Finite Group Databasebeta

← #44560895

Subgroups — #44560895

Order 32128.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#30632318C211yesmaximal, characteristic
2#30632315C211yesmaximal, characteristic
3#30632095C212yesmaximal, complemented
4#30632094C212yesmaximal, complemented
5#30632088C212yesmaximal, complemented
6#30632084C212yesmaximal, complemented
7#30631729C211yesmaximal, characteristic
8#30631723C211yesmaximal, characteristic
9C2×#22548646C211yesmaximal, characteristic
10#30631645C211yesmaximal, characteristic
11#428×C251C211yesmaximal, characteristic
12C22×C251⋊C8C2211yescharacteristic
13#22548715—21no—
14#22548714—21no—
15#22548714C2212yes—
16#22548714C2212yes—
17#22548707—21no—
18#22548706—21no—
19#22548706C2212yes—
20#22548706C2212yes—
21#22548694—22no—
22#22548692—22no—
23#22548674C2211yescharacteristic
24#22548674C2211yescharacteristic
25C2×C4⋊Dic251C2211yescharacteristic
26#22548649C2212yes—
27#22548648C2211yescharacteristic
28#22548648C2211yescharacteristic
29#22548647C2212yes—
30#22548646C2212yes—
31#22548646C2212yes—
32#22548646C2211yescharacteristic
33#22548646C2211yescharacteristic
34#22548625C2212yes—
35C8×Dic251C2212yes—
36C2×M16×C251C2211yescharacteristic
37#117×C251C2211yescharacteristic
38C2×C4⋊C4×C251C2211yescharacteristic
39#107×C251C2212yes—
40#106×C251C2212yes—
41C2×Dic502—22no—
42C2×Dic502—22no—
43C22×Dic251—21no—
44C2×C251⋊C8Q814yescomplemented
45C2×C251⋊C8C2311yescharacteristic
46C2×C251⋊C8C2311yescharacteristic
47C4⋊Dic251—22no—
48C4⋊Dic251—22no—
49C4⋊Dic251—21no—
50C4⋊Dic251D411yescharacteristic
51C4⋊Dic251D411yescharacteristic
52C4⋊Dic251C2312yes—
53C4⋊Dic251C2311yescharacteristic
54C4⋊Dic251C2311yescharacteristic
55Dic251⋊C4—22no—
56Dic251⋊C4—22no—
57Dic251⋊C4—22no—
58Dic251⋊C4—22no—
59C22⋊Dic251—21no—
60C22⋊Dic251D412yes—
61C4×Dic251—21no—
62C4×Dic251C2312yes—
63C22×C4×C251—21no—
64C22×C4×C251C2311yescharacteristic
65M16×C251Q814yescomplemented
66C2×C8×C251C2312yes—
67C4⋊C4×C251—21no—
68C4⋊C4×C251C2311yescharacteristic
69C4⋊C4×C251C2312yes—
70C4⋊C4×C251C2311yescharacteristic
71C22⋊C4×C251—21no—
72C42×C251—21no—
73Dic502—82no—
74Dic502—82no—
75C2×Dic251—21no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C2×Dic251—81no—
79C2×Dic251C2×D412yes—
80C2×Dic251C2×D412yes—
81C251⋊C8C2×Q812yes—
82C251⋊C8C2×Q812yes—
83C23×C251C2×D411yescharacteristic
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—22no—
87C2×C4×C251—81no—
88C2×C4×C251C2×D411yescharacteristic
89C2×C4×C251C2×Q812yes—
90C2×C4×C251C2×Q812yes—
91C2×C4×C251C2411yescharacteristic
92C8×C251C2×Q814yes—
93C22×C251—21no—
94C22×C251—21no—
95C22×C251C4⋊Q812yes—
96C22×C251C22×D411yescharacteristic
97C4×C251—41no—
98C4×C251—42no—
99C4×C251—41no—
100C4×C251C4⋊Q812yes—
101C4×C251C22×Q811yescharacteristic
102C4×C251C22×Q811yescharacteristic
103Dic251—22no—
104Dic251—22no—
105Dic251—41no—
106Dic251—42no—
107Dic251—41no—
108C2×C251—22no—
109C2×C251C2×C4⋊Q811yescharacteristic
110C2×C251#57711yescharacteristic
111C2×C251#57711yescharacteristic
112C251#316311yescharacteristic, complemented, Sylow
113#3163—2511nomaximal, Sylow
114#544—2511no—
115#544—2511no—
116#501—2512no—
117#501—2512no—
118#499—2512no—
119#498—2512no—
120#428—2511no—
121#428D25111yescharacteristic
122#427—2511no—
123C2×#106—2511no—
124#405—2511no—
125C2×M16C2×D25111yescharacteristic
126C22×C8—2511no—
127C4⋊Q8—5021no—
128C4⋊Q8—5021no—
129C4⋊Q8—2512no—
130C4⋊Q8—2512no—
131#125—5021no—
132#125—5021no—
133#125—2512no—
134#125—2512no—
135C22⋊Q8—5022no—
136C22⋊Q8—5022no—
137#117—2511no—
138#117—2511no—
139#117C2×D25111yescharacteristic
140C2×C4⋊C4—2511no—
141C2×C4⋊C4C2×D25111yescharacteristic
142#107—2512no—
143#107—2511no—
144#107—2511no—
145#107C2×D25112yes—
146#106—2512no—
147#106—2512no—
148#106—2512no—
149#106—2511no—
150#106—2511no—
151#106C2×D25112yes—
152#97—2512no—
153C4×C8—2512no—
154C2×Q8—5022no—
155C2×Q8—5022no—
156C22×C4—21no—
157C22×C4—5021no—
158C22×C4C22×D25111yescharacteristic
159M16Dic50214yescomplemented
160C2×C8—2514no—
161C2×C8—2511no—
162C2×C8—2511no—
163C2×C8C22×D25112yes—
164C4⋊C4—21no—
165C4⋊C4—5022no—
166C4⋊C4—5022no—
167C4⋊C4—5022no—
168C4⋊C4—5022no—
169C4⋊C4—5022no—
170C4⋊C4—5022no—
171C4⋊C4—5021no—
172C4⋊C4—2512no—
173C4⋊C4—2511no—
174C4⋊C4—2511no—
175C4⋊C4—2511no—
176C4⋊C4—2511no—
177C4⋊C4C22×D25112yes—
178C4⋊C4C22×D25111yescharacteristic
179C4⋊C4C22×D25111yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—5021no—
182C22⋊C4—2512no—
183C42—21no—
184C42—5021no—
185C42—2512no—
186C23D4×D25111yescharacteristic
187Q8—20082no—
188Q8—20082no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—20081no—
194C2×C4—5021no—
195C2×C4—5021no—
196C2×C4—5022no—
197C2×C4—2512no—
198C2×C4—2512no—
199C2×C4D4×D25111yescharacteristic
200C2×C4Q8×D25112yes—
201C2×C4C2×Dic50212yes—
202C2×C4C23×D25111yescharacteristic
203C8—2512no—
204C8—2512no—
205C8C2×Dic50214yes—
206C22—21no—
207C22—21no—
208C22#2254871412yes—
209C22C2×D4×D25111yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—10042no—
214C4—10041no—
215C4—10041no—
216C4—5022no—
217C4—5022no—
218C4#2254871412yes—
219C4C2×Q8×D25111yescharacteristic
220C4C22×Dic50211yescharacteristic
221C2—22no—
222C2C2×#2254871411yescharacteristic
223C2#3063249611yescharacteristic
224C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radicalC23×C251C2×D4
hypercenter#428D251
hypocenterC251#3163
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #30632095C2semidirect
4. #30632094C2semidirect
5. #30632088C2semidirect
6. #30632084C2semidirect
44. C2×C251⋊C8Q8semidirect
65. M16×C251Q8semidirect
112. C251#3163semidirect
159. M16Dic502semidirect

Maximal subgroups

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

subgroupindex
#306323182
#306323152
#306320952
#306320942
#306320882
#306320842
#306317292
#306317232
C2×#225486462
#306316452
#428×C2512
#3163251