16#15692212—21no—
17C2×#13108053—21no—
18C2×C4×Dic101C2211yescharacteristic
19C2×C4×Dic101C2211yescharacteristic
20C42×D101—21no—
21#15692184—21no—
22#15692184—21no—
23#15692183—22no—
24#15692181—22no—
25#15692162—21no—
26C2×M16×C101C2211yescharacteristic
27#117×C101C2211yescharacteristic
28C2×C42×C101C2211yescharacteristic
29#13108074—21no—
30#13108074—21no—
31#13108074D412yes—
32#13108074Q812yes—
33C2×Dic202C2×C411yescharacteristic
34C2×D404C2×C411yescharacteristic
35C2×C22⋊D101—21no—
36C22×Dic101—21no—
37C22×Dic101D411yescharacteristic
38C22×Dic101D411yescharacteristic
39C2×C4×D101—21no—
40C2×C4×D101—21no—
41C2×C4×D101—21no—
42#13108058—21no—
43#13108058—21no—
44C2×#11717579—21no—
45#13108053—24no—
46#13108053—21no—
47#13108053—21no—
48#13108051—21no—
49#13108051—21no—
50C4×Dic101—21no—
51C4×Dic101—21no—
52C4×Dic101—21no—
53C4×Dic101—21no—
54C4×Dic101C2×C411yescharacteristic
55C4×Dic101C2×C411yescharacteristic
56C4×Dic101D411yescharacteristic
57C4×Dic101D411yescharacteristic
58C22×C4×C101—21no—
59C22×C4×C101C2311yescharacteristic
60M16×C101—21no—
61M16×C101—21no—
62C2×C8×C101—21no—
63C4⋊C4×C101C2×C411yescharacteristic
64C4⋊C4×C101C2×C411yescharacteristic
65C22⋊C4×C101—21no—
66C42×C101—21no—
67C42×C101—21no—
68C42×C101—21no—
69C23×C101C2×D411yescharacteristic
70C22×D101—21no—
71C2×C4×C101—22no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—41no—
75C2×C4×C101—41no—
76C2×C4×C101C22×C411yescharacteristic
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D411yescharacteristic
79C2×C4×C101C2×Q811yescharacteristic
80C2×C4×C101#4111yescharacteristic
81C2×C4×C101#4111yescharacteristic
82C22⋊D101—22no—
83C22⋊D101—22no—
84C2×Dic101—21no—
85C2×Dic101—22no—
86C2×Dic101—22no—
87C2×Dic101—41no—
88C2×Dic101—41no—
89C2×Dic101—41no—
90C2×Dic101C22⋊C411yescharacteristic
91C2×Dic101C22⋊C411yescharacteristic
92C2×Dic101C22⋊C411yescharacteristic
93C2×Dic101C22⋊C411yescharacteristic
94D404—21no—
95D404C4⋊C412yescomplemented
96C4×D101—22no—
97C4×D101—22no—
98C4×D101—42no—
99C4×D101—42no—
100Dic202—21no—
101Dic202C4⋊C412yescomplemented
102C8×C101—41no—
103#11717579—41no—
104C22×C101—21no—
105C22×C101—21no—
106C22×C101C2×C22⋊C411yescharacteristic
107C22×C101#12111yescharacteristic
108C22×C101C22⋊Q811yescharacteristic
109C2×D101—22no—
110C2×D101—41no—
111C2×D101—41no—
112C4×C101—42no—
113C4×C101—41no—
114C4×C101—41no—
115C4×C101C2×C4⋊C411yescharacteristic
116C4×C101#11711yescharacteristic
117C4×C101#12111yescharacteristic
118C4×C101C22⋊Q811yescharacteristic
119Dic101—22no—
120Dic101—21no—
121Dic101—21no—
122Dic101—21no—
123Dic101—42no—
124Dic101#10412yescomplemented
125C2×C101—21no—
126C2×C101—21no—
127C2×C101#38011yescharacteristic
128C2×C101C2×#10411yescharacteristic
129C2×C101#42111yescharacteristic
130D101—42no—
131#1806—1011nomaximal, Sylow
132C101#180611yescharacteristic, complemented, Sylow
133C4×#41—1011no—
134#423—1011no—
135C2×#104—1011no—
136C2×#104—1011no—
137C4×C4⋊C4—1011no—
138#339—1011no—
139#339D10111yescharacteristic, complemented
140C2×#41—1011no—
141C2×M16—1011no—
142C2×M16C2×D10111yescharacteristic
143C4×Q8—1011no—
144C4×Q8—1011no—
145C4×D4—1011no—
146C4×D4—1011no—
147C4×D4—2021no—
148C4×D4—2021no—
149#117—2021no—
150#117C2×D10111yescharacteristic
151C2×C4⋊C4—2021no—
152C2×C42—1011no—
153C2×C42—1011no—
154C2×C42—2021no—
155C2×C42C2×D10111yescharacteristic
156C4⋊C8—2021no—
157C4⋊C8—2021no—
158#104—2022no—
159#104—2022no—
160#95—2021no—
161#41—1012no—
162#41—1012no—
163#41—2021no—
164#41—2021no—
165C2×Q8—1011no—
166C2×D4—1011no—
167C2×D4—2021no—
168C22×C4—21no—
169C22×C4—1011no—
170C22×C4—1011no—
171C22×C4—2021no—
172C22×C4—2021no—
173C22×C4—2021no—
174C22×C4—2021no—
175C22×C4C22×D10111yescharacteristic
176M16—21no—
177M16—21no—
178M16—2021no—
179M16—2021no—
180C2×C8—21no—
181C2×C8—2021no—
182C4⋊C4—2021no—
183C4⋊C4—2021no—
184C4⋊C4—2024no—
185C4⋊C4C4×D10111yescharacteristic
186C4⋊C4C4×D10111yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—2021no—
189C22⋊C4—2021no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—1011no—
194C42—1011no—
195C42—1011no—
196C42—1011no—
197C42—2021no—
198C42—2021no—
199C42—2021no—
200C42—2021no—
201C23—2021no—
202C23D4×D10111yescharacteristic
203Q8—1012no—
204Q8—2021no—
205D4—1012no—
206D4—2022no—
207D4—2022no—
208D4—2021no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—1011no—
215C2×C4—1011no—
216C2×C4—1011no—
217C2×C4—1011no—
218C2×C4—4042no—
219C2×C4—4042no—
220C2×C4—4041no—
221C2×C4—4041no—
222C2×C4—4041no—
223C2×C4—2022no—
224C2×C4—2022no—
225C2×C4—2021no—
226C2×C4—2022no—
227C2×C4—2022no—
228C2×C4C2×C4×D10111yescharacteristic
229C2×C4D4×D10111yescharacteristic
230C2×C4D4×D10111yescharacteristic
231C2×C4Q8×D10111yescharacteristic
232C2×C4#1310807411yescharacteristic
233C2×C4#1310807411yescharacteristic
234C8—41no—
235C8—4041no—
236C22—21no—
237C22—21no—
238C22—4041no—
239C22—4041no—
240C22—2022no—
241C22C22⋊C4×D10111yescharacteristic
242C22#1569222811yescharacteristic
243C22#1569223311yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—1012no—
248C4—4042no—
249C4—2022no—
250C4—2021no—
251C4—2021no—
252C4—2021no—
253C4C4⋊C4×D10111yescharacteristic
254C4#1569221211yescharacteristic
255C4#1569222811yescharacteristic
256C4#1569223311yescharacteristic
257C2—21no—
258C2—21no—
259C2—4042no—
260C2#2026503611yescharacteristic
261C2#104×D10111yescharacteristic
262C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#339×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC2×#13108074C22
cosocleC22×C4×C101C23
hypercenter#339D101
hypocenterC101#1806
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #20265200C2semidirect
5. C4×#13108053C2semidirect
6. #20264936C2semidirect
7. #339×C101C2semidirect
10. #15692220C4semidirect
11. #15692220C4semidirect
12. #15692217C4semidirect
13. #15692217C4semidirect
95. D404C4⋊C4semidirect
101. Dic202C4⋊C4semidirect
124. Dic101#104semidirect
132. C101#1806semidirect
139. #339D101semidirect

Maximal subgroups

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

subgroupindex
#202656452
#202652002
C2×#156921832
C2×#156921812
C4×#131080532
#202649362
#339×C1012
#1806101