Finite Group Databasebeta

← #27184168

Subgroups — #27184168

Order 12928.

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

normal subgroups only

This group has 350 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#20265902C211yesmaximal, characteristic, complemented
2#20265901C211yesmaximal, characteristic
3#20265570C212yesmaximal
4#20265401C212yesmaximal
5#20265385C212yesmaximal, complemented
6#20265384C211yesmaximal, characteristic, complemented
7#20265379C211yesmaximal, characteristic
8#20265307C211yesmaximal, characteristic, complemented
9#20265284C212yesmaximal, complemented
10C4×#13108063C211yesmaximal, characteristic
11#494×C101C211yesmaximal, characteristic, complemented
12C2×#13108064C2212yes—
13C2×#13108063C2211yescharacteristic
14C2×#13108061C2212yes—
15#15692253C2212yes—
16#15692250—22no—
17#15692246—21no—
18#15692245—22no—
19#15692244C2212yes—
20#15692235—21no—
21Q8×Dic101C2212yes—
22Q8×Dic101C2212yes—
23C4×Dic202C2211yescharacteristic
24C4×Dic202C2211yescharacteristic
25#15692220—21no—
26C4×D404C2211yescharacteristic
27#15692213—21no—
28#15692187C2211yescharacteristic
29C4⋊#11717579C2211yescharacteristic
30#15692184C2212yes—
31#15692180C2211yescharacteristic
32#15692179C2212yes—
33#15692179C2212yes—
34#15692177C2211yescharacteristic
35#15692165C2212yes—
36C2×Q16×C101C2212yes—
37C2×Q16×C101C2212yes—
38C4⋊Q8×C101C2211yescharacteristic
39C4⋊Q8×C101C2211yescharacteristic
40C4×C8×C101C2211yescharacteristic
41C2×Dic202D411yescharacteristic
42C2×Dic202C2311yescharacteristic
43C2×D404C2311yescharacteristic
44C2×C4×D101—21no—
45#13108064—42no—
46#13108063—22no—
47#13108061—42no—
48C2×#11717579C2312yes—
49C4⋊Dic101—22no—
50C4⋊Dic101—22no—
51C4⋊Dic101D411yescharacteristic
52C4⋊Dic101C2311yescharacteristic
53C4⋊Dic101C2311yescharacteristic
54#13108053—21no—
55#13108053—22no—
56#13108053—22no—
57#13108053D412yes—
58#13108051—21no—
59#13108051—22no—
60#13108051—22no—
61C4×Dic101—21no—
62C4×Dic101—22no—
63C4×Dic101—22no—
64C4×Dic101C2312yes—
65C2×Q8×C101C2312yes—
66C2×Q8×C101C2312yes—
67Q16×C101—24no—
68Q16×C101—42no—
69C2×C8×C101C2311yescharacteristic
70C2×C8×C101C2311yescharacteristic
71C4⋊C4×C101—21no—
72C4⋊C4×C101—21no—
73C4⋊C4×C101—21no—
74C4⋊C4×C101—21no—
75C42×C101C2311yescharacteristic
76C22×D101—21no—
77Q8×C101—22no—
78Q8×C101—22no—
79Q8×C101—22no—
80Q8×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2×D411yescharacteristic
85C2×C4×C101C2411yescharacteristic
86C2×Dic101—21no—
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101C2×D412yes—
90C2×Dic101C2×D412yes—
91D404—21no—
92D404—21no—
93C4×D101—42no—
94Dic202—22no—
95Dic202—21no—
96Dic202—21no—
97C8×C101—21no—
98C8×C101#4112yes—
99#11717579—22no—
100C22×C101C22×D411yescharacteristic
101C2×D101—41no—
102C2×D101—41no—
103C2×D101—41no—
104C4×C101—21no—
105C4×C101—42no—
106C4×C101—42no—
107C4×C101#13712yes—
108C4×C101C2×#4111yescharacteristic
109C4×C101H(5,𝔽2)11yescharacteristic
110Dic101—22no—
111Dic101—22no—
112Dic101—41no—
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#494D10111yescharacteristic, complemented
125#464—1012no—
126#462—1012no—
127#462—1011no—
128#461—1011no—
129#446—1011no—
130#441—1012no—
131C4×SD16—1011no—
132C2×Q16—1012no—
133C2×Q16C2×D10112yes—
134C2×Q16C2×D10112yes—
135C2×SD16—1012no—
136C2×SD16—1011no—
137C4⋊Q8—1012no—
138C4⋊Q8C2×D10111yescharacteristic
139C4⋊Q8C2×D10111yescharacteristic
140#126—2022no—
141#125—2022no—
142#125—2021no—
143#124—1012no—
144C22⋊Q8—2021no—
145C4×Q8—1011no—
146C4×Q8—1012no—
147C4×Q8—1012no—
148C4×Q8—1011no—
149C4×Q8—2021no—
150C4×D4—1011no—
151#117—2021no—
152#106—1011no—
153C4⋊C8—1012no—
154C4⋊C8—1011no—
155#103—1012no—
156#103—1012no—
157#103—1011no—
158#102—1011no—
159#97—1012no—
160C4×C8C2×D10111yescharacteristic
161C2×Q8—1011no—
162C2×Q8—1011no—
163C2×Q8C22×D10112yes—
164C2×Q8C22×D10112yes—
165C2×D4—1011no—
166C22×C4—2021no—
167Q16—24no—
168Q16—42no—
169Q16—4042no—
170SD16—4042no—
171SD16—2022no—
172C2×C8—1012no—
173C2×C8C22×D10111yescharacteristic
174C2×C8C22×D10111yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—1012no—
180C4⋊C4—1011no—
181C4⋊C4—1011no—
182C4⋊C4—1011no—
183C4⋊C4—2022no—
184C4⋊C4—2022no—
185C4⋊C4—2022no—
186C4⋊C4—2022no—
187C4⋊C4—2021no—
188C22⋊C4—2022no—
189C22⋊C4—2022no—
190C22⋊C4—2021no—
191C42—1012no—
192C42—2022no—
193C42—2022no—
194C42—2021no—
195C42C22×D10111yescharacteristic
196C23—2021no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—22no—
201Q8—2021no—
202Q8—2021no—
203Q8—2022no—
204D4—2021no—
205D4—2021no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—1012no—
209C2×C4—1012no—
210C2×C4—4042no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4—2021no—
214C2×C4D4×D10111yescharacteristic
215C2×C4D4×D10111yescharacteristic
216C2×C4C23×D10111yescharacteristic
217C8—21no—
218C8—2022no—
219C8#1310807312yes—
220C22—4041no—
221C22—4041no—
222C22—4041no—
223C22C2×D4×D10111yescharacteristic
224C4—21no—
225C4—42no—
226C4—42no—
227C4—4041no—
228C4—4041no—
229C4—4041no—
230C4—2022no—
231C4—2022no—
232C4#1569228712yes—
233C4C2×#1310807311yescharacteristic
234C4#1569230011yescharacteristic
235C2—8081no—
236C2#2026584711yescharacteristic
237C2C2×#1569228711yescharacteristic
238C2#2026599011yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#494×C101C2
socleC22×C101C22×D4
prime radicalC2×D404C23
hypercenter#494D101
hypocenterC101#3398
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #20265902C2semidirect
5. #20265385C2semidirect
6. #20265384C2semidirect
8. #20265307C2semidirect
9. #20265284C2semidirect
11. #494×C101C2semidirect
120. C101#3398semidirect
124. #494D101semidirect

Maximal subgroups

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

subgroupindex
#202659022
#202659012
#202655702
#202654012
#202653852
#202653842
#202653792
#202653072
#202652842
C4×#131080632
#494×C1012
#3398101