Finite Group Databasebeta

← C2×#30631878 #44559754

Subgroups — C2×#30631878 #44559754

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×#22548699C212yesmaximal, complemented
3C2×#22548696C211yesmaximal, characteristic, complemented
4#30631878C218yesmaximal, direct product factor
5C2×#22548629C212yesmaximal, complemented
6C2×#99×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—
15C23×Dic251C2211yescharacteristic, complemented
16#22548699—24no—
17#22548699C2218yescomplemented
18#22548696—22no—
19#22548696C2214yescomplemented
20C2×C4⋊Dic251—22no—
21C2×Dic251⋊C4—21no—
22C2×C22⋊Dic251—21no—
23C2×C22⋊Dic251C2211yescharacteristic, complemented
24C2×(C2×D251)⋊C4—22no—
25#22548629C2218yescomplemented
26C22×D4×C251C2211yescharacteristic
27C2×C22⋊C4×C251C2212yescomplemented
28#99×C251C2214yescomplemented
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×Dic251—21no—
51C22×Dic251—21no—
52C22×Dic251—22no—
53C22×Dic251—21no—
54C22×Dic251D412yes—
55C22×Dic251C2312yescomplemented
56C2×C4×D251—22no—
57C4⋊Dic251—28no—
58Dic251⋊C4—24no—
59C22⋊Dic251—24no—
60C22⋊Dic251—21no—
61C22⋊Dic251C2312yescomplemented
62(C2×D251)⋊C4—28no—
63C24×C251D411yescharacteristic
64C24×C251C2311yescharacteristic
65C2×D4×C251—22no—
66C2×D4×C251—22no—
67C2×D4×C251D412yes—
68C2×D4×C251C2312yes—
69C22×C4×C251—22no—
70C22×C4×C251D411yescharacteristic
71C22⋊C4×C251—22no—
72C22⋊C4×C251C2314yescomplemented
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—22no—
91C2×Dic251—22no—
92C2×Dic251—42no—
93C2×Dic251—41no—
94C2×Dic251—42no—
95C2×Dic251—41no—
96C2×Dic251C2×D414yes—
97C4×D251—28no—
98C23×C251—21no—
99C23×C251—21no—
100C23×C251—42no—
101C23×C251—42no—
102C23×C251C2×D412yes—
103C23×C251C2×D412yes—
104C23×C251C2×D411yescharacteristic
105C23×C251C2×D412yes—
106C23×C251C2×D411yescharacteristic
107C23×C251C2411yescharacteristic
108D4×C251—24no—
109D4×C251—24no—
110C2×C4×C251—21no—
111C2×C4×C251—21no—
112C2×C4×C251—24no—
113C2×C4×C251—44no—
114C2×C4×C251C2×D412yes—
115C22×C251—24no—
116C22×C251—22no—
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—44no—
136Dic251—24no—
137Dic251—42no—
138Dic251—42no—
139C2×C251—21no—
140C2×C251—22no—
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—2511no—
151C2×#123—2512no—
152#458—2518no—
153C2×#99—2512no—
154C2×#99D25111yescharacteristic, complemented
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×C4—2511no—
165#123—5024no—
166#123—5022no—
167#123—2514no—
168#123—2518no—
169C2×C4⋊C4—5022no—
170C2×C4⋊C4—5021no—
171C2×C22⋊C4—5022no—
172C2×C22⋊C4—5021no—
173C2×C22⋊C4—2511no—
174C2×C22⋊C4C2×D25112yescomplemented
175#99—2518no—
176#99C2×D25114yescomplemented
177C24—5021no—
178C24—2512no—
179C24D100411yescharacteristic
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×D4D100412yes—
201C2×D4C22×D25112yes—
202C22×C4—22no—
203C22×C4—5022no—
204C22×C4—5022no—
205C22×C4—5022no—
206C22×C4—5021no—
207C22×C4—5021no—
208C22×C4—5021no—
209C22×C4—5021no—
210C22×C4—2512no—
211C22×C4—2512no—
212C22×C4D100411yescharacteristic
213C4⋊C4—5028no—
214C4⋊C4—5024no—
215C22⋊C4—22no—
216C22⋊C4—5028no—
217C22⋊C4—5024no—
218C22⋊C4—5021no—
219C22⋊C4—2512no—
220C22⋊C4C22×D25114yescomplemented
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×D100411yescharacteristic
234C23C2×D100412yes—
235C23C2×D100411yescharacteristic
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—44no—
250C2×C4—10042no—
251C2×C4—10041no—
252C2×C4—10042no—
253C2×C4—10041no—
254C2×C4—5028no—
255C2×C4—5024no—
256C2×C4—5022no—
257C2×C4—5022no—
258C2×C4—5024no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—2514no—
262C2×C4C2×D100412yes—
263C22—24no—
264C22—22no—
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#2254868511yescharacteristic
278C22#2254868512yes—
279C22#2254868511yescharacteristic
280C22C2×D4×D25112yes—
281C22C22×D100411yescharacteristic
282C4—22no—
283C4—44no—
284C4—10042no—
285C4—10042no—
286C4—5024no—
287C2—22no—
288C2—21no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—10044no—
293C2#3063187812yesdirect product factor
294C2C2×#2254868511yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC23×C251C24
FrattiniC23C23×D251
FittingC2×#99×C251C2
socleC22×C251#120
prime radicalC2×#22548760C2
hypercenterC2×#99D251
hypocenterC251C2×#458
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
2. C2×#22548699C2semidirect
3. C2×#22548696C2semidirect
4. #30631878C2direct (N × H)
5. C2×#22548629C2semidirect
6. C2×#99×C251C2semidirect
15. C23×Dic251C22semidirect
17. #22548699C22semidirect
19. #22548696C22semidirect
23. C2×C22⋊Dic251C22semidirect
25. #22548629C22semidirect
27. C2×C22⋊C4×C251C22semidirect
28. #99×C251C22semidirect
55. C22×Dic251C23semidirect
61. C22⋊Dic251C23semidirect
72. C22⋊C4×C251C23semidirect
144. C2×C251#458semidirect
147. C251C2×#458semidirect
154. C2×#99D251semidirect
174. C2×C22⋊C4C2×D251semidirect
176. #99C2×D251semidirect
220. C22⋊C4C22×D251semidirect
293. C2#30631878direct (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×#225486992
C2×#225486962
#306318782
C2×#225486292
C2×#99×C2512
C2×#458251