Finite Group Databasebeta

← #27184165

Subgroups — #27184165

Order 12928.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265906C211yesmaximal, characteristic, complemented
2#20265900C211yesmaximal, characteristic
3#20265571C212yesmaximal
4#20265557C211yesmaximal, characteristic, complemented
5#20265400C212yesmaximal
6#20265388C211yesmaximal, characteristic, complemented
7#20265388C212yesmaximal, complemented
8#20265383C211yesmaximal, characteristic
9#20265285C212yesmaximal, complemented
10C4×#13108063C211yesmaximal, characteristic
11#446×C101C211yesmaximal, characteristic, complemented
12C2×Dic404C2212yes—
13C2×Dic404C2212yes—
14C2×Dic404C2212yes—
15C2×#13108063C2211yescharacteristic
16C2×#13108063C2212yes—
17#15692255C2211yescharacteristic
18#15692255C2211yescharacteristic
19#15692255C2212yes—
20#15692250—22no—
21#15692246—21no—
22#15692246—22no—
23#15692243C2212yes—
24#15692231—21no—
25Q8×Dic101—21no—
26C4×Dic202C2211yescharacteristic
27C4×Dic202C2212yes—
28C4×Dic202C2212yes—
29C4×D404C2211yescharacteristic
30#15692213—21no—
31#15692187C2211yescharacteristic
32#15692180C2211yescharacteristic
33#15692180C2212yes—
34#15692180C2212yes—
35#15692177C2211yescharacteristic
36C4×Q8×C101C2211yescharacteristic
37C4⋊C8×C101C2211yescharacteristic
38C4⋊C8×C101C2212yes—
39#97×C101C2212yes—
40C4×C8×C101C2211yescharacteristic
41C2×Dic202C2312yes—
42C2×Dic202C2311yescharacteristic
43C2×Dic202C2312yes—
44C2×D404C2311yescharacteristic
45C2×C4×D101—21no—
46Dic404—24no—
47Dic404—42no—
48Dic404—42no—
49#13108063—22no—
50#13108063—42no—
51C4⋊Dic101—21no—
52C4⋊Dic101—22no—
53C4⋊Dic101—21no—
54C4⋊Dic101—22no—
55C4⋊Dic101—21no—
56C4⋊Dic101—21no—
57C4⋊Dic101C2311yescharacteristic
58C4⋊Dic101C2311yescharacteristic
59#13108053—22no—
60#13108053—22no—
61#13108053—21no—
62#13108051—22no—
63#13108051—21no—
64#13108051—22no—
65C4×Dic101—22no—
66C4×Dic101—22no—
67C4×Dic101—21no—
68C2×Q8×C101D411yescharacteristic
69C2×C8×C101C2311yescharacteristic
70C2×C8×C101C2312yes—
71C2×C8×C101C2311yescharacteristic
72C4⋊C4×C101D411yescharacteristic
73C4⋊C4×C101D412yes—
74C42×C101C2311yescharacteristic
75C42×C101C2312yes—
76C22×D101—21no—
77Q8×C101—22no—
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2×D411yescharacteristic
80C2×C4×C101C2×D412yes—
81C2×C4×C101C2×D411yescharacteristic
82C2×C4×C101C2411yescharacteristic
83C2×Dic101—21no—
84C2×Dic101—21no—
85C2×Dic101—22no—
86C2×Dic101—21no—
87C2×Dic101—22no—
88D404—21no—
89D404—21no—
90C4×D101—42no—
91Dic202—21no—
92Dic202—22no—
93Dic202—22no—
94Dic202—22no—
95Dic202—22no—
96Dic202—21no—
97C8×C101—22no—
98C8×C101—21no—
99C8×C101#4112yes—
100C22×C101C22×D411yescharacteristic
101C2×D101—41no—
102C2×D101—41no—
103C2×D101—41no—
104C4×C101—22no—
105C4×C101—22no—
106C4×C101—21no—
107C4×C101#13712yes—
108C4×C101C2×#4111yescharacteristic
109C4×C101H(5,𝔽2)11yescharacteristic
110Dic101—41no—
111Dic101—42no—
112Dic101—42no—
113Dic101—41no—
114Dic101—41no—
115C2×C101#55011yescharacteristic
116C2×C101C2×#13711yescharacteristic
117C2×C101#57811yescharacteristic
118D101—81no—
119#3398—1011nomaximal, Sylow
120C101#339811yescharacteristic, complemented, Sylow
121#557—1011no—
122#556—1011no—
123#497—1012no—
124#494—1011no—
125#464—1012no—
126#462—1012no—
127#462—1011no—
128#461—1011no—
129#446D10111yescharacteristic, complemented
130#441—1012no—
131C4×SD16—1011no—
132C2×Q16—1012no—
133C2×Q16—1012no—
134C2×Q16—1012no—
135C2×SD16—1012no—
136C2×SD16—1011no—
137C4⋊Q8—1011no—
138C4⋊Q8—1011no—
139C4⋊Q8—1012no—
140#126—2022no—
141#125—2021no—
142#125—2022no—
143#124—1012no—
144C22⋊Q8—2021no—
145C4×Q8—1012no—
146C4×Q8—1011no—
147C4×Q8—1012no—
148C4×Q8—2021no—
149C4×Q8C2×D10111yescharacteristic
150C4×D4—1011no—
151#117—2021no—
152#106—1011no—
153C4⋊C8C2×D10111yescharacteristic
154C4⋊C8C2×D10112yes—
155#103—1011no—
156#103—1012no—
157#103—1012no—
158#102—1011no—
159#97C2×D10112yes—
160C4×C8C2×D10111yescharacteristic
161C2×Q8—1012no—
162C2×Q8—1011no—
163C2×Q8—1012no—
164C2×Q8D40411yescharacteristic
165C2×D4—1011no—
166C22×C4—2021no—
167Q16—4042no—
168Q16—4042no—
169Q16—2024no—
170SD16—4042no—
171SD16—2022no—
172C2×C8C22×D10111yescharacteristic
173C2×C8C22×D10112yes—
174C2×C8C22×D10111yescharacteristic
175C4⋊C4—1011no—
176C4⋊C4—1011no—
177C4⋊C4—2022no—
178C4⋊C4—2022no—
179C4⋊C4—2021no—
180C4⋊C4—2021no—
181C4⋊C4—2022no—
182C4⋊C4—2021no—
183C4⋊C4—2022no—
184C4⋊C4—2021no—
185C4⋊C4—2021no—
186C4⋊C4D40411yescharacteristic
187C4⋊C4D40412yes—
188C22⋊C4—2022no—
189C22⋊C4—2021no—
190C22⋊C4—2022no—
191C42—2022no—
192C42—2022no—
193C42—2021no—
194C42C22×D10111yescharacteristic
195C42C22×D10112yes—
196C23—2021no—
197Q8—22no—
198Q8—2021no—
199Q8—2022no—
200Q8—2022no—
201Q8—2022no—
202Q8—2022no—
203Q8—2021no—
204D4—2021no—
205D4—2021no—
206C2×C4—4042no—
207C2×C4—2021no—
208C2×C4—2021no—
209C2×C4—2022no—
210C2×C4—2021no—
211C2×C4—2022no—
212C2×C4C2×D40412yes—
213C2×C4C2×D40412yes—
214C2×C4C2×D40411yescharacteristic
215C2×C4C2×D40411yescharacteristic
216C2×C4C23×D10111yescharacteristic
217C8—22no—
218C8—21no—
219C8#1310807512yes—
220C22—4041no—
221C22—4041no—
222C22—4041no—
223C22C22×D40411yescharacteristic
224C4—22no—
225C4—21no—
226C4—22no—
227C4—4041no—
228C4—4042no—
229C4—4041no—
230C4—4042no—
231C4—4041no—
232C4#1569228812yes—
233C4C2×#1310807511yescharacteristic
234C4#1569230111yescharacteristic
235C2—8081no—
236C2#2026584911yescharacteristic
237C2C2×#1569228811yescharacteristic
238C2#2026599411yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#446×C101C2
socleC22×C101C22×D4
prime radicalC2×D404C23
hypercenter#446D101
hypocenterC101#3398
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #20265906C2semidirect
4. #20265557C2semidirect
6. #20265388C2semidirect
7. #20265388C2semidirect
9. #20265285C2semidirect
11. #446×C101C2semidirect
120. C101#3398semidirect
129. #446D101semidirect

Maximal subgroups

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

subgroupindex
#202659062
#202659002
#202655712
#202655572
#202654002
#202653882
#202653832
#202652852
C4×#131080632
#446×C1012
#3398101