Finite Group Databasebeta

← #44560890

Subgroups — #44560890

Order 32128.

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#30632314C212yesmaximal
2#30632094C212yesmaximal, complemented
3#30632093C212yesmaximal, complemented
4#30632089C211yesmaximal, characteristic, complemented
5#30632087C211yesmaximal, characteristic, complemented
6#30632084C211yesmaximal, characteristic, complemented
7#30631726C212yesmaximal
8#30631722C211yesmaximal, characteristic
9#106×D251C211yesmaximal, characteristic
10#30631641C211yesmaximal, characteristic
11#498×C251C211yesmaximal, characteristic, complemented
12C2×#16860793C2211yescharacteristic
13C2×C8×D251C2211yescharacteristic
14#22548714C2212yes—
15#22548713—22no—
16#22548707C2212yes—
17#22548706C2212yes—
18#22548705—22no—
19#22548693—22no—
20#22548691—22no—
21#22548673C2212yes—
22#22548672C2211yescharacteristic
23C4⋊C4×D251C2212yes—
24#22548649C2212yes—
25#22548648C2212yes—
26#22548648C2212yes—
27#22548647C2211yescharacteristic
28#22548647C2211yescharacteristic
29#22548646C2212yes—
30#22548646C2212yes—
31#22548626C2211yescharacteristic
32#22548625C2211yescharacteristic
33C8×Dic251C2211yescharacteristic
34C4⋊Q8×C251C2212yes—
35#106×C251C2211yescharacteristic
36#106×C251C2211yescharacteristic
37#106×C251C2212yes—
38C4×C8×C251C2211yescharacteristic
39C2×Dic502—22no—
40C2×C4×D251—22no—
41C2×C4×D251C2311yescharacteristic
42#16860793Q814yescomplemented
43C2×C251⋊C8C2311yescharacteristic
44C2×C251⋊C8C2311yescharacteristic
45C8×D251Q814yescomplemented
46C4⋊Dic251—22no—
47C4⋊Dic251C2312yes—
48C4⋊Dic251C2312yes—
49Dic251⋊C4—22no—
50Dic251⋊C4—22no—
51Dic251⋊C4—22no—
52Dic251⋊C4—22no—
53Dic251⋊C4—22no—
54Dic251⋊C4D412yes—
55(C2×D251)⋊C4—22no—
56(C2×D251)⋊C4D412yes—
57C4×Dic251—22no—
58C4×Dic251C2311yescharacteristic
59C2×Q8×C251—22no—
60C2×C8×C251C2311yescharacteristic
61C2×C8×C251C2311yescharacteristic
62C4⋊C4×C251—22no—
63C4⋊C4×C251C2312yes—
64C4⋊C4×C251C2312yes—
65C42×C251C2311yescharacteristic
66C22×D251C2×D411yescharacteristic
67Dic502—82no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251C2×D412yes—
71C2×Dic251C2×D411yescharacteristic
72C4×D251—82no—
73C4×D251C2×Q812yes—
74C4×D251C2×Q812yes—
75C251⋊C8C2×Q812yes—
76C251⋊C8C2×Q812yes—
77Q8×C251—82no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251C2×D412yes—
81C2×C4×C251C2411yescharacteristic
82C8×C251C2×Q812yes—
83C8×C251C2×Q812yes—
84C22×C251C22×D411yescharacteristic
85C2×D251—22no—
86C2×D251C4⋊Q812yes—
87C4×C251—22no—
88C4×C251—42no—
89C4×C251—42no—
90C4×C251C22×Q811yescharacteristic
91C4×C251C22×Q811yescharacteristic
92Dic251—22no—
93Dic251—42no—
94Dic251—42no—
95Dic251C4⋊Q812yes—
96C2×C251C2×C4⋊Q811yescharacteristic
97C2×C251#57712yes—
98D251—22no—
99C251#316311yescharacteristic, complemented, Sylow
100#3163—2511nomaximal, Sylow
101#544—2512no—
102#501—2512no—
103#501—2512no—
104#499—2511no—
105#499—2511no—
106#498—2511no—
107#498D25111yescharacteristic, complemented
108#428—2512no—
109#427—2511no—
110C2×#106—2511no—
111#405—2511no—
112C2×M16—2511no—
113C22×C8—2511no—
114C4⋊Q8—5022no—
115C4⋊Q8—2512no—
116C4⋊Q8C2×D25112yes—
117#125—5022no—
118#125—2512no—
119#125—2512no—
120C22⋊Q8—5022no—
121C22⋊Q8—5022no—
122#117—2512no—
123#117—2511no—
124C2×C4⋊C4—2512no—
125#107—2512no—
126#107—2512no—
127#107—2512no—
128#106—2511no—
129#106—2511no—
130#106—2512no—
131#106—2512no—
132#106C2×D25111yescharacteristic
133#106C2×D25111yescharacteristic
134#106C2×D25112yes—
135#97—2511no—
136#97—2511no—
137C4×C8—2511no—
138C4×C8C2×D25111yescharacteristic
139C2×Q8—22no—
140C2×Q8—5022no—
141C22×C4—5022no—
142C22×C4—2511no—
143M16—2514no—
144C2×C8—2511no—
145C2×C8—2511no—
146C2×C8—2514no—
147C2×C8C22×D25111yescharacteristic
148C2×C8C22×D25111yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—5022no—
151C4⋊C4—5022no—
152C4⋊C4—5022no—
153C4⋊C4—5022no—
154C4⋊C4—5022no—
155C4⋊C4—5022no—
156C4⋊C4—2512no—
157C4⋊C4—2512no—
158C4⋊C4—2512no—
159C4⋊C4C22×D25112yes—
160C4⋊C4C22×D25112yes—
161C22⋊C4—5022no—
162C22⋊C4—2512no—
163C42—5022no—
164C42—2511no—
165C42C22×D25111yescharacteristic
166C23—2511no—
167Q8—82no—
168Q8—20082no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—20082no—
172C2×C4—5022no—
173C2×C4—5022no—
174C2×C4—2512no—
175C2×C4—2511no—
176C2×C4—2512no—
177C2×C4—2512no—
178C2×C4D4×D25112yes—
179C2×C4C23×D25111yescharacteristic
180C8—2512no—
181C8—2512no—
182C8Q8×D25112yes—
183C8Q8×D25112yes—
184C22—5022no—
185C22—2512no—
186C22C2×D4×D25111yescharacteristic
187C4—22no—
188C4—42no—
189C4—42no—
190C4—10042no—
191C4—10042no—
192C4—5022no—
193C4—2512no—
194C4C2×Q8×D25111yescharacteristic
195C4C2×Q8×D25111yescharacteristic
196C2—5022no—
197C2C4⋊Q8×D25111yescharacteristic
198C2#3063249312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#498×C251C2
socleC22×C251C22×D4
prime radicalC22×D251C2×D4
hypercenter#498D251
hypocenterC251#3163
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #30632094C2semidirect
3. #30632093C2semidirect
4. #30632089C2semidirect
5. #30632087C2semidirect
6. #30632084C2semidirect
11. #498×C251C2semidirect
42. #16860793Q8semidirect
45. C8×D251Q8semidirect
99. C251#3163semidirect
107. #498D251semidirect

Maximal subgroups

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

subgroupindex
#306323142
#306320942
#306320932
#306320892
#306320872
#306320842
#306317262
#306317222
#106×D2512
#306316412
#498×C2512
#3163251