Finite Group Databasebeta

← #27187276

Subgroups — #27187276

Order 12928.

312 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 70 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#20266180C211yesmaximal, characteristic, complemented
2C2×#15692316C212yesmaximal, complemented
3C2×#15692315C212yesmaximal
4C4×#13108083C211yesmaximal, characteristic, complemented
5C2×#15692241C211yesmaximal, characteristic
6C2×#13108083C2212yes—
7C2×#13108083C2212yes—
8#15692322—21no—
9C4⋊#11717581—21no—
10#15692316—24no—
11#15692315—24no—
12#15692310—21no—
13C4×#11717581—21no—
14C22×Dic202C2211yescharacteristic
15C22×C22⋊D101C2211yescharacteristic
16#15692241—21no—
17#15692241—21no—
18#15692241—21no—
19#15692241—21no—
20C2×C22⋊Dic101C411yescharacteristic
21C2×#13108051C411yescharacteristic
22C2×#13108051C411yescharacteristic
23C2×C4×Dic101C2211yescharacteristic
24C2×C22⋊C4×C101C411yescharacteristic
25#13108083—22no—
26#13108083—22no—
27#13108083—22no—
28#13108083D414yes—
29C2×#11717581—22no—
30C2×#11717581—22no—
31C23×D101C2×C411yescharacteristic
32C2×Dic202—22no—
33C2×Dic202—21no—
34C2×Dic202—21no—
35C2×Dic202—22no—
36C2×C22⋊D101—22no—
37C2×C22⋊D101—21no—
38C2×C22⋊D101—21no—
39C2×C22⋊D101—22no—
40C22×Dic101C2×C411yescharacteristic
41C22×Dic101C2×C411yescharacteristic
42C22×Dic101C2311yescharacteristic
43C22⋊Dic101—21no—
44C22⋊Dic101—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—21no—
49C4×Dic101D411yescharacteristic
50C4×Dic101D411yescharacteristic
51C4×Dic101D411yescharacteristic
52C4×Dic101D411yescharacteristic
53C24×C101C2×C411yescharacteristic
54C22×C4×C101C2×C411yescharacteristic
55C22×C4×C101C2×C411yescharacteristic
56C22⋊C4×C101—21no—
57C22⋊C4×C101—21no—
58C23×C101—21no—
59C23×C101—22no—
60C23×C101—41no—
61C23×C101—41no—
62C23×C101C22×C411yescharacteristic
63C22×D101—21no—
64C22×D101—22no—
65C22×D101—41no—
66C22×D101—41no—
67C2×C4×C101—21no—
68C2×C4×C101—22no—
69C2×C4×C101—41no—
70C2×C4×C101C22⋊C411yescharacteristic
71C2×C4×C101C22⋊C411yescharacteristic
72C22⋊D101—42no—
73C22⋊D101—41no—
74C22⋊D101—41no—
75C2×Dic101—21no—
76C2×Dic101—22no—
77C2×Dic101—41no—
78C2×Dic101C22⋊C411yescharacteristic
79C2×Dic101C22⋊C411yescharacteristic
80C2×Dic101C2×D411yescharacteristic
81C2×Dic101C2×D412yes—
82C2×Dic101C2×D411yescharacteristic
83C2×Dic101#4112yes—
84Dic202—42no—
85Dic202—41no—
86Dic202—41no—
87#11717581—24no—
88#11717581—42no—
89C22×C101—21no—
90C22×C101—21no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101—41no—
94C22×C101—82no—
95C22×C101C2×C22⋊C411yescharacteristic
96C22×C101C4×D412yes—
97C2×D101—41no—
98C2×D101—41no—
99C2×D101—41no—
100C2×D101—82no—
101C4×C101—21no—
102C4×C101—21no—
103C4×C101—41no—
104Dic101—21no—
105Dic101—21no—
106Dic101—41no—
107Dic101#12112yes—
108Dic101#12411yescharacteristic
109Dic101#12711yescharacteristic
110C2×C101—22no—
111C2×C101—81no—
112C2×C101#39011yescharacteristic
113C2×C101#41311yescharacteristic
114C2×C101#41311yescharacteristic
115D101—81no—
116#1967—1011nomaximal, Sylow
117C101#196711yescharacteristic, complemented, Sylow
118C2×#124—1011no—
119#423—1011no—
120C2×#101—1012no—
121C2×#100—1012no—
122C4×M16—1011no—
123C22×Q8—1011no—
124C22×D4—1011no—
125C2×M16—1012no—
126C2×M16—1012no—
127#124—2021no—
128#124—2021no—
129#124—2021no—
130#124—2021no—
131C2×C22⋊C4—1011no—
132C2×C22⋊C4—1011no—
133C2×C22⋊C4—1011no—
134C2×C22⋊C4#8793811yescharacteristic
135C2×C42—1011no—
136C4⋊C8—2021no—
137C4⋊C8—2021no—
138#101—2024no—
139#100—2024no—
140#97—2021no—
141C4×C8—2021no—
142C24—1011no—
143C24C2×#8793811yescharacteristic
144C2×Q8—2022no—
145C2×Q8—2021no—
146C2×Q8—2021no—
147C2×Q8—2022no—
148C2×D4—2022no—
149C2×D4—2021no—
150C2×D4—2021no—
151C2×D4—2022no—
152C22×C4—1011no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4C2×#8793811yescharacteristic
156C22×C4C2×#8793811yescharacteristic
157M16—1014no—
158M16—2022no—
159M16—2022no—
160M16—2022no—
161C2×C8—2022no—
162C2×C8—2022no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C22⋊C4—2021no—
169C22⋊C4—2021no—
170C22⋊C4—2021no—
171C42—1011no—
172C42—1011no—
173C42—1011no—
174C42—1011no—
175C23—21no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—4041no—
180C23—4041no—
181C23—2022no—
182C23—2021no—
183C23C22×#8793811yescharacteristic
184Q8—4041no—
185Q8—4042no—
186Q8—4041no—
187D4—4041no—
188D4—4042no—
189D4—4041no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—1011no—
194C2×C4—1011no—
195C2×C4—1012no—
196C2×C4—1012no—
197C2×C4—1011no—
198C2×C4—1011no—
199C2×C4—4041no—
200C2×C4—2021no—
201C2×C4—2022no—
202C2×C4C22⋊#8793811yescharacteristic
203C2×C4C22⋊#8793811yescharacteristic
204C8—4042no—
205C8—2024no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—82no—
212C22—8082no—
213C22—4041no—
214C22—4041no—
215C22—4041no—
216C22C2×C22⋊#8793811yescharacteristic
217C22D4×#8793812yes—
218C4—21no—
219C4—21no—
220C4—41no—
221C4—1011no—
222C4—1012no—
223C4—1011no—
224C4—4041no—
225C4—2021no—
226C4—2021no—
227C2—22no—
228C2—81no—
229C2—8081no—
230C2#2026612911yescharacteristic
231C2#2026616311yescharacteristic
232C2#2026616411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C22⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×C22⋊D101C22
cosocleC22×Dic101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101#1967
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20266180C2semidirect
2. C2×#15692316C2semidirect
4. C4×#13108083C2semidirect
117. C101#1967semidirect

Maximal subgroups

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

subgroupindex
#202661802
C2×#156923162
C2×#156923152
C4×#131080832
C2×#156922412
#1967101