Finite Group Databasebeta

← #27183062

Subgroups — #27183062

Order 12928.

398 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265892C211yesmaximal, characteristic, complemented
2#20265831C211yesmaximal, characteristic, complemented
3#20265528C211yesmaximal, characteristic, complemented
4#20265520C212yesmaximal, complemented
5#20265467C211yesmaximal, characteristic, complemented
6#20265405C212yesmaximal, complemented
7#20265382C211yesmaximal, characteristic, complemented
8#20265319C212yesmaximal, complemented
9C8×D404C211yesmaximal, characteristic, complemented
10#20265170C212yesmaximal, complemented
11C4×D8×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2212yes—
13C2×#13108062C2212yes—
14C2×C8×D101C2212yes—
15#15692255C2211yescharacteristic
16#15692255C2211yescharacteristic
17#15692241—22no—
18#15692239—22no—
19#15692235C2212yes—
20#15692229—21no—
21#15692229—21no—
22#15692226C2212yes—
23C4×D404C2211yescharacteristic
24#15692214—21no—
25#15692213C2212yes—
26C2×C4⋊Dic101—21no—
27#15692186C2212yes—
28C4⋊#11717579C2211yescharacteristic
29#15692180C2212yes—
30#15692180C2212yes—
31#15692178C2212yes—
32#15692176C2212yes—
33#15692167C2212yes—
34C2×D8×C101C2211yescharacteristic
35C4×D4×C101C2211yescharacteristic
36C4×D4×C101C2211yescharacteristic
37#107×C101C2211yescharacteristic
38#102×C101C2211yescharacteristic
39#102×C101C2211yescharacteristic
40C4×C8×C101C2211yescharacteristic
41#13108073—24no—
42#13108073—42no—
43C2×Dic202—21no—
44C2×Dic202C2312yes—
45C2×D404D411yescharacteristic
46C2×C22⋊D101—22no—
47C2×C22⋊D101—22no—
48C22×Dic101—22no—
49C2×C4×D101C2312yes—
50#13108062—42no—
51C2×#11717579C2312yes—
52C8×D101—42no—
53C4⋊Dic101—21no—
54C4⋊Dic101—21no—
55C4⋊Dic101—21no—
56C4⋊Dic101—22no—
57C4⋊Dic101D411yescharacteristic
58C4⋊Dic101C2312yes—
59C22⋊Dic101—22no—
60#13108051—22no—
61#13108051—22no—
62#13108051D412yes—
63C4×Dic101—22no—
64C2×D4×C101C2311yescharacteristic
65C2×D4×C101C2311yescharacteristic
66C22×C4×C101—21no—
67C22×C4×C101—21no—
68D8×C101—22no—
69C2×C8×C101C2311yescharacteristic
70C2×C8×C101C2311yescharacteristic
71C4⋊C4×C101C2311yescharacteristic
72C4⋊C4×C101C2311yescharacteristic
73C22⋊C4×C101—21no—
74C22⋊C4×C101—21no—
75C42×C101C2311yescharacteristic
76C23×C101—21no—
77C23×C101—21no—
78C22×D101C2×D412yes—
79D4×C101—21no—
80D4×C101—21no—
81D4×C101—21no—
82D4×C101#4112yes—
83C2×C4×C101—21no—
84C2×C4×C101—21no—
85C2×C4×C101—42no—
86C2×C4×C101—42no—
87C2×C4×C101C2×D411yescharacteristic
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—44no—
91C22⋊D101—82no—
92C2×Dic101—22no—
93C2×Dic101—22no—
94C2×Dic101—44no—
95C2×Dic101C2×D412yes—
96D404—22no—
97C4×D101—22no—
98C4×D101—22no—
99Dic202—22no—
100Dic202—22no—
101Dic202—81no—
102C8×C101—21no—
103C8×C101—21no—
104#11717579—22no—
105C22×C101—22no—
106C22×C101—41no—
107C22×C101—41no—
108C22×C101—41no—
109C22×C101—41no—
110C22×C101—41no—
111C22×C101C22×D411yescharacteristic
112C2×D101—22no—
113C2×D101—22no—
114C2×D101—22no—
115C4×C101—21no—
116C4×C101—41no—
117C4×C101—41no—
118C4×C101#13512yes—
119C4×C101C2×#4111yescharacteristic
120C4×C101H(5,𝔽2)11yescharacteristic
121Dic101—22no—
122Dic101—42no—
123Dic101—42no—
124C2×C101—42no—
125C2×C101—81no—
126C2×C101#54611yescharacteristic
127C2×C101C2×#13511yescharacteristic
128C2×C101#57711yescharacteristic
129D101—42no—
130#3303—1011nomaximal, Sylow
131C101#330311yescharacteristic, complemented, Sylow
132#555—1011no—
133#547—1011no—
134#487—1011no—
135#485—1012no—
136#476—1011no—
137#465—1012no—
138#461—1011no—
139#449—1012no—
140C4×D8D10111yescharacteristic, complemented
141C8×D4—1011no—
142#416—1012no—
143C2×#41—1012no—
144C2×SD16—1012no—
145C2×D8C2×D10111yescharacteristic
146C22×C8—1012no—
147C4⋊Q8—1011no—
148C4⋊Q8—1011no—
149#124—2022no—
150#123—2022no—
151C22⋊Q8—1012no—
152#121—1012no—
153#121—2021no—
154#121—2021no—
155C4×D4—1011no—
156C4×D4C2×D10111yescharacteristic
157C4×D4C2×D10111yescharacteristic
158#117—1012no—
159#117—2021no—
160C2×C4⋊C4—2021no—
161#107C2×D10111yescharacteristic
162#106—1012no—
163C4⋊C8—1011no—
164#103—1012no—
165#103—1012no—
166#103—1012no—
167#102—1012no—
168#102C2×D10111yescharacteristic
169#102C2×D10111yescharacteristic
170C22⋊C8—1012no—
171C4×C8C2×D10111yescharacteristic
172#41—4042no—
173#41—2024no—
174C2×Q8—1012no—
175C2×Q8—2021no—
176C2×D4—1011no—
177C2×D4—2022no—
178C2×D4—2022no—
179C2×D4C22×D10111yescharacteristic
180C2×D4C22×D10111yescharacteristic
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—1012no—
184C22×C4—2022no—
185SD16—4042no—
186D8—22no—
187C2×C8—1012no—
188C2×C8—4042no—
189C2×C8C22×D10111yescharacteristic
190C2×C8C22×D10111yescharacteristic
191C4⋊C4—1011no—
192C4⋊C4—1012no—
193C4⋊C4—2022no—
194C4⋊C4—2021no—
195C4⋊C4—2021no—
196C4⋊C4—2021no—
197C4⋊C4C22×D10111yescharacteristic
198C4⋊C4C22×D10111yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—1012no—
202C22⋊C4—2022no—
203C22⋊C4—2022no—
204C22⋊C4—2022no—
205C42—2022no—
206C42C22×D10111yescharacteristic
207C23—21no—
208C23—21no—
209C23—1012no—
210Q8—8081no—
211Q8—2022no—
212Q8—2022no—
213D4—21no—
214D4—21no—
215D4—21no—
216D4—8082no—
217D4—4044no—
218D4—2022no—
219D4#1310807412yes—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—1012no—
225C2×C4—4044no—
226C2×C4—2022no—
227C2×C4—2022no—
228C2×C4—2022no—
229C2×C4—2022no—
230C2×C4D4×D10111yescharacteristic
231C2×C4D4×D10111yescharacteristic
232C2×C4C23×D10111yescharacteristic
233C8—21no—
234C8—21no—
235C8—2022no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—2022no—
243C22—2022no—
244C22—2022no—
245C22C2×D4×D10111yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—4042no—
250C4—4042no—
251C4—2022no—
252C4#1569227412yes—
253C4C2×#1310807411yescharacteristic
254C4#1569230111yescharacteristic
255C2—42no—
256C2—81no—
257C2—4042no—
258C2#2026582311yescharacteristic
259C2C2×#1569227411yescharacteristic
260C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×D8×C101C2
socleC22×C101C22×D4
hypercenterC4×D8D101
hypocenterC101#3303
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20265892C2semidirect
2. #20265831C2semidirect
3. #20265528C2semidirect
4. #20265520C2semidirect
5. #20265467C2semidirect
6. #20265405C2semidirect
7. #20265382C2semidirect
8. #20265319C2semidirect
9. C8×D404C2semidirect
10. #20265170C2semidirect
11. C4×D8×C101C2semidirect
131. C101#3303semidirect
140. C4×D8D101semidirect

Maximal subgroups

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

subgroupindex
#202658922
#202658312
#202655282
#202655202
#202654672
#202654052
#202653822
#202653192
C8×D4042
#202651702
C4×D8×C1012
#3303101