Finite Group Databasebeta

← #27172820

Subgroups — #27172820

Order 12928.

230 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265752C211yesmaximal, characteristic
2#101×D101C212yesmaximal, complemented
3#20264976C211yesmaximal, characteristic, complemented
4#20264973C212yesmaximal, complemented
5#356×C101C211yesmaximal, characteristic, complemented
6C2×#13108075C411yescharacteristic
7C2×Q8×D101C2211yescharacteristic
8M16×D101—22no—
9#15692253C411yescharacteristic
10#15692244C2211yescharacteristic
11#15692243C411yescharacteristic
12#15692240—22no—
13#15692233—22no—
14#15692212C411yescharacteristic
15#15692174C2212yes—
16#15692173—22no—
17C4⋊Q8×C101C2211yescharacteristic
18#101×C101C2212yes—
19#13108075—42no—
20C2×Dic202C2×C411yescharacteristic
21C2×Dic202D412yes—
22Q8×D101—42no—
23C2×D404—21no—
24C2×D404C2×C411yescharacteristic
25C2×C4×D101—21no—
26C2×C4×D101C2×C411yescharacteristic
27C2×C4×D101D412yes—
28#13108058—22no—
29#13108057—42no—
30C8×D101—42no—
31#13108053—21no—
32#13108053—22no—
33#13108051—21no—
34#13108051—22no—
35#13108051—22no—
36C4×Dic101C2×C411yescharacteristic
37C2×Q8×C101C2×C411yescharacteristic
38C2×Q8×C101C2311yescharacteristic
39M16×C101—22no—
40C4⋊C4×C101—22no—
41C42×C101C2×C411yescharacteristic
42C22×D101—21no—
43C22×D101C22⋊C411yescharacteristic
44Q8×C101—41no—
45Q8×C101—41no—
46C2×C4×C101—21no—
47C2×C4×C101—21no—
48C2×C4×C101C22×C411yescharacteristic
49C2×C4×C101C2×D412yes—
50C2×Dic101—21no—
51C2×Dic101C22⋊C412yes—
52C2×Dic101C22⋊C411yescharacteristic
53D404—41no—
54D404—42no—
55C4×D101—22no—
56C4×D101—22no—
57C4×D101—22no—
58C4×D101—42no—
59Dic202—41no—
60Dic202—42no—
61C8×C101—42no—
62#11717579—42no—
63C22×C101C2×C22⋊C411yescharacteristic
64C2×D101—41no—
65C2×D101—42no—
66C2×D101—41no—
67C2×D101#9912yes—
68C4×C101—22no—
69C4×C101—21no—
70C4×C101—41no—
71C4×C101—41no—
72Dic101—21no—
73Dic101—22no—
74Dic101—41no—
75C2×C101—21no—
76C2×C101C2×#9911yescharacteristic
77D101—22no—
78D101—81no—
79#2133—1011nomaximal, Sylow
80C101#213311yescharacteristic, complemented, Sylow
81#536—1011no—
82C2×#101—1012no—
83#356—1011no—
84#356D10111yescharacteristic, complemented
85#355—1012no—
86C2×#41—1011no—
87C22×Q8—1011no—
88C2×M16—2022no—
89C4⋊Q8—1011no—
90C4⋊Q8C2×D10111yescharacteristic
91#124—1011no—
92#124—1011no—
93#123—2022no—
94C22⋊Q8—2022no—
95#117—1011no—
96#101—1012no—
97#101—2022no—
98#101C2×D10112yes—
99#41—4042no—
100C2×Q8—1011no—
101C2×Q8—1012no—
102C2×Q8—4042no—
103C2×Q8C4×D10111yescharacteristic
104C2×Q8C22×D10111yescharacteristic
105C2×D4—1011no—
106C2×D4—2021no—
107C22×C4—1011no—
108C22×C4—1012no—
109C22×C4—2021no—
110M16—22no—
111M16—4042no—
112M16—2022no—
113C2×C8—4042no—
114C4⋊C4—22no—
115C4⋊C4—2021no—
116C4⋊C4—2022no—
117C22⋊C4—2022no—
118C22⋊C4—2021no—
119C22⋊C4—2022no—
120C42—1011no—
121C42C4×D10111yescharacteristic
122C23—1011no—
123C23—2021no—
124Q8—41no—
125Q8—41no—
126Q8—4042no—
127Q8—4041no—
128D4—4041no—
129D4—4042no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—1012no—
133C2×C4—1011no—
134C2×C4—4042no—
135C2×C4—2022no—
136C2×C4—2022no—
137C2×C4—2022no—
138C2×C4—2021no—
139C2×C4C2×C4×D10111yescharacteristic
140C2×C4D4×D10112yes—
141C8—42no—
142C8—4042no—
143C22—1012no—
144C22—4042no—
145C22—4041no—
146C22—4041no—
147C22C22⋊C4×D10111yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—4041no—
153C4—2021no—
154C4—2022no—
155C2—21no—
156C2—8081no—
157C2—2022no—
158C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×Q8C22×D101
Fitting#356×C101C2
socleC2×C101C2×#99
prime radicalC2×#13108075C4
cosocleC2×Q8×C101C23
hypercenter#356D101
hypocenterC101#2133
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #101×D101C2semidirect
3. #20264976C2semidirect
4. #20264973C2semidirect
5. #356×C101C2semidirect
80. C101#2133semidirect
84. #356D101semidirect

Maximal subgroups

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

subgroupindex
#202657522
#101×D1012
#202649762
#202649732
#356×C1012
#2133101