Finite Group Databasebeta

← #28389684

Subgroups — #28389684

Order 14464.

216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 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#20298946C211yesmaximal, characteristic, complemented
2#20298798C211yesmaximal, characteristic, complemented
3#20298793C211yesmaximal, characteristic
4#20298344C211yesmaximal, characteristic, complemented
5Q8⋊#11744751C211yesmaximal, characteristic, complemented
6D4⋊#11744751C211yesmaximal, characteristic, complemented
7#556×C113C211yesmaximal, characteristic, complemented
8C2×#14268634—21no—
9C2×#14268632—21no—
10#16813619C2211yescharacteristic
11C4⋊#11744751C2211yescharacteristic
12C4⋊#11744751C2211yescharacteristic
13#16813542—21no—
14#16813542—21no—
15#16813542—22no—
16C4×#11744751C2211yescharacteristic
17C4⋊Q8×C113C2211yescharacteristic
18#126×C113—22no—
19#124×C113—21no—
20C22⋊Q8×C113—21no—
21C4×Q8×C113—21no—
22C4×Q8×C113C2211yescharacteristic
23C4×D4×C113C2211yescharacteristic
24#117×C113—21no—
25C2×D452—21no—
26C2×D452—21no—
27C2×D452—21no—
28#14268634—42no—
29#14268632—42no—
30C2×#11744751—21no—
31C2×#11744751—21no—
32C2×#11744751—21no—
33C2×Q8×C113—21no—
34C2×Q8×C113D411yescharacteristic
35C2×D4×C113D411yescharacteristic
36C22×C4×C113—21no—
37C4⋊C4×C113—22no—
38C4⋊C4×C113—21no—
39C4⋊C4×C113—21no—
40C4⋊C4×C113D411yescharacteristic
41C4⋊C4×C113D411yescharacteristic
42C4⋊C4×C113D412yes—
43C22⋊C4×C113—22no—
44C22⋊C4×C113—22no—
45C22⋊C4×C113—21no—
46C42×C113—22no—
47C42×C113—21no—
48C42×C113C2311yescharacteristic
49C23×C113—21no—
50C22×D113—41no—
51Q8×C113—21no—
52Q8×C113—21no—
53Q8×C113—41no—
54Q8×C113—41no—
55D4×C113—21no—
56D4×C113—21no—
57C2×C4×C113—21no—
58C2×C4×C113—21no—
59C2×C4×C113—21no—
60C2×C4×C113—22no—
61C2×C4×C113—41no—
62C2×C4×C113—41no—
63C2×C4×C113C2×D411yescharacteristic
64C2×C4×C113C2×D411yescharacteristic
65C2×C4×C113C2×D411yescharacteristic
66D452—42no—
67D452—41no—
68D452—41no—
69D452—41no—
70D452—41no—
71#11744751—41no—
72#11744751—41no—
73#11744751—41no—
74C22×C113—41no—
75C22×C113—41no—
76C22×C113—41no—
77C22×C113#12011yescharacteristic
78C2×D113—81no—
79C2×D113—81no—
80C2×D113—81no—
81C4×C113—21no—
82C4×C113—41no—
83C4×C113—41no—
84C4×C113—42no—
85C4×C113—41no—
86C4×C113#13511yescharacteristic
87C4×C113#13511yescharacteristic
88C4×C113AGL(1,ℤ/8)11yescharacteristic
89C4×C113AGL(1,ℤ/8)11yescharacteristic
90C2×C113—81no—
91C2×C113#44911yescharacteristic
92C2×C113#44911yescharacteristic
93C2×C113#45411yescharacteristic
94D113—161no—
95#1628—1131nomaximal, Sylow
96C113#162811yescharacteristic, complemented, Sylow
97#556D11311yescharacteristic, complemented
98#486—1131no—
99#460—1131no—
100#459—1131no—
101#332—1131no—
102Q8⋊C8—1131no—
103D4⋊C8—1131no—
104C2×SD16—2261no—
105C2×D8—2261no—
106C4⋊Q8C2×D11311yescharacteristic
107#127—1131no—
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C2×D11311yescharacteristic
113C4×D4C2×D11311yescharacteristic
114#117—21no—
115C4⋊C8—1131no—
116C4⋊C8—1131no—
117#102—2261no—
118#102—2261no—
119#102—2262no—
120C4×C8—1131no—
121C2×Q8—21no—
122C2×Q8C22⋊D11311yescharacteristic
123C2×D4—2261no—
124C2×D4—2261no—
125C2×D4—2261no—
126C2×D4C22⋊D11311yescharacteristic
127C22×C4—21no—
128SD16—4522no—
129D8—4522no—
130C2×C8—2261no—
131C2×C8—2261no—
132C2×C8—2261no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D11311yescharacteristic
137C4⋊C4C22⋊D11311yescharacteristic
138C4⋊C4C22⋊D11312yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C22×D11311yescharacteristic
145C23—21no—
146C23—4521no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—4522no—
154D4—4521no—
155D4—4521no—
156D4—4521no—
157D4—4521no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×C22⋊D11311yescharacteristic
165C2×C4C2×C22⋊D11311yescharacteristic
166C2×C4C2×C22⋊D11311yescharacteristic
167C8—4521no—
168C8—4521no—
169C8—4521no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—9041no—
174C22—9041no—
175C22—9041no—
176C22#1681359111yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—41no—
181C4—42no—
182C4#1681364211yescharacteristic
183C4#1681364211yescharacteristic
184C4#1681364711yescharacteristic
185C4#1681364711yescharacteristic
186C2—81no—
187C2—18081no—
188C2#2029874311yescharacteristic
189C2#2029874311yescharacteristic
190C2#2029877111yescharacteristic
subgroupG/H
centerC22#16813591
commutator, cosocleC42×C113C23
FrattiniC42C22×D113
Fitting#556×C113C2
socleC22×C113#120
prime radical#20298793C2
hypercenter#556D113
hypocenterC113#1628
LCSG ▷ C42×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20298946C2semidirect
2. #20298798C2semidirect
4. #20298344C2semidirect
5. Q8⋊#11744751C2semidirect
6. D4⋊#11744751C2semidirect
7. #556×C113C2semidirect
96. C113#1628semidirect
97. #556D113semidirect

Maximal subgroups

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

subgroupindex
#202989462
#202987982
#202987932
#202983442
Q8⋊#117447512
D4⋊#117447512
#556×C1132
#1628113