Finite Group Databasebeta

← #27181510

Subgroups — #27181510

Order 12928.

380 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 129 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#20265730C211yesmaximal, characteristic
2C2×#15692253C211yesmaximal, characteristic
3#20265544C214yesmaximal, complemented
4#20265526C214yesmaximal, complemented
5#20265187C211yesmaximal, characteristic
6#20265186C211yesmaximal, characteristic
7#20265185C211yesmaximal, characteristic
8M16×Dic101C211yesmaximal, characteristic
9#419×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11C2×#13108058C2211yescharacteristic
12#15692254C2212yes—
13#15692253—22no—
14#15692253C2212yes—
15#15692253C2212yes—
16#15692246C2212yes—
17#15692232—22no—
18C4×Dic202—22no—
19#15692214C2211yescharacteristic
20C2×#13108053—22no—
21C2×C4×Dic101C2211yescharacteristic
22#15692180C2214yes—
23#15692179C2214yes—
24#15692178C2214yes—
25#15692165C2212yes—
26C8×Dic101C2212yes—
27C22×Q8×C101C2211yescharacteristic
28C2×M16×C101C2211yescharacteristic
29#117×C101C2211yescharacteristic
30#103×C101C2214yes—
31C2×Dic202—22no—
32C2×Dic202—22no—
33C2×Dic202—21no—
34C2×Dic202—22no—
35C2×Dic202C2312yes—
36C22×Dic101—21no—
37C22×Dic101D412yes—
38#13108058—22no—
39C2×#11717579C2312yes—
40C4⋊Dic101—22no—
41C4⋊Dic101C2312yes—
42#13108053—24no—
43#13108053—24no—
44#13108053—22no—
45#13108053—22no—
46C22⋊Dic101—21no—
47C4×Dic101—21no—
48C4×Dic101D412yes—
49C4×Dic101C2311yescharacteristic
50C4×Dic101C2311yescharacteristic
51C2×Q8×C101—21no—
52C2×Q8×C101—22no—
53C2×Q8×C101—22no—
54C2×Q8×C101C2312yes—
55C22×C4×C101—21no—
56C22×C4×C101C2311yescharacteristic
57M16×C101—22no—
58C2×C8×C101C2312yes—
59C4⋊C4×C101C2312yes—
60C22⋊C4×C101—21no—
61C42×C101—21no—
62C23×C101C2×D411yescharacteristic
63Q8×C101—22no—
64Q8×C101—22no—
65Q8×C101—42no—
66C2×C4×C101—22no—
67C2×C4×C101—22no—
68C2×C4×C101—42no—
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101#4112yes—
71C2×C4×C101#4112yes—
72C2×C4×C101C2411yescharacteristic
73C2×Dic101—24no—
74C2×Dic101—22no—
75C2×Dic101—22no—
76C2×Dic101—42no—
77C2×Dic101C2×D412yes—
78C2×Dic101C2×D412yes—
79Dic202—22no—
80Dic202—22no—
81Dic202—42no—
82Dic202—44no—
83C8×C101—22no—
84#11717579—22no—
85C22×C101—22no—
86C22×C101#12412yes—
87C22×C101C22×D411yescharacteristic
88C4×C101—42no—
89C4×C101—42no—
90C4×C101#12412yes—
91C4×C101C2×#4111yescharacteristic
92C4×C101C2×#4111yescharacteristic
93Dic101—22no—
94Dic101—42no—
95Dic101—42no—
96Dic101#13714yes—
97C2×C101—22no—
98C2×C101C2×#12411yescharacteristic
99C2×C101C2×#13712yes—
100#3139—1011nomaximal, Sylow
101C101#313911yescharacteristic, complemented, Sylow
102#533—1011no—
103C2×C4⋊Q8—1011no—
104#491—1014no—
105#487—1014no—
106#419—1011no—
107#419—1011no—
108#419—1011no—
109#419D10111yescharacteristic
110C4×M16—1011no—
111C22×Q8—1011no—
112C22×Q8C2×D10111yescharacteristic
113C2×M16—1011no—
114C2×M16C2×D10111yescharacteristic
115C4⋊Q8—1012no—
116C4⋊Q8—1012no—
117C4⋊Q8—1012no—
118C4⋊Q8—2022no—
119#125—1012no—
120C22⋊Q8—2022no—
121C4×Q8—2022no—
122#117—1011no—
123#117C2×D10111yescharacteristic
124C2×C4⋊C4—2022no—
125C2×C42—1011no—
126#103—1014no—
127#103—1014no—
128#103—1014no—
129#103C2×D10114yes—
130#97—1012no—
131C4×C8—1012no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—1012no—
136C2×Q8—2022no—
137C2×Q8—2022no—
138C2×Q8—2022no—
139C2×Q8—2021no—
140C2×Q8C22×D10112yes—
141C22×C4—21no—
142C22×C4—1012no—
143C22×C4—2021no—
144C22×C4C22×D10111yescharacteristic
145M16—22no—
146M16—2022no—
147C2×C8—1012no—
148C2×C8C22×D10112yes—
149C4⋊C4—1012no—
150C4⋊C4—2024no—
151C4⋊C4—2024no—
152C4⋊C4—2022no—
153C4⋊C4—2022no—
154C4⋊C4—2022no—
155C4⋊C4C22×D10112yes—
156C22⋊C4—21no—
157C22⋊C4—2021no—
158C42—21no—
159C42—1012no—
160C42—1011no—
161C42—1011no—
162C42—2021no—
163C23D4×D10111yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—42no—
167Q8—4044no—
168Q8—4042no—
169Q8—2022no—
170Q8—2022no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—1012no—
175C2×C4—1012no—
176C2×C4—4042no—
177C2×C4—2024no—
178C2×C4—2022no—
179C2×C4—2022no—
180C2×C4D4×D10111yescharacteristic
181C2×C4#1310807312yes—
182C2×C4#1310807412yes—
183C2×C4C23×D10111yescharacteristic
184C8—22no—
185C8—2022no—
186C22—22no—
187C22#1569224112yes—
188C22C2×D4×D10111yescharacteristic
189C4—42no—
190C4—42no—
191C4—1014no—
192C4—4042no—
193C4—4042no—
194C4—2022no—
195C4#1569224112yes—
196C4C2×#1310807311yescharacteristic
197C4C2×#1310807411yescharacteristic
198C2—22no—
199C2C2×#1569224111yescharacteristic
200C2#137×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#419×C101C2
socleC22×C101C22×D4
prime radicalC23×C101C2×D4
hypercenter#419D101
hypocenterC101#3139
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20265544C2semidirect
4. #20265526C2semidirect
101. C101#3139semidirect

Maximal subgroups

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

subgroupindex
#202657302
C2×#156922532
#202655442
#202655262
#202651872
#202651862
#202651852
M16×Dic1012
#419×C1012
#3139101