Finite Group Databasebeta

← #44560823

Subgroups — #44560823

Order 32128.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#22548747C214yesmaximal, complemented
2#30632238C211yesmaximal, characteristic
3C4⋊Q8×D251C211yesmaximal, characteristic, complemented
4#30632080C214yesmaximal, complemented
5#30632066C211yesmaximal, characteristic, complemented
6#30632061C211yesmaximal, characteristic, complemented
7#30632060C211yesmaximal, characteristic, complemented
8C4×#16860793C211yesmaximal, characteristic, complemented
9#494×C251C211yesmaximal, characteristic, complemented
10C2×#16860811C2212yes—
11C2×Q8×D251C2212yes—
12#22548747—216no—
13C2×#16860800C2214yes—
14C2×#16860799C2214yes—
15C2×#16860797C2214yes—
16C2×#16860793C2212yes—
17#22548715C2211yescharacteristic
18#22548714—22no—
19#22548713C2212yes—
20#22548712C2211yescharacteristic
21#22548704C2212yes—
22#22548690—22no—
23#22548677—22no—
24C4⋊C4×D251—22no—
25C42×D251C2211yescharacteristic
26#22548625C2212yes—
27C4×C251⋊C8C2211yescharacteristic
28C2×Q16×C251C2214yes—
29C4⋊Q8×C251C2211yescharacteristic
30C4⋊Q8×C251C2211yescharacteristic
31C4×C8×C251C2211yescharacteristic
32#16860811—28no—
33C2×Dic502—22no—
34C2×Dic502C2312yes—
35Q8×D251—28no—
36C2×D1004—22no—
37C2×D1004—22no—
38C2×D1004C2312yes—
39C2×C4×D251—22no—
40C2×C4×D251—22no—
41C2×C4×D251D412yes—
42C2×C4×D251C2311yescharacteristic
43#16860800—28no—
44#16860799—28no—
45#16860797—28no—
46#16860793D418yescomplemented
47C2×C251⋊C8C2312yes—
48C4⋊Dic251—22no—
49Dic251⋊C4—24no—
50(C2×D251)⋊C4—24no—
51C4×Dic251D412yes—
52C4×Dic251C2311yescharacteristic
53C2×Q8×C251C2312yes—
54C2×Q8×C251C2312yes—
55Q16×C251—28no—
56C2×C8×C251C2312yes—
57C4⋊C4×C251—22no—
58C4⋊C4×C251—22no—
59C42×C251C2311yescharacteristic
60C22×D251—22no—
61C22×D251C2×D411yescharacteristic
62Dic502—24no—
63Dic502—44no—
64D1004—24no—
65D1004—44no—
66D1004—44no—
67C2×Dic251—22no—
68C2×Dic251C2×D412yes—
69C2×Dic251C2×D411yescharacteristic
70C4×D251—24no—
71C4×D251—44no—
72C4×D251—44no—
73C4×D251C2×D414yes—
74C251⋊C8C2×D414yes—
75Q8×C251—24no—
76Q8×C251—24no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251C2×D412yes—
80C2×C4×C251C2411yescharacteristic
81C8×C251C2×D414yes—
82C22×C251C22×D411yescharacteristic
83C2×D251—22no—
84C2×D251—42no—
85C2×D251—44no—
86C2×D251#12712yes—
87C4×C251—42no—
88C4×C251—42no—
89C4×C251#13714yescomplemented
90C4×C251C22×D412yes—
91Dic251—22no—
92Dic251—42no—
93Dic251#12712yes—
94C2×C251C2×#12711yescharacteristic
95C2×C251C2×#13712yes—
96D251—22no—
97D251—82no—
98C251#315211yescharacteristic, complemented, Sylow
99#3152—2511nomaximal, Sylow
100C2×#137—2514no—
101#532—2511no—
102C2×C4⋊Q8—2511no—
103#497—2514no—
104#494—2511no—
105#494D25111yescharacteristic, complemented
106#492—2511no—
107#492—2511no—
108C4×M16—2511no—
109C2×#41—2512no—
110C22×Q8—2512no—
111#137—50216no—
112C2×Q16—2514no—
113C2×Q16C2×D25114yes—
114C2×SD16—2514no—
115C2×SD16—2514no—
116C2×M16—2512no—
117C4⋊Q8—5022no—
118C4⋊Q8—2512no—
119C4⋊Q8—2511no—
120C4⋊Q8C2×D25111yescharacteristic
121C4⋊Q8C2×D25111yescharacteristic
122#127—2511no—
123#124—2512no—
124#121—5022no—
125C4×D4—5022no—
126C2×C4⋊C4—5022no—
127C2×C42—2511no—
128#97—2512no—
129C4×C8—2511no—
130C4×C8C2×D25111yescharacteristic
131#41—5028no—
132C2×Q8—5022no—
133C2×Q8—5028no—
134C2×Q8—2512no—
135C2×Q8C22×D25112yes—
136C2×Q8C22×D25112yes—
137C2×D4—5022no—
138C2×D4—5022no—
139C2×D4—2512no—
140C22×C4—5022no—
141C22×C4—5022no—
142C22×C4—2512no—
143C22×C4—2511no—
144Q16—28no—
145Q16—5028no—
146SD16—5028no—
147SD16—5028no—
148M16—2518no—
149C2×C8—2512no—
150C2×C8C22×D25112yes—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—5024no—
154C4⋊C4—5022no—
155C22⋊C4—5024no—
156C42—2512no—
157C42—2511no—
158C42C22×D25111yescharacteristic
159C23—5022no—
160C23—2511no—
161Q8—24no—
162Q8—24no—
163Q8—10044no—
164Q8—5024no—
165D4—10044no—
166D4—10044no—
167D4—5024no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—10044no—
171C2×C4—10044no—
172C2×C4—5022no—
173C2×C4—5024no—
174C2×C4—2512no—
175C2×C4—2511no—
176C2×C4—2514no—
177C2×C4D4×D25112yes—
178C2×C4C23×D25111yescharacteristic
179C8—2514no—
180C8D4×D25114yes—
181C22—10042no—
182C22—10044no—
183C22—5022no—
184C22—2512no—
185C22C2×D4×D25111yescharacteristic
186C4—42no—
187C4—42no—
188C4—10042no—
189C4—5022no—
190C4—2512no—
191C4#2254874714yescomplemented
192C4C2×D4×D25112yes—
193C2—20082no—
194C2—5022no—
195C2#127×D25111yescharacteristic
196C2C2×#2254874712yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#494×C251C2
socleC22×C251C22×D4
prime radical#30632238C2
hypercenter#494D251
hypocenterC251#3152
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. C2×#22548747C2semidirect
3. C4⋊Q8×D251C2semidirect
4. #30632080C2semidirect
5. #30632066C2semidirect
6. #30632061C2semidirect
7. #30632060C2semidirect
8. C4×#16860793C2semidirect
9. #494×C251C2semidirect
46. #16860793D4semidirect
89. C4×C251#137semidirect
98. C251#3152semidirect
105. #494D251semidirect
191. C4#22548747semidirect

Maximal subgroups

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

subgroupindex
C2×#225487472
#306322382
C4⋊Q8×D2512
#306320802
#306320662
#306320612
#306320602
C4×#168607932
#494×C2512
#3152251