Finite Group Databasebeta

← C2×#30631875 #44559750

Subgroups — C2×#30631875 #44559750

Order 32128.

1018 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1018 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, complemented
2C2×#22548685C212yesmaximal, complemented
3C2×#22548683C211yesmaximal, characteristic, complemented
4#30631875C218yesmaximal, direct product factor
5C2×#22548629C212yesmaximal, complemented
6C2×#99×C251C211yesmaximal, characteristic, complemented
7C24×D251C2211yescharacteristic, complemented
8#22548760—24no—
9#22548760—22no—
10#22548760C2214yescomplemented
11C2×#16860810—21no—
12C2×#16860809—22no—
13C22×D1004—22no—
14C22×C22⋊D251—21no—
15C22×C22⋊D251—21no—
16C22×C22⋊D251C2212yes—
17C2×D4×D251—22no—
18#22548685—24no—
19#22548685C2218yescomplemented
20#22548683—22no—
21#22548683C2214yescomplemented
22C2×C22⋊Dic251C2211yescharacteristic, complemented
23C2×(C2×D251)⋊C4—22no—
24C2×(C2×D251)⋊C4—21no—
25#22548629C2218yescomplemented
26C22×D4×C251C2211yescharacteristic
27C2×C22⋊C4×C251C2212yescomplemented
28#99×C251C2214yescomplemented
29C23×D251—21no—
30C23×D251—21no—
31C23×D251—22no—
32C23×D251—21no—
33C23×D251—22no—
34C23×D251—44no—
35C23×D251D412yescomplemented
36C23×D251C2312yescomplemented
37#16860810—24no—
38#16860810—24no—
39#16860809—28no—
40#16860809—28no—
41C2×Dic502—21no—
42C2×D1004—21no—
43C2×D1004—28no—
44C2×D1004—48no—
45C2×C22⋊D251—22no—
46C2×C22⋊D251—24no—
47C2×C22⋊D251—22no—
48C2×C22⋊D251—22no—
49C2×C22⋊D251—22no—
50C2×C22⋊D251—22no—
51C2×C22⋊D251—24no—
52C2×C22⋊D251—24no—
53C2×C22⋊D251—44no—
54C2×C22⋊D251D414yescomplemented
55C2×C22⋊D251C2314yes—
56D4×D251—28no—
57D4×D251—28no—
58C22×Dic251—21no—
59C22×Dic251—21no—
60C22×Dic251D412yescomplemented
61C2×C4×D251—22no—
62C22⋊Dic251—21no—
63C22⋊Dic251C2312yescomplemented
64(C2×D251)⋊C4—28no—
65(C2×D251)⋊C4—24no—
66C24×C251D411yescharacteristic, complemented
67C24×C251C2311yescharacteristic
68C2×D4×C251—22no—
69C2×D4×C251—22no—
70C2×D4×C251D412yescomplemented
71C2×D4×C251C2312yes—
72C22×C4×C251—22no—
73C22×C4×C251D411yescharacteristic, complemented
74C22⋊C4×C251—22no—
75C22⋊C4×C251C2314yescomplemented
76C22×D251—24no—
77C22×D251—28no—
78C22×D251—22no—
79C22×D251—22no—
80C22×D251—24no—
81C22×D251—24no—
82C22×D251—22no—
83C22×D251—44no—
84C22×D251—44no—
85C22×D251—48no—
86C22×D251—44no—
87C22×D251—42no—
88C22×D251—41no—
89C22×D251—42no—
90C22×D251—41no—
91C22×D251—44no—
92C22×D251—82no—
93C22×D251C2×D414yescomplemented
94Dic502—24no—
95D1004—24no—
96D1004—416no—
97C22⋊D251—28no—
98C22⋊D251—24no—
99C22⋊D251—24no—
100C22⋊D251—24no—
101C22⋊D251—24no—
102C22⋊D251—48no—
103C2×Dic251—24no—
104C2×Dic251—22no—
105C2×Dic251—22no—
106C2×Dic251—22no—
107C2×Dic251—42no—
108C2×Dic251C2×D414yescomplemented
109C4×D251—28no—
110C23×C251—21no—
111C23×C251—21no—
112C23×C251—42no—
113C23×C251—42no—
114C23×C251C2×D412yes—
115C23×C251C2×D412yes—
116C23×C251C2×D411yescharacteristic
117C23×C251C2×D411yescharacteristic
118C23×C251C2×D412yescomplemented
119C23×C251C2411yescharacteristic
120D4×C251—24no—
121D4×C251—24no—
122C2×C4×C251—21no—
123C2×C4×C251—21no—
124C2×C4×C251—24no—
125C2×C4×C251—44no—
126C2×C4×C251C2×D412yescomplemented
127C22×C251—24no—
128C22×C251—22no—
129C22×C251—22no—
130C22×C251—42no—
131C22×C251—41no—
132C22×C251—41no—
133C22×C251—42no—
134C22×C251—42no—
135C22×C251—81no—
136C22×C251#12011yescharacteristic
137C22×C251#12012yes—
138C22×C251#12011yescharacteristic
139C22×C251C22×D412yes—
140C22×C251C22×D411yescharacteristic
141C2×D251—24no—
142C2×D251—28no—
143C2×D251—24no—
144C2×D251—42no—
145C2×D251—44no—
146C2×D251—48no—
147C2×D251—44no—
148C2×D251—42no—
149C2×D251—44no—
150C2×D251—44no—
151C2×D251—44no—
152C2×D251—82no—
153C2×D251—84no—
154C4×C251—22no—
155C4×C251—44no—
156Dic251—24no—
157Dic251—42no—
158C2×C251—21no—
159C2×C251—22no—
160C2×C251—22no—
161C2×C251—21no—
162C2×C251—42no—
163C2×C251B(4,𝔽2)12yescomplemented
164C2×C251C2×#12011yescharacteristic
165D251—44no—
166D251—44no—
167C251C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
168C2×B(4,𝔽2)—2511nomaximal, Sylow
169C2×H(5,𝔽2)—2511no—
170C2×#120—2511no—
171C2×#120—2512no—
172B(4,𝔽2)—2518no—
173C2×#99—2512no—
174C2×#99D25111yescharacteristic, complemented
175C25—2511no—
176H(5,𝔽2)—5024no—
177H(5,𝔽2)—5022no—
178H(5,𝔽2)—2514no—
179C2×#41—5022no—
180C2×#41—5021no—
181C22×D4—5022no—
182C22×D4—5022no—
183C22×D4—5021no—
184C22×D4—5021no—
185C22×D4—2512no—
186C22×D4C2×D25111yescharacteristic
187#120—5024no—
188#120—5022no—
189#120—2514no—
190#120—2518no—
191C2×C22⋊C4—5022no—
192C2×C22⋊C4—5021no—
193C2×C22⋊C4—2511no—
194C2×C22⋊C4C2×D25112yescomplemented
195#99—2518no—
196#99C2×D25114yescomplemented
197C24—10044no—
198C24—5021no—
199C24—5022no—
200C24—5021no—
201C24—5022no—
202C24—5021no—
203C24—2512no—
204C24—2512no—
205C24D100411yescharacteristic, complemented
206C24C22×D25111yescharacteristic
207#41—5024no—
208#41—5028no—
209#41—5028no—
210#41—5024no—
211C2×Q8—5021no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—10048no—
215C2×D4—10044no—
216C2×D4—5028no—
217C2×D4—5021no—
218C2×D4—5028no—
219C2×D4—5028no—
220C2×D4—5024no—
221C2×D4—5022no—
222C2×D4—5024no—
223C2×D4—5022no—
224C2×D4—5022no—
225C2×D4—5022no—
226C2×D4—5022no—
227C2×D4—5024no—
228C2×D4—2514no—
229C2×D4—2514no—
230C2×D4D100412yescomplemented
231C2×D4C22×D25112yes—
232C22×C4—22no—
233C22×C4—5022no—
234C22×C4—5021no—
235C22×C4—5021no—
236C22×C4—2512no—
237C22×C4D100411yescharacteristic, complemented
238C22⋊C4—22no—
239C22⋊C4—5028no—
240C22⋊C4—5021no—
241C22⋊C4—5024no—
242C22⋊C4—2512no—
243C22⋊C4C22×D25114yescomplemented
244C23—21no—
245C23—21no—
246C23—42no—
247C23—42no—
248C23—20082no—
249C23—10044no—
250C23—10044no—
251C23—10042no—
252C23—10048no—
253C23—10044no—
254C23—10044no—
255C23—10041no—
256C23—10042no—
257C23—10041no—
258C23—5024no—
259C23—5028no—
260C23—5024no—
261C23—5022no—
262C23—5024no—
263C23—5022no—
264C23—5022no—
265C23—2514no—
266C23D4×D25112yes—
267C23D4×D25112yes—
268C23C2×D100411yescharacteristic
269C23C2×D100411yescharacteristic
270C23C2×D100412yescomplemented
271C23C23×D25111yescharacteristic
272Q8—5024no—
273D4—24no—
274D4—24no—
275D4—100416no—
276D4—10048no—
277D4—5024no—
278D4—5028no—
279D4—5024no—
280D4—5024no—
281D4—5024no—
282D4—5024no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—44no—
287C2×C4—10042no—
288C2×C4—5028no—
289C2×C4—5022no—
290C2×C4—5024no—
291C2×C4—5022no—
292C2×C4—5022no—
293C2×C4—2514no—
294C2×C4C2×D100412yescomplemented
295C22—24no—
296C22—22no—
297C22—22no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—81no—
304C22—20082no—
305C22—20084no—
306C22—10044no—
307C22—10044no—
308C22—10042no—
309C22—10044no—
310C22—10048no—
311C22—10044no—
312C22—10044no—
313C22—10042no—
314C22—5028no—
315C22—5024no—
316C22—5024no—
317C22#2254868511yescharacteristic
318C22#2254868512yes—
319C22#2254868511yescharacteristic
320C22C2×D4×D25112yes—
321C22C22×D100411yescharacteristic
322C4—22no—
323C4—44no—
324C4—10042no—
325C4—5024no—
326C2—22no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—42no—
331C2—10044no—
332C2—10044no—
333C2#3063187512yesdirect product factor
334C2C2×#2254868511yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC23×C251C24
FrattiniC23C23×D251
FittingC2×#99×C251C2
socleC22×C251#120
hypercenterC2×#99D251
hypocenterC251C2×B(4,𝔽2)
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#22548760C2semidirect
2. C2×#22548685C2semidirect
3. C2×#22548683C2semidirect
4. #30631875C2direct (N × H)
5. C2×#22548629C2semidirect
6. C2×#99×C251C2semidirect
7. C24×D251C22semidirect
10. #22548760C22semidirect
19. #22548685C22semidirect
21. #22548683C22semidirect
22. C2×C22⋊Dic251C22semidirect
25. #22548629C22semidirect
27. C2×C22⋊C4×C251C22semidirect
28. #99×C251C22semidirect
35. C23×D251D4semidirect
36. C23×D251C23semidirect
54. C2×C22⋊D251D4semidirect
60. C22×Dic251D4semidirect
63. C22⋊Dic251C23semidirect
66. C24×C251D4semidirect
70. C2×D4×C251D4semidirect
73. C22×C4×C251D4semidirect
75. C22⋊C4×C251C23semidirect
93. C22×D251C2×D4semidirect
108. C2×Dic251C2×D4semidirect
118. C23×C251C2×D4semidirect
126. C2×C4×C251C2×D4semidirect
163. C2×C251B(4,𝔽2)semidirect
167. C251C2×B(4,𝔽2)semidirect
174. C2×#99D251semidirect
194. C2×C22⋊C4C2×D251semidirect
196. #99C2×D251semidirect
205. C24D1004semidirect
230. C2×D4D1004semidirect
237. C22×C4D1004semidirect
243. C22⋊C4C22×D251semidirect
270. C23C2×D1004semidirect
294. C2×C4C2×D1004semidirect
333. C2#30631875direct (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×#225486852
C2×#225486832
#306318752
C2×#225486292
C2×#99×C2512
C2×B(4,𝔽2)251