Finite Group Databasebeta

← #44562969

Subgroups — #44562969

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#30632390C212yesmaximal, complemented
2#30632014C212yesmaximal, complemented
3#30632002C212yesmaximal, complemented
4#30631984C212yesmaximal, complemented
5#30631941C212yesmaximal, complemented
6C4×D2008C211yesmaximal, characteristic, complemented
7#30631766C211yesmaximal, characteristic, complemented
8#30631726C212yesmaximal, complemented
9#499×C251C211yesmaximal, characteristic, complemented
10C2×D2008C2211yescharacteristic
11C2×#16860793C2212yes—
12#22548710—22no—
13#22548700—22no—
14#22548690—22no—
15#22548690C2212yes—
16#22548690C2212yes—
17C4×D1004C2212yes—
18C4×D1004C2211yescharacteristic
19#22548677—22no—
20#22548673C2212yes—
21C4⋊C4×D251C2212yes—
22#22548649C2211yescharacteristic
23#22548648C2212yes—
24#22548646C2212yes—
25C4⋊(C251⋊C8)C2211yescharacteristic
26#22548637C2212yes—
27#22548635C2212yes—
28#22548635C2212yes—
29(C2×D251)⋊C8C2212yes—
30#125×C251C2212yes—
31#107×C251C2211yescharacteristic
32#107×C251C2211yescharacteristic
33#106×C251C2212yes—
34C4×C8×C251C2211yescharacteristic
35C2×D1004—22no—
36C2×D1004—22no—
37C2×D1004D411yescharacteristic
38C2×D1004C2312yes—
39C2×C4×D251—22no—
40C2×C4×D251—22no—
41C2×C4×D251C2312yes—
42D2008—22no—
43#16860793—24no—
44C2×C251⋊C8C2312yes—
45C4⋊Dic251D411yescharacteristic
46C4⋊Dic251C2312yes—
47Dic251⋊C4—22no—
48(C2×D251)⋊C4—22no—
49(C2×D251)⋊C4—22no—
50(C2×D251)⋊C4—22no—
51(C2×D251)⋊C4—22no—
52(C2×D251)⋊C4D412yes—
53C4×Dic251—22no—
54C2×C8×C251C2311yescharacteristic
55C2×C8×C251C2311yescharacteristic
56C4⋊C4×C251—22no—
57C4⋊C4×C251—22no—
58C4⋊C4×C251C2312yes—
59C4⋊C4×C251C2312yes—
60C42×C251C2311yescharacteristic
61C22×D251—22no—
62C22×D251C2×D412yes—
63D1004—22no—
64D1004—22no—
65D1004—41no—
66D1004—44no—
67D1004—82no—
68C2×Dic251—22no—
69C2×Dic251C2×D412yes—
70C4×D251—22no—
71C4×D251—22no—
72C4×D251—82no—
73C4×D251—82no—
74C251⋊C8—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C2×C4×C251C2411yescharacteristic
80C8×C251—21no—
81C8×C251#4112yes—
82C22×C251C22×D411yescharacteristic
83C2×D251—22no—
84C2×D251—22no—
85C2×D251—22no—
86C2×D251—42no—
87C2×D251—42no—
88C2×D251—42no—
89C4×C251—21no—
90C4×C251—21no—
91C4×C251—42no—
92C4×C251—42no—
93C4×C251C2×#4111yescharacteristic
94C4×C2512−1+411yescharacteristic
95Dic251—22no—
96Dic251—42no—
97C2×C251#54711yescharacteristic
98C2×C251#57511yescharacteristic
99C2×C251#57611yescharacteristic
100D251—42no—
101D251—82no—
102C251#335511yescharacteristic, complemented, Sylow
103#3355—2511nomaximal, Sylow
104#553—2512no—
105#499D25111yescharacteristic, complemented
106#482—2512no—
107#480—2512no—
108#478—2512no—
109#469—2512no—
110C4×D8—2511no—
111#436—2511no—
112#428—2512no—
113C2×D8—2511no—
114C2×M16—2512no—
115#126—5022no—
116#125C2×D25112yes—
117#123—5022no—
118#121—5022no—
119#121—2512no—
120#121—2512no—
121C4×D4—5022no—
122C4×D4—2512no—
123C4×D4—2511no—
124#117—2512no—
125C2×C4⋊C4—2512no—
126#107—2511no—
127#107—2512no—
128#107C2×D25111yescharacteristic
129#107C2×D25111yescharacteristic
130#106—2512no—
131#106C2×D25112yes—
132C4⋊C8—2511no—
133#102—2512no—
134#102—2512no—
135#102—2512no—
136C22⋊C8—2512no—
137C4×C8C2×D25111yescharacteristic
138C2×D4—5022no—
139C2×D4—5022no—
140C2×D4—2511no—
141C2×D4—2512no—
142C22×C4—5022no—
143C22×C4—5022no—
144C22×C4—2512no—
145D8—5022no—
146M16—5024no—
147C2×C8—2512no—
148C2×C8C22×D25111yescharacteristic
149C2×C8C22×D25111yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—5022no—
153C4⋊C4—2512no—
154C4⋊C4—2511no—
155C4⋊C4C22×D25112yes—
156C4⋊C4C22×D25112yes—
157C22⋊C4—5022no—
158C22⋊C4—5022no—
159C22⋊C4—5022no—
160C22⋊C4—5022no—
161C22⋊C4—2512no—
162C42—5022no—
163C42C22×D25111yescharacteristic
164C23—5022no—
165C23—2512no—
166D4—20082no—
167D4—10041no—
168D4—10044no—
169D4—5022no—
170D4—5022no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—20082no—
174C2×C4—20082no—
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#1686081112yes—
185C22—10042no—
186C22—10042no—
187C22—10042no—
188C22—5022no—
189C22—5022no—
190C22—5022no—
191C22C2×D4×D25111yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—42no—
196C4—10042no—
197C4—5022no—
198C4C2×#1686081111yescharacteristic
199C4#2254876311yescharacteristic
200C2—20082no—
201C2—10042no—
202C2#3063234011yescharacteristic
203C2#3063248411yescharacteristic
204C2#3063248911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#499×C251C2
socleC22×C251C22×D4
hypercenter#499D251
hypocenterC251#3355
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #30632390C2semidirect
2. #30632014C2semidirect
3. #30632002C2semidirect
4. #30631984C2semidirect
5. #30631941C2semidirect
6. C4×D2008C2semidirect
7. #30631766C2semidirect
8. #30631726C2semidirect
9. #499×C251C2semidirect
102. C251#3355semidirect
105. #499D251semidirect

Maximal subgroups

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

subgroupindex
#306323902
#306320142
#306320022
#306319842
#306319412
C4×D20082
#306317662
#306317262
#499×C2512
#3355251