Finite Group Databasebeta

← #27169479

Subgroups — #27169479

Order 12928.

280 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 63 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×#15692240C211yesmaximal, characteristic
2#20265139C211yesmaximal, characteristic, complemented
3C22⋊C8×D101C211yesmaximal, characteristic, complemented
4#20264903C211yesmaximal, characteristic, complemented
5#20264900C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#324×C101C211yesmaximal, characteristic, complemented
8C22×D404C411yescharacteristic
9C22×C4×D101C2211yescharacteristic
10C2×#13108057—21no—
11C2×C8×D101—21no—
12#15692240—21no—
13#15692240—21no—
14#15692240—21no—
15#15692240—21no—
16C2×#13108053C411yescharacteristic
17C2×#13108051C411yescharacteristic
18C2×#13108051C411yescharacteristic
19C2×#13108051C2211yescharacteristic
20C22⋊#11717579C2211yescharacteristic
21C22⋊#11717579C2211yescharacteristic
22#15692167—21no—
23#15692167—21no—
24C2×C4⋊C4×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26C22⋊C8×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28C23×D101C2×C411yescharacteristic
29C2×D404—21no—
30C2×D404—21no—
31C2×D404—42no—
32C22×Dic101C2×C411yescharacteristic
33C22×Dic101C2×C411yescharacteristic
34C2×C4×D101—42no—
35C2×C4×D101D411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39#13108057—42no—
40C2×#11717579—21no—
41C2×#11717579—21no—
42C8×D101—42no—
43#13108053—21no—
44#13108053—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—21no—
51C22×C4×C101C2×C411yescharacteristic
52C22×C4×C101C2×C411yescharacteristic
53C22×C4×C101C2311yescharacteristic
54C2×C8×C101—21no—
55C2×C8×C101—21no—
56C4⋊C4×C101—21no—
57C4⋊C4×C101—21no—
58C23×C101C22×C411yescharacteristic
59C22×D101—21no—
60C22×D101—21no—
61C22×D101—22no—
62C22×D101—21no—
63C22×D101—21no—
64C22×D101—21no—
65C22×D101—41no—
66C22×D101—41no—
67C22×D101—41no—
68C22×D101C22⋊C411yescharacteristic
69C22×D101C22⋊C411yescharacteristic
70C2×C4×C101—21no—
71C2×C4×C101—21no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—41no—
75C2×C4×C101—41no—
76C2×C4×C101C2×D411yescharacteristic
77C2×C4×C101C2×D411yescharacteristic
78C2×Dic101—21no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C22⋊C411yescharacteristic
83D404—42no—
84D404—42no—
85C4×D101—22no—
86C4×D101—22no—
87C4×D101—41no—
88C4×D101—41no—
89C8×C101—41no—
90C8×C101—41no—
91#11717579—41no—
92#11717579—41no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101C2×C22⋊C411yescharacteristic
96C22×C101#13111yescharacteristic
97C22×C101#13111yescharacteristic
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—21no—
101C2×D101—21no—
102C2×D101—21no—
103C2×D101—42no—
104C2×D101—42no—
105C2×D101—41no—
106C2×D101—41no—
107C2×D101—41no—
108C2×D101—81no—
109C2×D101—81no—
110C2×D101#9912yes—
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—41no—
114C4×C101—41no—
115Dic101—21no—
116Dic101—21no—
117Dic101—41no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101#40811yescharacteristic
121C2×C101C2×#9911yescharacteristic
122C2×C101#41311yescharacteristic
123D101—22no—
124D101—41no—
125D101—81no—
126#1468—1011nomaximal, Sylow
127C101#146811yescharacteristic, complemented, Sylow
128C2×#123—1011no—
129#407—1011no—
130C2×C22⋊C8—1011no—
131#324—1011no—
132#324D10111yescharacteristic, complemented
133#323—1011no—
134#323—1011no—
135C22×D4—1011no—
136C23×C4—1011no—
137C2×M16—2021no—
138C22×C8—2021no—
139#123—2021no—
140#123—2021no—
141#123—2021no—
142#123—2021no—
143C2×C4⋊C4—1011no—
144C2×C4⋊C4C2×D10111yescharacteristic
145C2×C22⋊C4—1011no—
146C2×C22⋊C4—1011no—
147C2×C22⋊C4—1011no—
148C22⋊C8—1011no—
149C22⋊C8—1011no—
150C22⋊C8—2021no—
151C22⋊C8—2021no—
152C22⋊C8C2×D10111yescharacteristic
153C22⋊C8C2×D10111yescharacteristic
154C24—1011no—
155C24—1011no—
156C2×D4—4042no—
157C2×D4—2021no—
158C2×D4—2021no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—1011no—
163C22×C4—1011no—
164C22×C4—1011no—
165C22×C4—4042no—
166C22×C4C4×D10111yescharacteristic
167C22×C4C4×D10111yescharacteristic
168C22×C4C22×D10111yescharacteristic
169M16—4042no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—4042no—
173C2×C8—2021no—
174C2×C8—2021no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—2021no—
178C4⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C22⋊C4—2021no—
184C22⋊C4—2021no—
185C23—1011no—
186C23—1011no—
187C23—4041no—
188C23—4041no—
189C23—4041no—
190C23—2022no—
191C23—2021no—
192C23—2021no—
193C23—2021no—
194C23—2021no—
195C23—2021no—
196C23C2×C4×D10111yescharacteristic
197D4—4042no—
198D4—4042no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—1011no—
206C2×C4—1011no—
207C2×C4—4041no—
208C2×C4—4041no—
209C2×C4—4041no—
210C2×C4—4041no—
211C2×C4—2022no—
212C2×C4—2022no—
213C2×C4—2021no—
214C2×C4D4×D10111yescharacteristic
215C2×C4D4×D10111yescharacteristic
216C8—41no—
217C8—41no—
218C8—4041no—
219C8—4041no—
220C22—21no—
221C22—21no—
222C22—8081no—
223C22—8081no—
224C22—1012no—
225C22—4042no—
226C22—4042no—
227C22—4041no—
228C22—4041no—
229C22—4041no—
230C22—2021no—
231C22—2021no—
232C22—2021no—
233C22—2021no—
234C22—2021no—
235C22C22⋊C4×D10111yescharacteristic
236C22#1569226111yescharacteristic
237C22#1569226311yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—4041no—
243C4—2021no—
244C4—2021no—
245C2—21no—
246C2—21no—
247C2—8081no—
248C2—4041no—
249C2—2022no—
250C2#2026514211yescharacteristic
251C2#99×D10111yescharacteristic
252C2#2026516011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×D404C4
cosocleC22×C4×C101C23
hypercenter#324D101
hypocenterC101#1468
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20265139C2semidirect
3. C22⋊C8×D101C2semidirect
4. #20264903C2semidirect
5. #20264900C2semidirect
6. #20264900C2semidirect
7. #324×C101C2semidirect
127. C101#1468semidirect
132. #324D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922402
#202651392
C22⋊C8×D1012
#202649032
#202649002
#324×C1012
#1468101