Finite Group Databasebeta

← #44549517

Subgroups — #44549517

Order 32128.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632340C211yesmaximal, characteristic
2#30632039C211yesmaximal, characteristic, complemented
3#30631978C211yesmaximal, characteristic, complemented
4#30631978C211yesmaximal, characteristic, complemented
5#30631414C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#332×C251C211yesmaximal, characteristic, complemented
8C2×#16860811—21no—
9#22548715C2211yescharacteristic
10#22548700—22no—
11#22548691—21no—
12#22548690—21no—
13C4×D1004C2211yescharacteristic
14C4×D1004C2211yescharacteristic
15C4⋊C4×D251—21no—
16#22548647—21no—
17#22548647—21no—
18#22548638—22no—
19#22548637—21no—
20#22548637—21no—
21C4×C251⋊C8C2211yescharacteristic
22C4⋊Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4⋊C8×C251C2211yescharacteristic
25#16860811—42no—
26#16860811—42no—
27C2×Dic502—21no—
28C2×D1004—22no—
29C2×D1004D411yescharacteristic
30C2×D1004D411yescharacteristic
31C2×C4×D251—22no—
32C2×C4×D251—21no—
33C2×C4×D251—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251D411yescharacteristic
38C4⋊Dic251D411yescharacteristic
39Dic251⋊C4—22no—
40(C2×D251)⋊C4—22no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43C2×Q8×C251—21no—
44C2×C8×C251—21no—
45C2×C8×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D412yes—
48C42×C251C2311yescharacteristic
49C22×D251—21no—
50C22×D251—21no—
51Dic502—42no—
52D1004—21no—
53D1004—21no—
54D1004—21no—
55D1004—44no—
56D1004SD1612yes—
57C2×Dic251—21no—
58C2×Dic251—21no—
59C2×Dic251—41no—
60C4×D251—44no—
61C4×D251—41no—
62C4×D251—41no—
63C4×D251—41no—
64C4×D251—41no—
65C251⋊C8—41no—
66Q8×C251—41no—
67Q8×C251—41no—
68C2×C4×C251—22no—
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C8×C251—41no—
73C8×C251—41no—
74C22×C251#12011yescharacteristic
75C2×D251—22no—
76C2×D251—41no—
77C2×D251—41no—
78C2×D251—41no—
79C2×D251—41no—
80C2×D251—41no—
81C4×C251—21no—
82C4×C251—42no—
83C4×C251C2×SD1611yescharacteristic
84C4×C251#13511yescharacteristic
85C4×C251AGL(1,ℤ/8)11yescharacteristic
86C4×C251#13711yescharacteristic
87Dic251—41no—
88Dic251—41no—
89Dic251—81no—
90C2×C251#45011yescharacteristic
91C2×C251#45211yescharacteristic
92C2×C251#45411yescharacteristic
93D251—42no—
94D251—81no—
95C251#163511yescharacteristic, complemented, Sylow
96#1635—2511nomaximal, Sylow
97#547—2511no—
98#487—2511no—
99#476—2511no—
100#476—2511no—
101#332D25111yescharacteristic, complemented
102D4⋊C8—2511no—
103D4⋊C8—2511no—
104C2×#41—5021no—
105C4⋊Q8—2511no—
106C4⋊Q8C2×D25111yescharacteristic
107#123—5022no—
108C22⋊Q8—5021no—
109#121—5021no—
110C4×D4—2511no—
111C4×D4—2511no—
112C2×C4⋊C4—5021no—
113#106—5021no—
114#106—5021no—
115C4⋊C8C2×D25111yescharacteristic
116C4⋊C8C2×D25111yescharacteristic
117#103—5022no—
118#102—5021no—
119#102—5021no—
120C4×C8—2511no—
121#41—10042no—
122#41—10042no—
123C2×Q8—21no—
124C2×Q8—5021no—
125C2×D4—5022no—
126C2×D4—2511no—
127C2×D4—2511no—
128C22×C4—5022no—
129C22×C4—5021no—
130C22×C4—5021no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—5021no—
134C4⋊C4—21no—
135C4⋊C4—5022no—
136C4⋊C4—5021no—
137C4⋊C4—5021no—
138C4⋊C4—2511no—
139C4⋊C4—2511no—
140C4⋊C4D100412yes—
141C22⋊C4—5022no—
142C22⋊C4—5021no—
143C22⋊C4—5021no—
144C42C22×D25111yescharacteristic
145C23—5021no—
146C23—5021no—
147Q8—41no—
148Q8—41no—
149Q8—10042no—
150D4—10044no—
151D4—5021no—
152D4—5021no—
153D4—5021no—
154D4—2512no—
155C2×C4—22no—
156C2×C4—10044no—
157C2×C4—10041no—
158C2×C4—10041no—
159C2×C4—10041no—
160C2×C4—10041no—
161C2×C4—10041no—
162C2×C4—5021no—
163C2×C4—5021no—
164C2×C4D4×D25111yescharacteristic
165C2×C4D4×D25111yescharacteristic
166C2×C4C2×D100411yescharacteristic
167C8—41no—
168C8—41no—
169C8—10041no—
170C22—10041no—
171C22—10041no—
172C22—10041no—
173C22—10041no—
174C22—10041no—
175C22—5022no—
176C22#2254868511yescharacteristic
177C4—21no—
178C4—42no—
179C4—20081no—
180C4—10041no—
181C4—10041no—
182C4SD16×D25111yescharacteristic
183C4#2254873611yescharacteristic
184C4#2254873911yescharacteristic
185C4#2254874711yescharacteristic
186C2—20081no—
187C2—10042no—
188C2#3063183911yescharacteristic
189C2#3063185211yescharacteristic
190C2#3063186211yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#332×C251C2
socleC22×C251#120
prime radical#30632340C2
hypercenter#332D251
hypocenterC251#1635
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #30632039C2semidirect
3. #30631978C2semidirect
4. #30631978C2semidirect
5. #30631414C2semidirect
6. #30631414C2semidirect
7. #332×C251C2semidirect
95. C251#1635semidirect
101. #332D251semidirect

Maximal subgroups

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

subgroupindex
#306323402
#306320392
#306319782
#306314142
#332×C2512
#1635251