Finite Group Databasebeta

← C2×#30631876 #44559752

Subgroups — C2×#30631876 #44559752

Order 32128.

794 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548760C211yesmaximal, characteristic
2C2×#22548698C212yesmaximal, complemented
3#30631876C218yesmaximal, direct product factor
4C2×#22548630C211yesmaximal, characteristic, complemented
5C2×#22548629C212yesmaximal, complemented
6C2×#123×C251C211yesmaximal, characteristic, complemented
7#22548760—24no—
8#22548760—22no—
9#22548760C2214yes—
10C2×#16860810—21no—
11C2×#16860809—22no—
12C22×C22⋊D251—21no—
13C22×C22⋊D251C2212yes—
14C2×D4×D251—22no—
15#22548698—24no—
16#22548698C2218yescomplemented
17C2×Dic251⋊C4—22no—
18C2×C22⋊Dic251C2212yescomplemented
19C2×(C2×D251)⋊C4—22no—
20#22548630C2214yescomplemented
21#22548629C2218yescomplemented
22C22×D4×C251C2211yescharacteristic
23C23×C4×C251C2211yescharacteristic, complemented
24#123×C251—22no—
25#123×C251C2214yescomplemented
26C2×C4⋊C4×C251—21no—
27C2×C22⋊C4×C251—21no—
28C2×C22⋊C4×C251C2211yescharacteristic, complemented
29C23×D251—21no—
30C23×D251D412yes—
31#16860810—24no—
32#16860810—24no—
33#16860809—28no—
34#16860809—28no—
35C2×Dic502—21no—
36C2×D1004—21no—
37C2×C22⋊D251—22no—
38C2×C22⋊D251—24no—
39C2×C22⋊D251—22no—
40C2×C22⋊D251—22no—
41C2×C22⋊D251—24no—
42C2×C22⋊D251—22no—
43C2×C22⋊D251—22no—
44C2×C22⋊D251D414yes—
45C2×C22⋊D251C2314yes—
46D4×D251—28no—
47D4×D251—28no—
48C22×Dic251—21no—
49C22×Dic251—22no—
50C22×Dic251D412yes—
51C2×C4×D251—22no—
52Dic251⋊C4—28no—
53C22⋊Dic251—22no—
54C22⋊Dic251C2314yescomplemented
55(C2×D251)⋊C4—28no—
56C24×C251D411yescharacteristic
57C24×C251C2311yescharacteristic
58C2×D4×C251—22no—
59C2×D4×C251—22no—
60C2×D4×C251D412yes—
61C2×D4×C251C2312yes—
62C22×C4×C251—21no—
63C22×C4×C251—21no—
64C22×C4×C251—21no—
65C22×C4×C251—22no—
66C22×C4×C251—22no—
67C22×C4×C251D411yescharacteristic
68C22×C4×C251C2312yescomplemented
69C4⋊C4×C251—24no—
70C22⋊C4×C251—24no—
71C22⋊C4×C251—21no—
72C22⋊C4×C251C2312yescomplemented
73C22×D251—22no—
74C22×D251—24no—
75C22×D251—22no—
76C22×D251—44no—
77C22×D251—82no—
78C22×D251C2×D414yes—
79Dic502—24no—
80D1004—24no—
81C22⋊D251—28no—
82C22⋊D251—24no—
83C22⋊D251—24no—
84C22⋊D251—24no—
85C22⋊D251—24no—
86C2×Dic251—24no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—24no—
90C2×Dic251—44no—
91C2×Dic251C2×D414yes—
92C4×D251—28no—
93C23×C251—21no—
94C23×C251—21no—
95C23×C251—42no—
96C23×C251—42no—
97C23×C251C2×D412yes—
98C23×C251C2×D412yes—
99C23×C251C2×D411yescharacteristic
100C23×C251C2×D412yes—
101C23×C251C2×D411yescharacteristic
102C23×C251C2411yescharacteristic
103D4×C251—24no—
104D4×C251—24no—
105C2×C4×C251—21no—
106C2×C4×C251—21no—
107C2×C4×C251—22no—
108C2×C4×C251—22no—
109C2×C4×C251—24no—
110C2×C4×C251—42no—
111C2×C4×C251—41no—
112C2×C4×C251—42no—
113C2×C4×C251—41no—
114C2×C4×C251C2×D412yes—
115C22×C251—22no—
116C22×C251—24no—
117C22×C251—22no—
118C22×C251—42no—
119C22×C251—41no—
120C22×C251—41no—
121C22×C251—42no—
122C22×C251—42no—
123C22×C251—81no—
124C22×C251#12011yescharacteristic
125C22×C251#12012yes—
126C22×C251#12011yescharacteristic
127C22×C251C22×D412yes—
128C22×C251C22×D411yescharacteristic
129C2×D251—24no—
130C2×D251—44no—
131C2×D251—44no—
132C2×D251—82no—
133C2×D251—84no—
134C4×C251—22no—
135C4×C251—42no—
136C4×C251—42no—
137Dic251—24no—
138Dic251—44no—
139C2×C251—22no—
140C2×C251—21no—
141C2×C251—22no—
142C2×C251—21no—
143C2×C251—42no—
144C2×C251#45812yescomplemented
145C2×C251C2×#12011yescharacteristic
146D251—44no—
147C251C2×#45811yescharacteristic, complemented, Sylow
148C2×#458—2511nomaximal, Sylow
149C2×H(5,𝔽2)—2511no—
150C2×#123—2512no—
151C2×#123D25111yescharacteristic, complemented
152#458—2518no—
153C2×#99—2511no—
154C2×#99—2512no—
155H(5,𝔽2)—5024no—
156H(5,𝔽2)—5022no—
157H(5,𝔽2)—2514no—
158C2×#41—5022no—
159C2×#41—5021no—
160C22×D4—5022no—
161C22×D4—5021no—
162C22×D4—2512no—
163C22×D4C2×D25111yescharacteristic
164C23×C4C2×D25111yescharacteristic, complemented
165#123—22no—
166#123—5024no—
167#123—2518no—
168#123C2×D25114yescomplemented
169C2×C4⋊C4—21no—
170C2×C4⋊C4—5022no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—5022no—
173C2×C22⋊C4—2512no—
174C2×C22⋊C4C2×D25111yescharacteristic, complemented
175#99—2514no—
176#99—2518no—
177C24—5021no—
178C24—2512no—
179C24C22⋊D25111yescharacteristic
180C24C22×D25111yescharacteristic
181#41—5024no—
182#41—5028no—
183#41—5028no—
184#41—5024no—
185C2×Q8—5021no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—5021no—
189C2×D4—5028no—
190C2×D4—5028no—
191C2×D4—5022no—
192C2×D4—5024no—
193C2×D4—5022no—
194C2×D4—5022no—
195C2×D4—5022no—
196C2×D4—5024no—
197C2×D4—5022no—
198C2×D4—2514no—
199C2×D4—2514no—
200C2×D4C22⋊D25112yes—
201C2×D4C22×D25112yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4—5021no—
210C22×C4—2512no—
211C22×C4C22⋊D25111yescharacteristic
212C22×C4C22×D25112yescomplemented
213C4⋊C4—24no—
214C4⋊C4—5028no—
215C22⋊C4—24no—
216C22⋊C4—21no—
217C22⋊C4—5028no—
218C22⋊C4—5022no—
219C22⋊C4—2514no—
220C22⋊C4C22×D25112yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—20082no—
226C23—10044no—
227C23—5024no—
228C23—5022no—
229C23—5022no—
230C23—2514no—
231C23D4×D25112yes—
232C23D4×D25112yes—
233C23C2×C22⋊D25111yescharacteristic
234C23C2×C22⋊D25112yes—
235C23C2×C22⋊D25111yescharacteristic
236C23C23×D25111yescharacteristic
237Q8—5024no—
238D4—24no—
239D4—24no—
240D4—5024no—
241D4—5028no—
242D4—5024no—
243D4—5024no—
244D4—5024no—
245D4—5024no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—10044no—
256C2×C4—5028no—
257C2×C4—5024no—
258C2×C4—5024no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—2514no—
262C2×C4C2×C22⋊D25112yes—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—20082no—
273C22—20084no—
274C22—10044no—
275C22—10044no—
276C22—5024no—
277C22#2254868311yescharacteristic
278C22#2254868312yes—
279C22#2254868311yescharacteristic
280C22C2×D4×D25112yes—
281C22C22×C22⋊D25111yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—10044no—
286C4—5024no—
287C2—21no—
288C2—22no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—10044no—
293C2#3063187612yesdirect product factor
294C2C2×#2254868311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC23×C251C24
FrattiniC23C23×D251
FittingC2×#123×C251C2
socleC22×C251#120
prime radicalC2×#22548760C2
hypercenterC2×#123D251
hypocenterC251C2×#458
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#22548698C2semidirect
3. #30631876C2direct (N × H)
4. C2×#22548630C2semidirect
5. C2×#22548629C2semidirect
6. C2×#123×C251C2semidirect
16. #22548698C22semidirect
18. C2×C22⋊Dic251C22semidirect
20. #22548630C22semidirect
21. #22548629C22semidirect
23. C23×C4×C251C22semidirect
25. #123×C251C22semidirect
28. C2×C22⋊C4×C251C22semidirect
54. C22⋊Dic251C23semidirect
68. C22×C4×C251C23semidirect
72. C22⋊C4×C251C23semidirect
144. C2×C251#458semidirect
147. C251C2×#458semidirect
151. C2×#123D251semidirect
164. C23×C4C2×D251semidirect
168. #123C2×D251semidirect
174. C2×C22⋊C4C2×D251semidirect
212. C22×C4C22×D251semidirect
220. C22⋊C4C22×D251semidirect
293. C2#30631876direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225486982
#306318762
C2×#225486302
C2×#225486292
C2×#123×C2512
C2×#458251