Finite Group Databasebeta

← #44550593

Subgroups — #44550593

Order 32128.

302 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 97 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#30632170C211yesmaximal, characteristic
2#30631729C211yesmaximal, characteristic, complemented
3#30631688C211yesmaximal, characteristic
4#30631557C211yesmaximal, characteristic, complemented
5#30631455C211yesmaximal, characteristic, complemented
6#30631447C211yesmaximal, characteristic, complemented
7C2×#104×C251C211yesmaximal, characteristic
8C2×#16860794C2211yescharacteristic
9Q8×Dic251—21no—
10D4×Dic251—22no—
11D4×Dic251—21no—
12#22548674—21no—
13#22548674C2211yescharacteristic
14C2×C4⋊Dic251—21no—
15C2×C4⋊Dic251—22no—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2211yescharacteristic
18#22548649—21no—
19#22548647—21no—
20#22548639—21no—
21#22548636—21no—
22C2×#41×C251C2211yescharacteristic
23C2×M16×C251C2211yescharacteristic
24C2×C42×C251C2211yescharacteristic
25#104×C251C414yescomplemented
26C22×Dic251—22no—
27C22×Dic251—21no—
28C22×Dic251D412yes—
29#16860794—21no—
30#16860794—21no—
31C2×C251⋊C8—21no—
32C4⋊Dic251—22no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251—44no—
38C4⋊Dic251D411yescharacteristic
39C4⋊Dic251D411yescharacteristic
40C22⋊Dic251—22no—
41C22⋊Dic251—21no—
42C4×Dic251—22no—
43C4×Dic251—21no—
44#41×C251—21no—
45#41×C251—21no—
46#41×C251C2×C412yes—
47#41×C251Q812yes—
48C2×Q8×C251D411yescharacteristic
49C2×D4×C251—21no—
50C2×D4×C251D411yescharacteristic
51C22×C4×C251—21no—
52C22×C4×C251—21no—
53C22×C4×C251C2311yescharacteristic
54M16×C251—21no—
55M16×C251C2×C412yescomplemented
56C2×C8×C251—21no—
57C42×C251—21no—
58C42×C251C2×C412yescomplemented
59C2×Dic251—21no—
60C2×Dic251—22no—
61C2×Dic251—21no—
62C2×Dic251—42no—
63C2×Dic251—42no—
64C2×Dic251—41no—
65C2×Dic251—82no—
66C251⋊C8—41no—
67C23×C251—21no—
68C23×C251C2×D411yescharacteristic
69Q8×C251—21no—
70Q8×C251C4⋊C412yescomplemented
71D4×C251—22no—
72D4×C251—22no—
73D4×C251—21no—
74D4×C251C4⋊C412yescomplemented
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—21no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251C22×C411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×Q811yescharacteristic
85C2×C4×C251#4111yescharacteristic
86C2×C4×C251#4111yescharacteristic
87C8×C251—22no—
88C22×C251—21no—
89C22×C251—21no—
90C22×C251—22no—
91C22×C251—41no—
92C22×C251—41no—
93C22×C251C4×D411yescharacteristic
94C22×C251#12011yescharacteristic
95C22×C251C22⋊Q811yescharacteristic
96C4×C251—24no—
97C4×C251—22no—
98C4×C251C2×C4⋊C411yescharacteristic
99C4×C251C4×D411yescharacteristic
100C4×C251C22⋊Q811yescharacteristic
101C4×C251#12311yescharacteristic
102Dic251—42no—
103Dic251—41no—
104Dic251—41no—
105Dic251—81no—
106C2×C251—21no—
107C2×C251—21no—
108C2×C251—42no—
109C2×C251#38511yescharacteristic
110C2×C251#45311yescharacteristic
111C2×C251#45611yescharacteristic
112C251#185911yescharacteristic, complemented, Sylow
113#1859—2511nomaximal, Sylow
114#520—2511no—
115#428—2511no—
116C2×#104D25111yescharacteristic
117#417—2511no—
118#384—2511no—
119#343—2511no—
120#339—2511no—
121C2×#41C2×D25111yescharacteristic
122C2×M16—2511no—
123C2×M16C2×D25111yescharacteristic
124C4×Q8—5021no—
125C4×D4—5021no—
126C4×D4—5022no—
127#117—5021no—
128#117—2511no—
129C2×C4⋊C4—5022no—
130C2×C4⋊C4—5021no—
131C2×C4⋊C4—2511no—
132C2×C4⋊C4—2511no—
133C2×C42C2×D25111yescharacteristic
134#107—5021no—
135#106—5021no—
136#104Dic25114yescomplemented
137#103—5021no—
138#102—5021no—
139#41—21no—
140#41—21no—
141#41C2×Dic25112yes—
142#41Dic50212yes—
143C2×Q8D100411yescharacteristic
144C2×D4—21no—
145C2×D4D100411yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—5022no—
149C22×C4—5021no—
150C22×C4—2512no—
151C22×C4C22×D25111yescharacteristic
152M16—21no—
153M16—5021no—
154M16—5021no—
155M16C2×Dic25112yescomplemented
156C2×C8—21no—
157C2×C8—5021no—
158C4⋊C4—10044no—
159C4⋊C4—5022no—
160C4⋊C4—5021no—
161C4⋊C4—5021no—
162C4⋊C4—5021no—
163C4⋊C4—5021no—
164C4⋊C4—2511no—
165C4⋊C4—2511no—
166C22⋊C4—5022no—
167C22⋊C4—5021no—
168C42—21no—
169C42—5022no—
170C42—5021no—
171C42C2×Dic25112yescomplemented
172C23—21no—
173C23D4×D25111yescharacteristic
174Q8—21no—
175Q8C4⋊Dic25112yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4C4⋊Dic25112yescomplemented
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—20082no—
187C2×C4—10042no—
188C2×C4—10041no—
189C2×C4—10042no—
190C2×C4—5021no—
191C2×C4—5022no—
192C2×C4—5021no—
193C2×C4C22×Dic25111yescharacteristic
194C2×C4D4×D25111yescharacteristic
195C2×C4C2×D100411yescharacteristic
196C2×C4C2×Dic50211yescharacteristic
197C2×C4#1686080911yescharacteristic
198C2×C4#1686080911yescharacteristic
199C8—22no—
200C8—10041no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22D4×Dic25111yescharacteristic
207C22#2254868511yescharacteristic
208C22#2254869211yescharacteristic
209C4—24no—
210C4—22no—
211C4—20081no—
212C4—10041no—
213C4—10042no—
214C4—10041no—
215C4C2×C4⋊Dic25111yescharacteristic
216C4D4×Dic25111yescharacteristic
217C4#2254869211yescharacteristic
218C4#2254869911yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#3063156211yescharacteristic
223C2#3063185611yescharacteristic
224C2#3063187111yescharacteristic
subgroupG/H
centerC22#22548685
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC2×#104×C251C2
socleC22×C251#120
prime radicalC2×#41×C251C22
cosocleC22×C4×C251C23
hypercenterC2×#104D251
hypocenterC251#1859
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #30631729C2semidirect
4. #30631557C2semidirect
5. #30631455C2semidirect
6. #30631447C2semidirect
25. #104×C251C4semidirect
55. M16×C251C2×C4semidirect
58. C42×C251C2×C4semidirect
70. Q8×C251C4⋊C4semidirect
74. D4×C251C4⋊C4semidirect
112. C251#1859semidirect
136. #104Dic251semidirect
155. M16C2×Dic251semidirect
171. C42C2×Dic251semidirect
175. Q8C4⋊Dic251semidirect
179. D4C4⋊Dic251semidirect

Maximal subgroups

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

subgroupindex
#306321702
#306317292
#306316882
#306315572
#306314552
#306314472
C2×#104×C2512
#1859251