Finite Group Databasebeta

← #44548775

Subgroups — #44548775

Order 32128.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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#30632219C211yesmaximal, characteristic
2C4⋊C8×D251C211yesmaximal, characteristic, complemented
3C4×#16860793C211yesmaximal, characteristic, complemented
4#30631419C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#16860793—21no—
9C2×C8×D251—21no—
10#22548708—21no—
11#22548706C411yescharacteristic
12#22548702C411yescharacteristic
13#22548697—21no—
14#22548695C411yescharacteristic
15#22548686C411yescharacteristic
16C4×Dic502C2211yescharacteristic
17C4×C22⋊D251—21no—
18#22548674—21no—
19C42×D251C2211yescharacteristic
20C4⋊(C251⋊C8)C2211yescharacteristic
21Dic251⋊C8—21no—
22#22548625—21no—
23C4×C251⋊C8C2211yescharacteristic
24C4×D4×C251C2211yescharacteristic
25C4⋊C8×C251C2211yescharacteristic
26C4×C8×C251C2211yescharacteristic
27C2×Dic502C2×C411yescharacteristic
28C2×C22⋊D251—21no—
29C2×C4×D251C2×C411yescharacteristic
30C2×C4×D251D411yescharacteristic
31C2×C4×D251D411yescharacteristic
32#16860793—24no—
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C8×D251—42no—
36C4⋊Dic251—21no—
37C4⋊Dic251—21no—
38C4⋊Dic251C2×C411yescharacteristic
39Dic251⋊C4—21no—
40Dic251⋊C4—21no—
41C22⋊Dic251—21no—
42C22⋊Dic251—21no—
43C22⋊Dic251—21no—
44(C2×D251)⋊C4—21no—
45C4×Dic251—21no—
46C4×Dic251C2×C411yescharacteristic
47C4×Dic251D411yescharacteristic
48C4×Dic251D411yescharacteristic
49C2×D4×C251C2×C411yescharacteristic
50C22×C4×C251—21no—
51C2×C8×C251—21no—
52C2×C8×C251—21no—
53C4⋊C4×C251C2×C411yescharacteristic
54C22⋊C4×C251—21no—
55C42×C251C2311yescharacteristic
56C22×D251C22⋊C411yescharacteristic
57Dic502—21no—
58Dic502—21no—
59C22⋊D251—42no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251C22⋊C411yescharacteristic
63C2×Dic251C22⋊C411yescharacteristic
64C2×Dic251C22⋊C411yescharacteristic
65C4×D251—22no—
66C4×D251—21no—
67C4×D251—21no—
68C4×D251—21no—
69C4×D251—21no—
70C251⋊C8—22no—
71C251⋊C8—41no—
72C23×C251—21no—
73D4×C251—21no—
74D4×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251C22×C411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C8×C251—22no—
82C8×C251—41no—
83C22×C251—41no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251C2×C22⋊C411yescharacteristic
87C2×D251—21no—
88C2×D251—21no—
89C2×D251#10412yes—
90C4×C251—21no—
91C4×C251—41no—
92C4×C251#13111yescharacteristic
93C4×C251#13111yescharacteristic
94C4×C251AGL(1,ℤ/8)11yescharacteristic
95C4×C251#13711yescharacteristic
96Dic251—21no—
97Dic251—21no—
98Dic251—21no—
99Dic251—41no—
100Dic251—41no—
101C2×C251—81no—
102C2×C251#40811yescharacteristic
103C2×C251#41711yescharacteristic
104C2×C251C2×#10411yescharacteristic
105D251—22no—
106C251#148311yescharacteristic, complemented, Sylow
107#1483—2511nomaximal, Sylow
108#529—2511no—
109C2×C4⋊C8—2511no—
110C4×M16—2511no—
111Q8⋊C8—2511no—
112Q8⋊C8—2511no—
113D4⋊C8—2511no—
114D4⋊C8D25111yescharacteristic, complemented
115C2×M16—5021no—
116C22×C8—5021no—
117#126—5021no—
118#125—2511no—
119#124—2511no—
120#123—5021no—
121C22⋊Q8—2511no—
122#121—2511no—
123C4×Q8—2511no—
124C4×D4—5021no—
125C4×D4C2×D25111yescharacteristic
126#117—5021no—
127C2×C42—2511no—
128C4⋊C8—5021no—
129C4⋊C8—2511no—
130C4⋊C8C2×D25111yescharacteristic
131#97—5021no—
132C4×C8—2511no—
133C4×C8C2×D25111yescharacteristic
134C2×Q8—2511no—
135C2×D4—5021no—
136C2×D4C4×D25111yescharacteristic
137C22×C4—21no—
138C22×C4—2511no—
139C22×C4—2511no—
140C22×C4—2511no—
141M16—5024no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—10042no—
145C2×C8—5021no—
146C2×C8—5021no—
147C4⋊C4—5021no—
148C4⋊C4—5021no—
149C4⋊C4—5021no—
150C4⋊C4—5021no—
151C4⋊C4—2511no—
152C4⋊C4C4×D25111yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—5021no—
155C22⋊C4—5021no—
156C22⋊C4—5021no—
157C22⋊C4—5021no—
158C42—5021no—
159C42—2511no—
160C42—2511no—
161C42—2511no—
162C42C22×D25111yescharacteristic
163C23—21no—
164C23—2511no—
165Q8—5021no—
166Q8—5021no—
167D4—21no—
168D4—21no—
169D4—10042no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—5022no—
174C2×C4—5021no—
175C2×C4—5021no—
176C2×C4—5021no—
177C2×C4—5021no—
178C2×C4—5021no—
179C2×C4—5021no—
180C2×C4—2511no—
181C2×C4—2511no—
182C2×C4—2511no—
183C2×C4C2×C4×D25111yescharacteristic
184C2×C4D4×D25111yescharacteristic
185C2×C4D4×D25111yescharacteristic
186C8—22no—
187C8—41no—
188C8—10041no—
189C8—5022no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—5021no—
194C22—5021no—
195C22—2512no—
196C22C22⋊C4×D25111yescharacteristic
197C4—21no—
198C4—41no—
199C4—10041no—
200C4—10041no—
201C4—5021no—
202C4—5021no—
203C4—5021no—
204C4#2254872111yescharacteristic
205C4#2254872311yescharacteristic
206C4#2254873911yescharacteristic
207C4#2254874411yescharacteristic
208C2—81no—
209C2—5022no—
210C2#3063165211yescharacteristic
211C2#3063168511yescharacteristic
212C2#104×D25111yescharacteristic
subgroupG/H
centerC2×C4D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
FittingD4⋊C8×C251C2
socleC22×C251C2×C22⋊C4
prime radical#22548686C4
cosocleC42×C251C23
hypercenterD4⋊C8D251
hypocenterC251#1483
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4⋊C8×D251C2semidirect
3. C4×#16860793C2semidirect
4. #30631419C2semidirect
5. #30631417C2semidirect
6. D4⋊(C251⋊C8)C2semidirect
7. D4⋊C8×C251C2semidirect
106. C251#1483semidirect
114. D4⋊C8D251semidirect

Maximal subgroups

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

subgroupindex
#306322192
C4⋊C8×D2512
C4×#168607932
#306314192
#306314172
D4⋊(C251⋊C8)2
D4⋊C8×C2512
#1483251