Finite Group Databasebeta

← #27181653

Subgroups — #27181653

Order 12928.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#20265733C211yesmaximal, characteristic
2#20265732C211yesmaximal, characteristic
3#20265587C211yesmaximal, characteristic, complemented
4#20265586C211yesmaximal, characteristic, complemented
5#20265585C212yesmaximal, complemented
6#20265584C212yesmaximal, complemented
7#20265583C211yesmaximal, characteristic, complemented
8#20265214C212yesmaximal
9#20265212C212yesmaximal
10#97×D101C211yesmaximal, characteristic
11#501×C101C211yesmaximal, characteristic, complemented
12C2×C8×D101C2212yes—
13#15692255C2211yescharacteristic
14#15692254C2211yescharacteristic
15#15692254C2211yescharacteristic
16#15692247C2211yescharacteristic
17#15692246C2212yes—
18#15692235—21no—
19#15692233—21no—
20#15692233—21no—
21#15692231—21no—
22Q8×Dic101—21no—
23#15692220—21no—
24#15692220—21no—
25#15692220—21no—
26#15692213C2212yes—
27#15692213C2211yescharacteristic
28#15692213C2211yescharacteristic
29C42×D101C2211yescharacteristic
30#15692189C2212yes—
31#15692188C2212yes—
32#15692188C2212yes—
33#15692187C2212yes—
34#15692186C2212yes—
35#15692186C2212yes—
36#15692166C2211yescharacteristic
37#15692165C2212yes—
38C4⋊Q8×C101C2211yescharacteristic
39#125×C101C2211yescharacteristic
40#107×C101C2212yes—
41#106×C101C2212yes—
42#97×C101C2211yescharacteristic
43C2×Dic202—21no—
44C2×Dic202—21no—
45C2×Dic202—21no—
46C2×C4×D101D411yescharacteristic
47C2×C4×D101D411yescharacteristic
48C2×C4×D101C2311yescharacteristic
49C2×#11717579C2312yes—
50C8×D101Q818yescomplemented
51C4⋊Dic101—21no—
52C4⋊Dic101—22no—
53C4⋊Dic101C2311yescharacteristic
54C4⋊Dic101C2312yes—
55C4⋊Dic101C2311yescharacteristic
56#13108053—21no—
57#13108053—21no—
58#13108053—22no—
59#13108053—21no—
60#13108053—21no—
61#13108051—22no—
62#13108051—21no—
63#13108051—21no—
64C4×Dic101—21no—
65C4×Dic101—21no—
66C4×Dic101—22no—
67C4×Dic101D411yescharacteristic
68C4×Dic101D411yescharacteristic
69C4×Dic101C2311yescharacteristic
70C2×Q8×C101—21no—
71C2×C8×C101C2312yes—
72C4⋊C4×C101—21no—
73C4⋊C4×C101—21no—
74C4⋊C4×C101—21no—
75C4⋊C4×C101C2311yescharacteristic
76C4⋊C4×C101C2312yes—
77C4⋊C4×C101C2311yescharacteristic
78C42×C101C2311yescharacteristic
79C22×D101C2×D411yescharacteristic
80Q8×C101—42no—
81C2×C4×C101—21no—
82C2×C4×C101—21no—
83C2×C4×C101—22no—
84C2×C4×C101C2×D411yescharacteristic
85C2×C4×C101C2×D411yescharacteristic
86C2×C4×C101C2411yescharacteristic
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101—22no—
90C2×Dic101C2×D411yescharacteristic
91C2×Dic101C2×D411yescharacteristic
92C2×Dic101C2×D411yescharacteristic
93C4×D101—41no—
94C4×D101—41no—
95C4×D101C2×Q814yes—
96Dic202—42no—
97Dic202—42no—
98Dic202—42no—
99C8×C101C2×Q814yes—
100#11717579C2×Q814yes—
101C22×C101C22×D411yescharacteristic
102C2×D101—21no—
103C2×D101—21no—
104C2×D101C4⋊Q812yes—
105C4×C101—21no—
106C4×C101—21no—
107C4×C101—41no—
108C4×C101—41no—
109C4×C101—42no—
110C4×C101C22×Q812yes—
111Dic101—21no—
112Dic101—21no—
113Dic101—41no—
114Dic101—41no—
115Dic101—42no—
116Dic101C4⋊Q812yes—
117C2×C101C2×C4⋊Q811yescharacteristic
118C2×C101#57511yescharacteristic
119C2×C101#57511yescharacteristic
120D101—22no—
121#3162—1011nomaximal, Sylow
122C101#316211yescharacteristic, complemented, Sylow
123#533—1011no—
124#533—1011no—
125#501—1011no—
126#501—1011no—
127#501—1012no—
128#501—1012no—
129#501—1011no—
130#501D10111yescharacteristic, complemented
131#427—1012no—
132#427—1012no—
133C2×#97—1011no—
134C22×C8—1012no—
135C4⋊Q8—1011no—
136C4⋊Q8—1011no—
137C4⋊Q8—1011no—
138C4⋊Q8C2×D10111yescharacteristic
139#125—1012no—
140#125—1011no—
141#125C2×D10111yescharacteristic
142C22⋊Q8—2021no—
143C22⋊Q8—2021no—
144C22⋊Q8—2021no—
145C22⋊Q8—2021no—
146C4×Q8—2021no—
147C4×Q8—2021no—
148C4×Q8—2021no—
149C4×Q8—2021no—
150#117—1011no—
151#117—1012no—
152#117—1011no—
153C2×C42—1011no—
154#107—1012no—
155#107—1012no—
156#107—1012no—
157#107C2×D10112yes—
158#106—1012no—
159#106—1012no—
160#106—1012no—
161#106C2×D10112yes—
162#97—1012no—
163#97—1011no—
164#97C2×D10111yescharacteristic
165C2×Q8—21no—
166C2×Q8—2021no—
167C2×Q8—2021no—
168C2×Q8—2021no—
169C22×C4—1011no—
170C22×C4—1011no—
171C22×C4—1011no—
172C2×C8—1012no—
173C2×C8—1018no—
174C2×C8C22×D10112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—1011no—
179C4⋊C4—1012no—
180C4⋊C4—1011no—
181C4⋊C4—2022no—
182C4⋊C4—2021no—
183C4⋊C4—2021no—
184C4⋊C4—2022no—
185C4⋊C4—2021no—
186C4⋊C4—2021no—
187C4⋊C4—2021no—
188C4⋊C4C22×D10111yescharacteristic
189C4⋊C4C22×D10112yes—
190C4⋊C4C22×D10111yescharacteristic
191C22⋊C4—2021no—
192C22⋊C4—2021no—
193C22⋊C4—2022no—
194C42—1011no—
195C42—1011no—
196C42—1011no—
197C42—2021no—
198C42—2021no—
199C42—2022no—
200C42C22×D10111yescharacteristic
201C23—1011no—
202Q8—42no—
203Q8—4042no—
204Q8—4042no—
205Q8—4042no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—1011no—
210C2×C4—1014no—
211C2×C4—1011no—
212C2×C4—1011no—
213C2×C4—4041no—
214C2×C4—4041no—
215C2×C4—2021no—
216C2×C4—2022no—
217C2×C4—2021no—
218C2×C4D4×D10111yescharacteristic
219C2×C4D4×D10111yescharacteristic
220C2×C4C23×D10111yescharacteristic
221C8—1014no—
222C8Q8×D10114yes—
223C22—1012no—
224C22—2021no—
225C22—2021no—
226C22C2×D4×D10111yescharacteristic
227C4—21no—
228C4—21no—
229C4—42no—
230C4—41no—
231C4—41no—
232C4—1012no—
233C4—4041no—
234C4—4042no—
235C4—4041no—
236C4—2021no—
237C4—2021no—
238C4C2×Q8×D10112yes—
239C2—2022no—
240C2C4⋊Q8×D10111yescharacteristic
241C2#2026597311yescharacteristic
242C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#501×C101C2
socleC22×C101C22×D4
prime radicalC22×D101C2×D4
hypercenter#501D101
hypocenterC101#3162
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #20265587C2semidirect
4. #20265586C2semidirect
5. #20265585C2semidirect
6. #20265584C2semidirect
7. #20265583C2semidirect
11. #501×C101C2semidirect
50. C8×D101Q8semidirect
122. C101#3162semidirect
130. #501D101semidirect

Maximal subgroups

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

subgroupindex
#202657332
#202657322
#202655872
#202655862
#202655852
#202655842
#202655832
#202652142
#202652122
#97×D1012
#501×C1012
#3162101