Finite Group Databasebeta

← #44549388

Subgroups — #44549388

Order 32128.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#30632312C211yesmaximal, characteristic
2#30632022C211yesmaximal, characteristic, complemented
3#30631622C211yesmaximal, characteristic
4#30631425C211yesmaximal, characteristic, complemented
5#30631424C211yesmaximal, characteristic, complemented
6#30631410C211yesmaximal, characteristic
7#484×C251C211yesmaximal, characteristic, complemented
8#22548695—21no—
9#22548694C2211yescharacteristic
10#22548693—21no—
11#22548693—21no—
12#22548692—21no—
13#22548691—21no—
14C2×Dic251⋊C4—21no—
15C2×(C2×D251)⋊C4—21no—
16C2×(C2×D251)⋊C4C2211yescharacteristic
17C22⋊C4×D251—21no—
18#22548648—21no—
19#22548638—21no—
20C22⋊(C251⋊C8)C2211yescharacteristic
21C251⋊#95C2211yescharacteristic
22C22⋊Q8×C251C2211yescharacteristic
23C2×C4⋊C4×C251C2211yescharacteristic
24#107×C251—21no—
25#103×C251—21no—
26C22⋊C8×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28C2×Dic502—21no—
29C2×Dic502D411yescharacteristic
30C22×Dic251—21no—
31C22×Dic251D411yescharacteristic
32C2×C4×D251—21no—
33C2×C4×D251—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251D411yescharacteristic
37Dic251⋊C4—21no—
38Dic251⋊C4—21no—
39Dic251⋊C4—21no—
40Dic251⋊C4—42no—
41C22⋊Dic251—21no—
42(C2×D251)⋊C4—21no—
43(C2×D251)⋊C4—21no—
44(C2×D251)⋊C4—21no—
45(C2×D251)⋊C4—21no—
46(C2×D251)⋊C4—42no—
47C2×Q8×C251D411yescharacteristic
48C22×C4×C251—21no—
49C22×C4×C251C2311yescharacteristic
50C2×C8×C251—21no—
51C4⋊C4×C251—21no—
52C4⋊C4×C251—21no—
53C4⋊C4×C251—21no—
54C4⋊C4×C251D411yescharacteristic
55C22⋊C4×C251—21no—
56C22×D251—21no—
57C22×D251—21no—
58C22×D251—41no—
59C22×D251—41no—
60C22×D251Q1612yes—
61Dic502—41no—
62Dic502—81no—
63C2×Dic251—22no—
64C2×Dic251—21no—
65C2×Dic251—21no—
66C2×Dic251—21no—
67C2×Dic251—41no—
68C2×Dic251—41no—
69C2×Dic251—41no—
70C4×D251—81no—
71C4×D251—81no—
72C251⋊C8—41no—
73C23×C251C2×D411yescharacteristic
74Q8×C251—41no—
75C2×C4×C251—21no—
76C2×C4×C251—21no—
77C2×C4×C251—41no—
78C2×C4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251—41no—
81C2×C4×C251C2×D411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C8×C251—41no—
84C22×C251—21no—
85C22×C251—21no—
86C22×C251#12011yescharacteristic
87C22×C251C2×Q1611yescharacteristic
88C22×C251#13711yescharacteristic
89C2×D251—22no—
90C2×D251—41no—
91C2×D251—41no—
92C2×D251—42no—
93C2×D251—81no—
94C4×C251—21no—
95C4×C251—21no—
96C4×C251—41no—
97C4×C251—41no—
98C4×C251—81no—
99Dic251—42no—
100Dic251—41no—
101Dic251—41no—
102Dic251—41no—
103C2×C251—21no—
104C2×C251—21no—
105C2×C251#45111yescharacteristic
106C2×C251#45311yescharacteristic
107C2×C251#45811yescharacteristic
108D251—42no—
109C251#161411yescharacteristic, complemented, Sylow
110#1614—2511nomaximal, Sylow
111#543—2511no—
112#484—2511no—
113#484D25111yescharacteristic, complemented
114#399—2511no—
115#328—2511no—
116#328—2511no—
117#323—2511no—
118C22⋊Q8—5021no—
119C22⋊Q8—5021no—
120C22⋊Q8—5021no—
121C22⋊Q8—5021no—
122C22⋊Q8—5021no—
123C22⋊Q8—2511no—
124C22⋊Q8C2×D25111yescharacteristic
125C2×C4⋊C4—5021no—
126C2×C4⋊C4C2×D25111yescharacteristic
127C2×C22⋊C4—5021no—
128C2×C22⋊C4—5021no—
129C2×C22⋊C4—2511no—
130#107—21no—
131#107—5021no—
132#103—21no—
133#103—5021no—
134C22⋊C8—2511no—
135C22⋊C8C2×D25111yescharacteristic
136#95—2511no—
137C24—2511no—
138C2×Q8—5021no—
139C2×Q8—2511no—
140C2×Q8C22⋊D25111yescharacteristic
141C22×C4—21no—
142C22×C4—5021no—
143C22×C4—5021no—
144C22×C4—5021no—
145C22×C4—2511no—
146C22×C4C22×D25111yescharacteristic
147C2×C8—21no—
148C2×C8—5021no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—10042no—
153C4⋊C4—5021no—
154C4⋊C4—5021no—
155C4⋊C4—5021no—
156C4⋊C4—5021no—
157C4⋊C4—2511no—
158C4⋊C4C22⋊D25111yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—10042no—
161C22⋊C4—5021no—
162C22⋊C4—5021no—
163C22⋊C4—5021no—
164C22⋊C4—5021no—
165C22⋊C4—5021no—
166C23—10041no—
167C23—10041no—
168C23—5021no—
169C23—5021no—
170C23—2512no—
171C23D4×D25111yescharacteristic
172Q8—41no—
173Q8—20081no—
174Q8—10041no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—20081no—
182C2×C4—20081no—
183C2×C4—10041no—
184C2×C4—10041no—
185C2×C4—10041no—
186C2×C4—5022no—
187C2×C4—5021no—
188C2×C4—5021no—
189C2×C4—5021no—
190C2×C4D4×D25111yescharacteristic
191C2×C4C2×C22⋊D25111yescharacteristic
192C8—41no—
193C8—10041no—
194C22—21no—
195C22—21no—
196C22—20081no—
197C22—10042no—
198C22—10041no—
199C22—10041no—
200C22—5022no—
201C22#2254868311yescharacteristic
202C22Q16×D25111yescharacteristic
203C22#2254874711yescharacteristic
204C4—21no—
205C4—21no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4—10042no—
210C4—10041no—
211C4—10041no—
212C4—10041no—
213C2—21no—
214C2—21no—
215C2—10042no—
216C2#3063184111yescharacteristic
217C2#3063185311yescharacteristic
218C2#3063187611yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#484×C251C2
socleC22×C251#120
prime radicalC23×D251D4
hypercenter#484D251
hypocenterC251#1614
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #30632022C2semidirect
4. #30631425C2semidirect
5. #30631424C2semidirect
7. #484×C251C2semidirect
109. C251#1614semidirect
113. #484D251semidirect

Maximal subgroups

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

subgroupindex
#306323122
#306320222
#306316222
#306314252
#306314242
#306314102
#484×C2512
#1614251