Finite Group Databasebeta

← #44562972

Subgroups — #44562972

Order 32128.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#30632387C212yesmaximal, complemented
2#30632089C211yesmaximal, characteristic, complemented
3#30632009C212yesmaximal, complemented
4#30631999C212yesmaximal, complemented
5#30631987C212yesmaximal, complemented
6#30631937C212yesmaximal, complemented
7#30631766C211yesmaximal, characteristic, complemented
8#30631728C212yesmaximal, complemented
9C4×D8×C251C211yesmaximal, characteristic, complemented
10C2×#16860793C2212yes—
11#22548708—22no—
12#22548707C2212yes—
13#22548698—22no—
14#22548688—22no—
15#22548686C2212yes—
16#22548686C2212yes—
17C4×D1004C2211yescharacteristic
18C4×C22⋊D251—22no—
19#22548673C2212yes—
20C4⋊C4×D251C2212yes—
21#22548649C2211yescharacteristic
22#22548649C2211yescharacteristic
23#22548648C2212yes—
24#22548647C2212yes—
25#22548646C2212yes—
26C4⋊(C251⋊C8)C2211yescharacteristic
27#22548636C2212yes—
28#22548636C2212yes—
29(C2×D251)⋊C8C2212yes—
30C2×D8×C251C2211yescharacteristic
31C4×D4×C251C2212yes—
32#107×C251C2211yescharacteristic
33#102×C251C2212yes—
34C4×C8×C251C2211yescharacteristic
35C2×D1004D411yescharacteristic
36C2×C22⋊D251—22no—
37C2×C22⋊D251—22no—
38C2×C4×D251—22no—
39C2×C4×D251C2312yes—
40#16860793—24no—
41C2×C251⋊C8C2312yes—
42C4⋊Dic251D411yescharacteristic
43C4⋊Dic251C2312yes—
44C4⋊Dic251C2312yes—
45Dic251⋊C4—22no—
46Dic251⋊C4—22no—
47Dic251⋊C4—22no—
48C22⋊Dic251—22no—
49C22⋊Dic251—22no—
50(C2×D251)⋊C4—22no—
51(C2×D251)⋊C4D412yes—
52C4×Dic251—22no—
53C2×D4×C251C2312yes—
54C22×C4×C251—22no—
55D8×C251—22no—
56C2×C8×C251C2311yescharacteristic
57C2×C8×C251C2311yescharacteristic
58C4⋊C4×C251C2312yes—
59C22⋊C4×C251—22no—
60C42×C251C2311yescharacteristic
61C22×D251C2×D412yes—
62D1004—41no—
63C22⋊D251—44no—
64C22⋊D251—82no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251C2×D412yes—
68C4×D251—22no—
69C4×D251—22no—
70C4×D251—82no—
71C251⋊C8—22no—
72C23×C251—22no—
73D4×C251—22no—
74D4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—82no—
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C2×C4×C251C2411yescharacteristic
80C8×C251—21no—
81C8×C251#4112yes—
82C22×C251—42no—
83C22×C251—42no—
84C22×C251—42no—
85C22×C251C22×D411yescharacteristic
86C2×D251—22no—
87C2×D251—22no—
88C2×D251—22no—
89C4×C251—21no—
90C4×C251—21no—
91C4×C251—42no—
92C4×C251C2×#4111yescharacteristic
93C4×C2512−1+411yescharacteristic
94Dic251—22no—
95Dic251—42no—
96Dic251—42no—
97C2×C251—82no—
98C2×C251#54711yescharacteristic
99C2×C251#57511yescharacteristic
100C2×C251#57611yescharacteristic
101D251—42no—
102C251#335511yescharacteristic, complemented, Sylow
103#3355—2511nomaximal, Sylow
104#553—2512no—
105#499—2511no—
106#482—2512no—
107#480—2512no—
108#478—2512no—
109#469—2512no—
110C4×D8D25111yescharacteristic, complemented
111#436—2511no—
112#428—2512no—
113C2×D8C2×D25111yescharacteristic
114C2×M16—2512no—
115#126—5022no—
116#125—2512no—
117#123—5022no—
118#121—5022no—
119#121—2512no—
120#121—2512no—
121C4×D4—5022no—
122C4×D4—2511no—
123C4×D4C2×D25112yes—
124#117—2512no—
125C2×C4⋊C4—2512no—
126#107—2511no—
127#107—2511no—
128#107—2512no—
129#107C2×D25111yescharacteristic
130#106—2512no—
131#106—2512no—
132C4⋊C8—2511no—
133#102—2512no—
134#102—2512no—
135#102C2×D25112yes—
136C22⋊C8—2512no—
137C4×C8C2×D25111yescharacteristic
138C2×D4—5022no—
139C2×D4—5022no—
140C2×D4—2511no—
141C2×D4C22×D25112yes—
142C22×C4—22no—
143C22×C4—5022no—
144C22×C4—2512no—
145D8—22no—
146M16—5024no—
147C2×C8—2512no—
148C2×C8C22×D25111yescharacteristic
149C2×C8C22×D25111yescharacteristic
150C4⋊C4—5022no—
151C4⋊C4—5022no—
152C4⋊C4—5022no—
153C4⋊C4—2512no—
154C4⋊C4—2512no—
155C4⋊C4—2511no—
156C4⋊C4C22×D25112yes—
157C22⋊C4—22no—
158C22⋊C4—5022no—
159C22⋊C4—5022no—
160C22⋊C4—5022no—
161C22⋊C4—2512no—
162C42—5022no—
163C42C22×D25111yescharacteristic
164C23—22no—
165C23—2512no—
166D4—22no—
167D4—22no—
168D4—20082no—
169D4—10041no—
170D4—10044no—
171C2×C4—22no—
172C2×C4—82no—
173C2×C4—20082no—
174C2×C4—5022no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—5022no—
178C2×C4—2512no—
179C2×C4D4×D25111yescharacteristic
180C2×C4D4×D25111yescharacteristic
181C2×C4C23×D25111yescharacteristic
182C8—21no—
183C8—5022no—
184C8#1686081012yes—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—5022no—
189C22—5022no—
190C22—5022no—
191C22C2×D4×D25111yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—10042no—
196C4—10042no—
197C4—5022no—
198C4C2×#1686081011yescharacteristic
199C4#2254876211yescharacteristic
200C2—82no—
201C2—10042no—
202C2#3063234211yescharacteristic
203C2#3063248411yescharacteristic
204C2#3063248811yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×D8×C251C2
socleC22×C251C22×D4
hypercenterC4×D8D251
hypocenterC251#3355
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #30632387C2semidirect
2. #30632089C2semidirect
3. #30632009C2semidirect
4. #30631999C2semidirect
5. #30631987C2semidirect
6. #30631937C2semidirect
7. #30631766C2semidirect
8. #30631728C2semidirect
9. C4×D8×C251C2semidirect
102. C251#3355semidirect
110. C4×D8D251semidirect

Maximal subgroups

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

subgroupindex
#306323872
#306320892
#306320092
#306319992
#306319872
#306319372
#306317662
#306317282
C4×D8×C2512
#3355251