Finite Group Databasebeta

← #44562941

Subgroups — #44562941

Order 32128.

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

normal subgroups only

This group has 336 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#30632378C212yesmaximal
2#30632089C211yesmaximal, characteristic, complemented
3#30632027C212yesmaximal, complemented
4#30632021C212yesmaximal, complemented
5#30631994C212yesmaximal, complemented
6#30631942C212yesmaximal, complemented
7#30631766C211yesmaximal, characteristic
8#30631728C212yesmaximal, complemented
9C4×Q16×C251C211yesmaximal, characteristic, complemented
10C2×#16860793C2212yes—
11#22548709—22no—
12#22548707C2212yes—
13#22548705—22no—
14#22548700—22no—
15#22548691C2212yes—
16#22548691C2212yes—
17C4×D1004C2211yescharacteristic
18#22548673C2212yes—
19#22548672—22no—
20C4⋊C4×D251C2212yes—
21#22548649C2211yescharacteristic
22#22548649C2211yescharacteristic
23#22548648C2212yes—
24#22548647C2212yes—
25#22548646C2212yes—
26C4⋊(C251⋊C8)C2211yescharacteristic
27#22548639C2212yes—
28#22548639C2212yes—
29(C2×D251)⋊C8C2212yes—
30C2×Q16×C251C2211yescharacteristic
31C4×Q8×C251C2212yes—
32#107×C251C2211yescharacteristic
33#103×C251C2212yes—
34C4×C8×C251C2211yescharacteristic
35C2×D1004D411yescharacteristic
36C2×C4×D251—22no—
37C2×C4×D251C2312yes—
38#16860793—24no—
39C2×C251⋊C8C2312yes—
40C4⋊Dic251D411yescharacteristic
41C4⋊Dic251C2312yes—
42C4⋊Dic251C2312yes—
43Dic251⋊C4—22no—
44Dic251⋊C4—22no—
45Dic251⋊C4—22no—
46Dic251⋊C4—22no—
47Dic251⋊C4—22no—
48(C2×D251)⋊C4—22no—
49(C2×D251)⋊C4—22no—
50(C2×D251)⋊C4—22no—
51(C2×D251)⋊C4D412yes—
52C4×Dic251—22no—
53C2×Q8×C251C2312yes—
54Q16×C251—22no—
55C2×C8×C251C2311yescharacteristic
56C2×C8×C251C2311yescharacteristic
57C4⋊C4×C251—22no—
58C4⋊C4×C251C2312yes—
59C42×C251—22no—
60C42×C251C2311yescharacteristic
61C22×D251C2×D412yes—
62D1004—41no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251C2×D412yes—
66C4×D251—22no—
67C4×D251—22no—
68C4×D251—44no—
69C251⋊C8—22no—
70Q8×C251—22no—
71Q8×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C2×C4×C251C2411yescharacteristic
77C8×C251—21no—
78C8×C251#4112yes—
79C22×C251C22×D411yescharacteristic
80C2×D251—22no—
81C2×D251—22no—
82C2×D251—22no—
83C4×C251—21no—
84C4×C251—21no—
85C4×C251—42no—
86C4×C251—42no—
87C4×C251C2×#4111yescharacteristic
88C4×C2512−1+411yescharacteristic
89Dic251—22no—
90Dic251—42no—
91Dic251—42no—
92C2×C251#54711yescharacteristic
93C2×C251#57511yescharacteristic
94C2×C251#57811yescharacteristic
95D251—42no—
96C251#335211yescharacteristic, complemented, Sylow
97#3352—2511nomaximal, Sylow
98#552—2512no—
99#499—2511no—
100#485—2512no—
101#484—2512no—
102#479—2512no—
103#470—2512no—
104C4×Q16D25111yescharacteristic, complemented
105#436—2511no—
106#428—2512no—
107C2×Q16C2×D25111yescharacteristic
108C2×M16—2512no—
109#126—5022no—
110#125—5022no—
111#125—2512no—
112#123—5022no—
113C22⋊Q8—2512no—
114C22⋊Q8—2512no—
115C4×Q8C2×D25112yes—
116C4×D4—2511no—
117#117—5022no—
118#117—2512no—
119C2×C4⋊C4—2512no—
120#107—2511no—
121#107—2511no—
122#107—2512no—
123#107C2×D25111yescharacteristic
124#106—2512no—
125#106—2512no—
126C4⋊C8—2511no—
127#103—2512no—
128#103—2512no—
129#103C2×D25112yes—
130C22⋊C8—2512no—
131C4×C8C2×D25111yescharacteristic
132C2×Q8C22×D25112yes—
133C2×D4—2511no—
134C22×C4—5022no—
135C22×C4—2512no—
136Q16—22no—
137M16—5024no—
138C2×C8—2512no—
139C2×C8C22×D25111yescharacteristic
140C2×C8C22×D25111yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—5022no—
143C4⋊C4—5022no—
144C4⋊C4—5022no—
145C4⋊C4—5022no—
146C4⋊C4—5022no—
147C4⋊C4—2512no—
148C4⋊C4—2512no—
149C4⋊C4—2511no—
150C4⋊C4C22×D25112yes—
151C22⋊C4—5022no—
152C22⋊C4—5022no—
153C22⋊C4—5022no—
154C22⋊C4—2512no—
155C42—22no—
156C42—5022no—
157C42C22×D25111yescharacteristic
158C23—2512no—
159Q8—22no—
160Q8—22no—
161D4—10041no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—10044no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—2512no—
170C2×C4D4×D25111yescharacteristic
171C2×C4D4×D25111yescharacteristic
172C2×C4C23×D25111yescharacteristic
173C8—21no—
174C8—5022no—
175C8#1686081012yes—
176C22—5022no—
177C22—5022no—
178C22—5022no—
179C22C2×D4×D25111yescharacteristic
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—10042no—
185C4—10042no—
186C4—5022no—
187C4C2×#1686081011yescharacteristic
188C4#2254876211yescharacteristic
189C2—10042no—
190C2#3063234211yescharacteristic
191C2#3063248411yescharacteristic
192C2#3063250011yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×Q16×C251C2
socleC22×C251C22×D4
prime radicalC2×D1004D4
hypercenterC4×Q16D251
hypocenterC251#3352
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #30632089C2semidirect
3. #30632027C2semidirect
4. #30632021C2semidirect
5. #30631994C2semidirect
6. #30631942C2semidirect
8. #30631728C2semidirect
9. C4×Q16×C251C2semidirect
96. C251#3352semidirect
104. C4×Q16D251semidirect

Maximal subgroups

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

subgroupindex
#306323782
#306320892
#306320272
#306320212
#306319942
#306319422
#306317662
#306317282
C4×Q16×C2512
#3352251