Finite Group Databasebeta

← #49131786

Subgroups — #49131786

Order 36224.

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

normal subgroups only

This group has 340 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#31827092C212yesmaximal
2#31826872C212yesmaximal, complemented
3#31826871C212yesmaximal, complemented
4#31826867C211yesmaximal, characteristic, complemented
5#31826865C211yesmaximal, characteristic, complemented
6#31826862C211yesmaximal, characteristic, complemented
7#31826504C212yesmaximal
8#31826500C211yesmaximal, characteristic
9#106×D283C211yesmaximal, characteristic
10#31826419C211yesmaximal, characteristic
11#498×C283C211yesmaximal, characteristic, complemented
12C2×#17985375C2211yescharacteristic
13C2×C8×D283C2211yescharacteristic
14#22583609C2212yes—
15#22583608—22no—
16#22583602C2212yes—
17#22583601C2212yes—
18#22583600—22no—
19#22583588—22no—
20#22583586—22no—
21#22583568C2212yes—
22#22583567C2211yescharacteristic
23C4⋊C4×D283C2212yes—
24#22583544C2212yes—
25#22583543C2212yes—
26#22583543C2212yes—
27#22583542C2211yescharacteristic
28#22583542C2211yescharacteristic
29#22583541C2212yes—
30#22583541C2212yes—
31#22583521C2211yescharacteristic
32#22583520C2211yescharacteristic
33C8×Dic283C2211yescharacteristic
34C4⋊Q8×C283C2212yes—
35#106×C283C2211yescharacteristic
36#106×C283C2211yescharacteristic
37#106×C283C2212yes—
38C4×C8×C283C2211yescharacteristic
39C2×Dic566—22no—
40C2×C4×D283—22no—
41C2×C4×D283C2311yescharacteristic
42#17985375Q814yescomplemented
43C2×C283⋊C8C2311yescharacteristic
44C2×C283⋊C8C2311yescharacteristic
45C8×D283Q814yescomplemented
46C4⋊Dic283—22no—
47C4⋊Dic283C2312yes—
48C4⋊Dic283C2312yes—
49Dic283⋊C4—22no—
50Dic283⋊C4—22no—
51Dic283⋊C4—22no—
52Dic283⋊C4—22no—
53Dic283⋊C4—22no—
54Dic283⋊C4D412yes—
55(C2×D283)⋊C4—22no—
56(C2×D283)⋊C4D412yes—
57C4×Dic283—22no—
58C4×Dic283C2311yescharacteristic
59C2×Q8×C283—22no—
60C2×C8×C283C2311yescharacteristic
61C2×C8×C283C2311yescharacteristic
62C4⋊C4×C283—22no—
63C4⋊C4×C283C2312yes—
64C4⋊C4×C283C2312yes—
65C42×C283C2311yescharacteristic
66C22×D283C2×D411yescharacteristic
67Dic566—82no—
68C2×Dic283—22no—
69C2×Dic283—22no—
70C2×Dic283C2×D412yes—
71C2×Dic283C2×D411yescharacteristic
72C4×D283—82no—
73C4×D283C2×Q812yes—
74C4×D283C2×Q812yes—
75C283⋊C8C2×Q812yes—
76C283⋊C8C2×Q812yes—
77Q8×C283—82no—
78C2×C4×C283—22no—
79C2×C4×C283—22no—
80C2×C4×C283C2×D412yes—
81C2×C4×C283C2411yescharacteristic
82C8×C283C2×Q812yes—
83C8×C283C2×Q812yes—
84C22×C283C22×D411yescharacteristic
85C2×D283—22no—
86C2×D283C4⋊Q812yes—
87C4×C283—22no—
88C4×C283—42no—
89C4×C283—42no—
90C4×C283C22×Q811yescharacteristic
91C4×C283C22×Q811yescharacteristic
92Dic283—22no—
93Dic283—42no—
94Dic283—42no—
95Dic283C4⋊Q812yes—
96C2×C283C2×C4⋊Q811yescharacteristic
97C2×C283#57712yes—
98D283—22no—
99C283#316311yescharacteristic, complemented, Sylow
100#3163—2831nomaximal, Sylow
101#544—2832no—
102#501—2832no—
103#501—2832no—
104#499—2831no—
105#499—2831no—
106#498—2831no—
107#498D28311yescharacteristic, complemented
108#428—2832no—
109#427—2831no—
110C2×#106—2831no—
111#405—2831no—
112C2×M16—2831no—
113C22×C8—2831no—
114C4⋊Q8—2832no—
115C4⋊Q8—5662no—
116C4⋊Q8C2×D28312yes—
117#125—2832no—
118#125—2832no—
119#125—5662no—
120C22⋊Q8—5662no—
121C22⋊Q8—5662no—
122#117—2831no—
123#117—2832no—
124C2×C4⋊C4—2832no—
125#107—2832no—
126#107—2832no—
127#107—2832no—
128#106—2831no—
129#106—2831no—
130#106—2832no—
131#106—2832no—
132#106C2×D28311yescharacteristic
133#106C2×D28311yescharacteristic
134#106C2×D28312yes—
135#97—2831no—
136#97—2831no—
137C4×C8—2831no—
138C4×C8C2×D28311yescharacteristic
139C2×Q8—22no—
140C2×Q8—5662no—
141C22×C4—2831no—
142C22×C4—5662no—
143M16—2834no—
144C2×C8—2831no—
145C2×C8—2831no—
146C2×C8—2834no—
147C2×C8C22×D28311yescharacteristic
148C2×C8C22×D28311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—2832no—
151C4⋊C4—2832no—
152C4⋊C4—2832no—
153C4⋊C4—5662no—
154C4⋊C4—5662no—
155C4⋊C4—5662no—
156C4⋊C4—5662no—
157C4⋊C4—5662no—
158C4⋊C4—5662no—
159C4⋊C4C22×D28312yes—
160C4⋊C4C22×D28312yes—
161C22⋊C4—2832no—
162C22⋊C4—5662no—
163C42—2831no—
164C42—5662no—
165C42C22×D28311yescharacteristic
166C23—2831no—
167Q8—82no—
168Q8—22642no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—2831no—
172C2×C4—2832no—
173C2×C4—2832no—
174C2×C4—2832no—
175C2×C4—5662no—
176C2×C4—5662no—
177C2×C4—22642no—
178C2×C4D4×D28312yes—
179C2×C4C23×D28311yescharacteristic
180C8—2832no—
181C8—2832no—
182C8Q8×D28312yes—
183C8Q8×D28312yes—
184C22—2832no—
185C22—5662no—
186C22C2×D4×D28311yescharacteristic
187C4—22no—
188C4—42no—
189C4—42no—
190C4—2832no—
191C4—5662no—
192C4—11322no—
193C4—11322no—
194C4C2×Q8×D28311yescharacteristic
195C4C2×Q8×D28311yescharacteristic
196C2—5662no—
197C2C4⋊Q8×D28311yescharacteristic
198C2#3182727112yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#498×C283C2
socleC22×C283C22×D4
prime radicalC22×D283C2×D4
hypercenter#498D283
hypocenterC283#3163
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #31826872C2semidirect
3. #31826871C2semidirect
4. #31826867C2semidirect
5. #31826865C2semidirect
6. #31826862C2semidirect
11. #498×C283C2semidirect
42. #17985375Q8semidirect
45. C8×D283Q8semidirect
99. C283#3163semidirect
107. #498D283semidirect

Maximal subgroups

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

subgroupindex
#318270922
#318268722
#318268712
#318268672
#318268652
#318268622
#318265042
#318265002
#106×D2832
#318264192
#498×C2832
#3163283