Finite Group Databasebeta

← C2×#30631670 #44558839

Subgroups — C2×#30631670 #44558839

Order 32128.

756 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C22×#16860811C211yesmaximal, characteristic
2C2×M16×D251C212yesmaximal, complemented
3#30631670C218yesmaximal, direct product factor
4C2×#22548634C211yesmaximal, characteristic, complemented
5C2×#22548631C212yesmaximal, complemented
6C2×#101×C251C211yesmaximal, characteristic, complemented
7C2×#16860811—22no—
8C2×#16860811—24no—
9C2×#16860811—22no—
10C2×#16860811C414yes—
11C2×#16860811C2214yes—
12C22×D1004C412yes—
13C22×D1004C2211yescharacteristic
14C22×C4×D251C412yes—
15C22×C4×D251C2211yescharacteristic
16C2×#16860794C2212yes—
17C2×#16860793—22no—
18M16×D251—24no—
19M16×D251C2218yescomplemented
20C2×C8×D251—22no—
21#22548634C2214yescomplemented
22#22548631C2218yescomplemented
23C22×Q8×C251C2211yescharacteristic
24C2×M16×C251C2212yes—
25#101×C251C2214yescomplemented
26C23×D251C2×C412yes—
27C23×D251C2×C411yescharacteristic
28#16860811—28no—
29#16860811—28no—
30#16860811—28no—
31#16860811—28no—
32C2×D1004—24no—
33C2×D1004—24no—
34C2×D1004—22no—
35C2×D1004—22no—
36C2×D1004C2×C418yes—
37C2×D1004C2×C412yes—
38C2×D1004C2312yes—
39C22×Dic251C2×C411yescharacteristic
40C2×C4×D251—24no—
41C2×C4×D251—22no—
42C2×C4×D251—22no—
43C2×C4×D251C2×C418yes—
44C2×C4×D251C2×C412yes—
45C2×C4×D251D412yes—
46C2×C4×D251D414yes—
47C2×C4×D251D412yes—
48C2×C4×D251C2312yes—
49#16860794—22no—
50#16860794C2314yes—
51#16860793—28no—
52C2×C251⋊C8—22no—
53C8×D251—28no—
54C2×Q8×C251—22no—
55C2×Q8×C251—22no—
56C2×Q8×C251C2×C412yes—
57C2×Q8×C251C2312yes—
58C22×C4×C251C2×C412yes—
59C22×C4×C251C2311yescharacteristic
60M16×C251—22no—
61M16×C251C2314yes—
62C2×C8×C251—22no—
63C22×D251—22no—
64C22×D251—22no—
65C22×D251—42no—
66C22×D251—44no—
67C22×D251C22⋊C412yes—
68C22×D251C22⋊C412yes—
69C22×D251C22×C414yes—
70C22×D251C22×C412yes—
71D1004—216no—
72D1004—24no—
73D1004—24no—
74C2×Dic251C22⋊C412yes—
75C2×Dic251C22⋊C411yescharacteristic
76C2×Dic251C22⋊C411yescharacteristic
77C2×Dic251C22×C412yes—
78C4×D251—28no—
79C4×D251—28no—
80C4×D251C22⋊C414yes—
81C4×D251C22⋊C414yes—
82C4×D251C2×D414yes—
83C4×D251C2×D414yes—
84C251⋊C8—24no—
85C23×C251C22×C411yescharacteristic
86Q8×C251—24no—
87Q8×C251—24no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251C22×C414yes—
91C2×C4×C251C22×C411yescharacteristic
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2×D412yes—
94C2×C4×C251C2411yescharacteristic
95C8×C251—24no—
96C22×C251—22no—
97C22×C251C2×C22⋊C411yescharacteristic
98C22×C251C2×C22⋊C411yescharacteristic
99C22×C251C23×C411yescharacteristic
100C2×D251—24no—
101C2×D251—24no—
102C2×D251—42no—
103C2×D251—42no—
104C2×D251—44no—
105C2×D251—44no—
106C2×D251—44no—
107C2×D251C2×C22⋊C414yes—
108C4×C251—24no—
109C4×C251C2×C22⋊C412yes—
110C4×C251C22×D412yes—
111Dic251C2×C22⋊C412yes—
112Dic251C2×C22⋊C412yes—
113C2×C251—21no—
114C2×C251—21no—
115C2×C251#41312yescomplemented
116C2×C251C22×C22⋊C411yescharacteristic
117D251—24no—
118D251—44no—
119C251C2×#41311yescharacteristic, complemented, Sylow
120C2×#413—2511nomaximal, Sylow
121C22×#41—2511no—
122C22×M16—2512no—
123#413—2518no—
124C2×#101—2511no—
125C2×#101D25111yescharacteristic, complemented
126C2×#100—2512no—
127C2×#41—5022no—
128C2×#41—5024no—
129C2×#41—5022no—
130C2×#41—2514no—
131C2×#41—2514no—
132C22×Q8C2×D25111yescharacteristic
133C22×D4—2512no—
134C22×D4—2511no—
135C23×C4—2512no—
136C23×C4—2511no—
137C2×M16—5022no—
138C2×M16—5024no—
139C2×M16—2512no—
140C2×M16—2518no—
141C2×M16C2×D25112yes—
142C22×C8—5022no—
143#101—2514no—
144#101C2×D25114yescomplemented
145#100—2518no—
146C24—2512no—
147C24—2511no—
148#41—5028no—
149#41—5028no—
150#41—5028no—
151#41—5028no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8C4×D25112yes—
155C2×Q8C22×D25112yes—
156C2×D4—5024no—
157C2×D4—5024no—
158C2×D4—5022no—
159C2×D4—5022no—
160C2×D4—2518no—
161C2×D4—2512no—
162C2×D4—2512no—
163C22×C4—5024no—
164C22×C4—5022no—
165C22×C4—5022no—
166C22×C4—2518no—
167C22×C4—2511no—
168C22×C4—2512no—
169C22×C4—2512no—
170C22×C4—2514no—
171C22×C4—2512no—
172C22×C4—2512no—
173C22×C4C4×D25112yes—
174C22×C4C22×D25111yescharacteristic
175M16—22no—
176M16—5028no—
177M16—5022no—
178M16—2514no—
179M16C22×D25114yes—
180C2×C8—22no—
181C2×C8—5028no—
182C2×C8—5022no—
183C23—10044no—
184C23—10042no—
185C23—5022no—
186C23—5022no—
187C23—2514no—
188C23—2512no—
189C23—2512no—
190C23—2512no—
191C23C2×C4×D25111yescharacteristic
192Q8—24no—
193Q8—24no—
194D4—50216no—
195D4—5024no—
196D4—5024no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—5028no—
200C2×C4—5028no—
201C2×C4—2512no—
202C2×C4—2512no—
203C2×C4—2511no—
204C2×C4—2511no—
205C2×C4—2514no—
206C2×C4—2514no—
207C2×C4—2514no—
208C2×C4—2514no—
209C2×C4C2×C4×D25114yes—
210C2×C4C2×C4×D25111yescharacteristic
211C2×C4D4×D25112yes—
212C2×C4D4×D25112yes—
213C2×C4C23×D25111yescharacteristic
214C8—24no—
215C8—5024no—
216C22—22no—
217C22—10044no—
218C22—10044no—
219C22—10042no—
220C22—10042no—
221C22—10044no—
222C22—5024no—
223C22—5024no—
224C22—2514no—
225C22C22⋊C4×D25111yescharacteristic
226C22C22⋊C4×D25111yescharacteristic
227C22C22×C4×D25111yescharacteristic
228C4—24no—
229C4—2512no—
230C4—2512no—
231C4C22⋊C4×D25112yes—
232C4C2×D4×D25112yes—
233C2—21no—
234C2—21no—
235C2—10044no—
236C2—5024no—
237C2#3063167012yesdirect product factor
238C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC2×C4C23×D251
FittingC2×#101×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×#16860811C2
cosocleC2×C4×C251C24
hypercenterC2×#101D251
hypocenterC251C2×#413
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×M16×D251C2semidirect
3. #30631670C2direct (N × H)
4. C2×#22548634C2semidirect
5. C2×#22548631C2semidirect
6. C2×#101×C251C2semidirect
19. M16×D251C22semidirect
21. #22548634C22semidirect
22. #22548631C22semidirect
25. #101×C251C22semidirect
115. C2×C251#413semidirect
119. C251C2×#413semidirect
125. C2×#101D251semidirect
144. #101C2×D251semidirect
237. C2#30631670direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168608112
C2×M16×D2512
#306316702
C2×#225486342
C2×#225486312
C2×#101×C2512
C2×#413251