Finite Group Databasebeta

← #27186517

Subgroups — #27186517

Order 12928.

334 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 62 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#20266148C211yesmaximal, characteristic, complemented
2C2×(C2×#87938)⋊C4C211yesmaximal, characteristic, complemented
3#20266030C212yesmaximal, complemented
4#20266028C212yesmaximal, complemented
5#20265687C211yesmaximal, characteristic
6C23×#87938—21no—
7C2×#13108084—21no—
8#15692312C2212yes—
9C22⋊#11717581—21no—
10(C2×#87938)⋊C4—21no—
11(C2×#87938)⋊C4C2212yes—
12C2×#13108073C411yescharacteristic, complemented
13C22×C4×D101C2211yescharacteristic
14#15692237—21no—
15#15692235C2211yescharacteristic
16#15692234C411yescharacteristic, complemented
17#15692226C2211yescharacteristic
18#15692223—21no—
19D4×Dic101—21no—
20#15692215—21no—
21#15692213C411yescharacteristic, complemented
22#121×C101C411yescharacteristic, complemented
23C22×#87938—24no—
24C22×#87938—22no—
25C22×#87938—42no—
26C22×#87938—42no—
27#13108084—42no—
28C2×#11717581—22no—
29C23×D101D411yescharacteristic
30#13108073—42no—
31C2×Dic202C2×C411yescharacteristic, complemented
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C22×Dic101—21no—
35C22×Dic101D411yescharacteristic
36C2×C4×D101—42no—
37C2×C4×D101C2×C411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×C4×D101D411yescharacteristic
40C2×C4×D101C2311yescharacteristic
41C4⋊Dic101—21no—
42C4⋊Dic101C2×C411yescharacteristic, complemented
43#13108053—21no—
44C22⋊Dic101—21no—
45C22⋊Dic101—21no—
46#13108051—21no—
47#13108051—21no—
48C4×Dic101—21no—
49C2×D4×C101—21no—
50C2×D4×C101C2×C411yescharacteristic, complemented
51C22×C4×C101C2×C411yescharacteristic
52C4⋊C4×C101C2×C411yescharacteristic, complemented
53C22⋊C4×C101—21no—
54C23×C101—21no—
55C23×C101C22⋊C411yescharacteristic
56C22×D101—22no—
57C22×D101—22no—
58C22×D101—21no—
59C22×D101—21no—
60C22×D101C22⋊C411yescharacteristic
61C22×D101C2×D411yescharacteristic
62C2×#87938—24no—
63C2×#87938—48no—
64C2×#87938—42no—
65C2×#87938—42no—
66D4×C101—41no—
67D4×C101—42no—
68C2×C4×C101—21no—
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—21no—
77C2×Dic101—41no—
78C2×Dic101—42no—
79C2×Dic101C22⋊C411yescharacteristic
80C2×Dic101C2×D411yescharacteristic
81C4×D101—22no—
82C4×D101—22no—
83C4×D101—41no—
84C4×D101—41no—
85Dic202—41no—
86#11717581—42no—
87C22×C101—21no—
88C22×C101—21no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101#10412yes—
93C22×C101C2×C22⋊C411yescharacteristic
94C2×D101—24no—
95C2×D101—24no—
96C2×D101—21no—
97C2×D101—21no—
98C2×D101—21no—
99C2×D101—21no—
100C2×D101#9912yes—
101C2×D101AGL(1,ℤ/8)11yescharacteristic
102C2×D101#13711yescharacteristic
103#87938—44no—
104C4×C101—21no—
105C4×C101—21no—
106C4×C101—41no—
107Dic101—21no—
108Dic101—21no—
109Dic101—41no—
110Dic101—41no—
111C2×C101—22no—
112C2×C101—81no—
113C2×C101C2×#9911yescharacteristic
114C2×C101#41711yescharacteristic
115C2×C101C2×#10411yescharacteristic
116D101—21no—
117D101—22no—
118D101—21no—
119#1504—1011nomaximal, Sylow
120C101#150411yescharacteristic, complemented, Sylow
121#525—1011no—
122#407—1011no—
123C2×#95—1011no—
124#328—1012no—
125#327—1012no—
126C2×#41—1011no—
127C23×C4—1011no—
128C23×C4—2021no—
129C2×M16—2021no—
130#123—2021no—
131C22⋊Q8—1011no—
132C22⋊Q8—1011no—
133#121—1011no—
134#121#8793811yescharacteristic, complemented
135#120—2021no—
136C4×D4—2021no—
137C4×D4—2021no—
138#117—1011no—
139C22⋊C8—1012no—
140C22⋊C8—2021no—
141#95—1012no—
142#95—2021no—
143C24—1011no—
144#41—4042no—
145C2×Q8—1011no—
146C2×D4—21no—
147C2×D4—2021no—
148C2×D4—2021no—
149C2×D4C2×#8793811yescharacteristic, complemented
150C22×C4—1011no—
151C22×C4—1011no—
152C22×C4—1011no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—4042no—
156C22×C4—4042no—
157C22×C4—4042no—
158C22×C4—2021no—
159C22×C4—2022no—
160C22×C4—2024no—
161C22×C4C2×#8793811yescharacteristic
162M16—4042no—
163C2×C8—2022no—
164C4⋊C4—1011no—
165C4⋊C4—2021no—
166C4⋊C4—2021no—
167C4⋊C4C2×#8793811yescharacteristic, complemented
168C22⋊C4—21no—
169C22⋊C4—2021no—
170C22⋊C4—2021no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C42—2021no—
174C23—21no—
175C23—1011no—
176C23—1011no—
177C23—2022no—
178C23—2021no—
179C23—2021no—
180C23—2022no—
181C23C22⋊#8793811yescharacteristic
182Q8—4041no—
183D4—41no—
184D4—42no—
185D4—4042no—
186D4—4041no—
187D4—4041no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—1011no—
191C2×C4—1011no—
192C2×C4—4041no—
193C2×C4—4041no—
194C2×C4—4042no—
195C2×C4—4048no—
196C2×C4—4042no—
197C2×C4—4042no—
198C2×C4—4041no—
199C2×C4—2022no—
200C2×C4—2022no—
201C2×C4—2024no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4C22⋊#8793811yescharacteristic
205C2×C4C22×#8793811yescharacteristic
206C8—4042no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—1012no—
213C22—1011no—
214C22—1011no—
215C22—2024no—
216C22—2024no—
217C22—2021no—
218C22—2021no—
219C22—2021no—
220C22—2021no—
221C22#1569231912yes—
222C22C2×C22⋊#8793811yescharacteristic
223C4—21no—
224C4—21no—
225C4—41no—
226C4—4044no—
227C4—4041no—
228C4—4041no—
229C4—2021no—
230C4—2021no—
231C2—22no—
232C2—81no—
233C2—2021no—
234C2—2022no—
235C2—2021no—
236C2D101⋊#9911yescharacteristic
237C2#2026616811yescharacteristic
238C2C2×#1569231911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#121×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265687C2
cosocleC2×C4×D101C23
hypercenter#121#87938
hypocenterC101#1504
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20266148C2semidirect
2. C2×(C2×#87938)⋊C4C2semidirect
3. #20266030C2semidirect
4. #20266028C2semidirect
12. C2×#13108073C4semidirect
16. #15692234C4semidirect
21. #15692213C4semidirect
22. #121×C101C4semidirect
31. C2×Dic202C2×C4semidirect
42. C4⋊Dic101C2×C4semidirect
50. C2×D4×C101C2×C4semidirect
52. C4⋊C4×C101C2×C4semidirect
120. C101#1504semidirect
134. #121#87938semidirect
149. C2×D4C2×#87938semidirect
167. C4⋊C4C2×#87938semidirect

Maximal subgroups

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

subgroupindex
#202661482
C2×(C2×#87938)⋊C42
#202660302
#202660282
#202656872
#1504101