Finite Group Databasebeta

← #44550573

Subgroups — #44550573

Order 32128.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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#30632161C211yesmaximal, characteristic
2#30631727C211yesmaximal, characteristic, complemented
3C2×#22548631C211yesmaximal, characteristic
4#30631588C211yesmaximal, characteristic
5#30631448C211yesmaximal, characteristic, complemented
6#30631446C211yesmaximal, characteristic, complemented
7#428×C251C211yesmaximal, characteristic, complemented
8C22×D1004C2211yescharacteristic
9C2×#16860794C2211yescharacteristic
10C4×D1004—21no—
11C4×D1004—21no—
12#22548677—21no—
13#22548677—21no—
14#22548674C2211yescharacteristic
15C2×(C2×D251)⋊C4—22no—
16C22⋊C4×D251—22no—
17C2×C4×Dic251C2211yescharacteristic
18#22548648—21no—
19#22548646—21no—
20#22548631C414yescomplemented
21C2×M16×C251C2211yescharacteristic
22#117×C251C2211yescharacteristic
23C2×C4⋊C4×C251C2211yescharacteristic
24#107×C251—21no—
25#106×C251—21no—
26C23×D251D412yes—
27C2×D1004—21no—
28C2×D1004—21no—
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004C2×C412yes—
32C2×D1004Q812yes—
33C22×Dic251—21no—
34C2×C4×D251—22no—
35C2×C4×D251—22no—
36#16860794—21no—
37#16860794C2×C412yescomplemented
38C2×C251⋊C8—21no—
39C4⋊Dic251D411yescharacteristic
40C4⋊Dic251D411yescharacteristic
41C22⋊Dic251—21no—
42(C2×D251)⋊C4—22no—
43(C2×D251)⋊C4—22no—
44(C2×D251)⋊C4—44no—
45C4×Dic251—21no—
46C4×Dic251—21no—
47C4×Dic251—21no—
48C22×C4×C251—21no—
49C22×C4×C251C2311yescharacteristic
50M16×C251—21no—
51M16×C251C2×C412yescomplemented
52C2×C8×C251—21no—
53C4⋊C4×C251—21no—
54C4⋊C4×C251—21no—
55C4⋊C4×C251D411yescharacteristic
56C4⋊C4×C251D411yescharacteristic
57C22⋊C4×C251—21no—
58C42×C251—21no—
59C22×D251—22no—
60C22×D251—22no—
61C22×D251—42no—
62C22×D251—42no—
63C22×D251C4⋊C414yes—
64D1004—22no—
65D1004—22no—
66D1004—22no—
67D1004—22no—
68C2×Dic251—22no—
69C2×Dic251—21no—
70C2×Dic251—21no—
71C2×Dic251—41no—
72C2×Dic251—41no—
73C4×D251—82no—
74C4×D251—82no—
75C251⋊C8—22no—
76C23×C251C2×D411yescharacteristic
77C2×C4×C251—21no—
78C2×C4×C251—21no—
79C2×C4×C251—42no—
80C2×C4×C251—41no—
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×C251C2×C4⋊C411yescharacteristic
91C22×C251#12011yescharacteristic
92C22×C251#12311yescharacteristic
93C2×D251—24no—
94C2×D251—42no—
95C2×D251—42no—
96C2×D251—44no—
97C2×D251—82no—
98C4×C251—41no—
99C4×C251—41no—
100C4×C251—81no—
101C4×C251C4×D411yescharacteristic
102C4×C251C4×D411yescharacteristic
103C4×C251C22⋊Q811yescharacteristic
104C4×C251C22⋊Q811yescharacteristic
105Dic251—42no—
106Dic251—41no—
107Dic251—41no—
108C2×C251—21no—
109C2×C251—21no—
110C2×C251#38511yescharacteristic
111C2×C251#45311yescharacteristic
112C2×C251#45511yescharacteristic
113D251—44no—
114C251#185611yescharacteristic, complemented, Sylow
115#1856—2511nomaximal, Sylow
116#518—2511no—
117#428—2511no—
118#428D25111yescharacteristic, complemented
119C2×#100—2511no—
120#390—2511no—
121#339—2511no—
122#339—2511no—
123C22×D4—2511no—
124C2×M16—2511no—
125C2×M16C2×D25111yescharacteristic
126C4×D4—5021no—
127C4×D4—5021no—
128C4×D4—5021no—
129C4×D4—5021no—
130#117—2511no—
131#117C2×D25111yescharacteristic
132C2×C4⋊C4C2×D25111yescharacteristic
133C2×C22⋊C4—5022no—
134C2×C22⋊C4—5022no—
135C2×C42—2511no—
136#107—21no—
137#107—5021no—
138#106—21no—
139#106—5021no—
140#100—2514no—
141C24—2512no—
142C2×D4—5021no—
143C2×D4—5021no—
144C2×D4—5021no—
145C2×D4—5021no—
146C2×D4—2512no—
147C2×D4—2512no—
148C22×C4—21no—
149C22×C4—5022no—
150C22×C4—5022no—
151C22×C4—5021no—
152C22×C4C22×D25111yescharacteristic
153M16—21no—
154M16—5021no—
155M16—2512no—
156M16C4×D25112yescomplemented
157C2×C8—21no—
158C2×C8—5021no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—2511no—
162C4⋊C4—2511no—
163C4⋊C4C22⋊D25111yescharacteristic
164C4⋊C4C22⋊D25111yescharacteristic
165C22⋊C4—21no—
166C22⋊C4—10044no—
167C22⋊C4—5022no—
168C22⋊C4—5022no—
169C22⋊C4—5021no—
170C42—21no—
171C42—5021no—
172C42—5021no—
173C42—5021no—
174C23—10042no—
175C23—10042no—
176C23—5022no—
177C23—5022no—
178C23—2514no—
179C23D4×D25111yescharacteristic
180D4—5022no—
181D4—5022no—
182D4—5022no—
183D4—5022no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—20082no—
189C2×C4—20082no—
190C2×C4—10041no—
191C2×C4—10041no—
192C2×C4—5022no—
193C2×C4—5021no—
194C2×C4—5021no—
195C2×C4C2×C4×D25111yescharacteristic
196C2×C4D4×D25111yescharacteristic
197C2×C4C2×C22⋊D25111yescharacteristic
198C2×C4Q8×D25111yescharacteristic
199C2×C4#1686081011yescharacteristic
200C2×C4#1686081111yescharacteristic
201C8—22no—
202C8—5022no—
203C22—21no—
204C22—21no—
205C22—20082no—
206C22—10044no—
207C22—10042no—
208C22—10042no—
209C22—5024no—
210C22C4⋊C4×D25111yescharacteristic
211C22#2254868311yescharacteristic
212C22#2254870011yescharacteristic
213C4—41no—
214C4—41no—
215C4—81no—
216C4—10042no—
217C4—10041no—
218C4—10041no—
219C4C4×C22⋊D25111yescharacteristic
220C4#2254867711yescharacteristic
221C4#2254869111yescharacteristic
222C4#2254869311yescharacteristic
223C2—21no—
224C2—21no—
225C2—10044no—
226C2#3063155811yescharacteristic
227C2#3063185311yescharacteristic
228C2#3063186311yescharacteristic
subgroupG/H
centerC22#22548683
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#428×C251C2
socleC22×C251#120
prime radicalC22×D1004C22
cosocleC22×C4×C251C23
hypercenter#428D251
hypocenterC251#1856
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
2. #30631727C2semidirect
5. #30631448C2semidirect
6. #30631446C2semidirect
7. #428×C251C2semidirect
20. #22548631C4semidirect
37. #16860794C2×C4semidirect
51. M16×C251C2×C4semidirect
114. C251#1856semidirect
118. #428D251semidirect
156. M16C4×D251semidirect

Maximal subgroups

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

subgroupindex
#306321612
#306317272
C2×#225486312
#306315882
#306314482
#306314462
#428×C2512
#1856251