Finite Group Databasebeta

← C2×#30631952 #44560066

Subgroups — C2×#30631952 #44560066

Order 32128.

468 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22548740C211yesmaximal, characteristic
2C22×#16860797C211yesmaximal, characteristic, complemented
3C2×#22548722C211yesmaximal, characteristic, complemented
4#30631952C218yesmaximal, direct product factor
5C2×#22548650C211yesmaximal, characteristic, complemented
6C2×#22548634C211yesmaximal, characteristic, complemented
7C2×#22548631C211yesmaximal, characteristic, complemented
8C2×#137×C251C211yesmaximal, characteristic, complemented
9C22×D1004C2211yescharacteristic
10#22548740—22no—
11#22548740C2214yes—
12C2×#16860797—21no—
13C2×#16860797—21no—
14C2×#16860797—22no—
15C2×#16860797—21no—
16C2×#16860797C2214yescomplemented
17C2×#16860795—21no—
18#22548722—22no—
19#22548722C2214yescomplemented
20C2×#16860794—21no—
21C2×#16860794C2211yescharacteristic
22C2×#16860794C2211yescharacteristic
23C22×C251⋊C8—21no—
24C22×C251⋊C8C2211yescharacteristic
25#22548650C2214yescomplemented
26#22548634C2214yescomplemented
27#22548631C2214yescomplemented
28C2×#41×C251C2211yescharacteristic
29C22×Q8×C251C2211yescharacteristic
30#137×C251—22no—
31#137×C251C2214yescomplemented
32C2×Q16×C251—21no—
33C2×SD16×C251—21no—
34C2×M16×C251C2211yescharacteristic
35C23×D251—21no—
36C2×D1004—21no—
37C2×D1004—21no—
38C2×D1004—21no—
39C2×D1004—21no—
40C2×D1004—21no—
41C2×D1004C2312yes—
42#16860797—24no—
43#16860797—24no—
44#16860797—24no—
45#16860795—24no—
46#16860794—24no—
47#16860794—21no—
48#16860794—21no—
49#16860794C2312yes—
50#16860794C2312yes—
51C2×C251⋊C8—24no—
52C2×C251⋊C8—21no—
53C2×C251⋊C8—21no—
54C2×C251⋊C8D411yescharacteristic
55C2×C251⋊C8D411yescharacteristic
56C2×C251⋊C8D412yes—
57C2×C251⋊C8C2312yes—
58#41×C251—21no—
59#41×C251—21no—
60#41×C251D412yes—
61#41×C251C2312yes—
62C2×Q8×C251—21no—
63C2×Q8×C251—21no—
64C2×Q8×C251—21no—
65C2×Q8×C251—21no—
66C2×Q8×C251—21no—
67C2×Q8×C251D411yescharacteristic
68C2×Q8×C251C2312yes—
69C2×D4×C251—21no—
70C2×D4×C251D411yescharacteristic
71C22×C4×C251—21no—
72C22×C4×C251—21no—
73C22×C4×C251C2311yescharacteristic
74Q16×C251—24no—
75SD16×C251—24no—
76M16×C251—21no—
77M16×C251C2312yes—
78C2×C8×C251—21no—
79C22×D251—22no—
80C22×D251—41no—
81C22×D251—41no—
82C22×D251—82no—
83D1004—22no—
84D1004—22no—
85D1004—41no—
86D1004—41no—
87C251⋊C8—22no—
88C251⋊C8—22no—
89C251⋊C8C2×D412yes—
90C251⋊C8C2×D412yes—
91C23×C251—21no—
92C23×C251#4111yescharacteristic
93Q8×C251—21no—
94Q8×C251—22no—
95Q8×C251—22no—
96Q8×C251—41no—
97Q8×C251—41no—
98Q8×C251C2×D412yes—
99D4×C251—22no—
100D4×C251—22no—
101D4×C251—21no—
102D4×C251C2×D412yes—
103C2×C4×C251—21no—
104C2×C4×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—41no—
108C2×C4×C251—41no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251#4111yescharacteristic
114C2×C4×C251C2411yescharacteristic
115C8×C251—22no—
116C22×C251—22no—
117C22×C251—22no—
118C22×C251—42no—
119C22×C251#12111yescharacteristic
120C22×C251#12111yescharacteristic
121C22×C251C2×#4111yescharacteristic
122C2×D251—42no—
123C2×D251—82no—
124C2×D251—82no—
125C2×D251—82no—
126C4×C251—22no—
127C4×C251—42no—
128C4×C251#12111yescharacteristic
129C4×C251#12111yescharacteristic
130C4×C251C22×D411yescharacteristic
131C4×C251C22×D411yescharacteristic
132C2×C251—21no—
133C2×C251—21no—
134C2×C251—42no—
135C2×C251#47112yescomplemented
136C2×C251C2×#12111yescharacteristic
137D251—82no—
138C251C2×#47111yescharacteristic, complemented, Sylow
139C2×#471—2511nomaximal, Sylow
140C2×#137D25111yescharacteristic, complemented
141C2×AGL(1,ℤ/8)—2511no—
142C22×SD16—2511no—
143C2×#131—2511no—
144#471—2518no—
145C2×#108—2511no—
146C2×#101—2511no—
147C2×#100—2511no—
148C2×#41C2×D25111yescharacteristic
149C22×Q8C2×D25111yescharacteristic
150C22×D4—2511no—
151#137—22no—
152#137C2×D25114yescomplemented
153AGL(1,ℤ/8)—5022no—
154AGL(1,ℤ/8)—2514no—
155C2×Q16—21no—
156C2×SD16—21no—
157C2×SD16—5021no—
158C2×SD16—5021no—
159C2×SD16—5021no—
160C2×SD16—5022no—
161C2×SD16—2514no—
162C2×D8—5021no—
163#131—5022no—
164#131—2514no—
165C2×M16—5021no—
166C2×M16—2511no—
167C2×M16—2511no—
168C2×M16C2×D25111yescharacteristic
169C22×C8—5021no—
170C22×C8—2511no—
171#108—2514no—
172#101—2514no—
173#100—2514no—
174C24—5021no—
175#41—21no—
176#41—21no—
177#41C22⋊D25112yes—
178#41C22×D25112yes—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8C22⋊D25111yescharacteristic
185C2×Q8C22×D25112yes—
186C2×D4—21no—
187C2×D4—5021no—
188C2×D4—5021no—
189C2×D4—5021no—
190C2×D4—5021no—
191C2×D4—5021no—
192C2×D4—2512no—
193C2×D4C22⋊D25111yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4C22×D25111yescharacteristic
197Q16—24no—
198SD16—24no—
199SD16—5024no—
200SD16—5024no—
201SD16—5024no—
202D8—5024no—
203M16—21no—
204M16—5021no—
205M16—5021no—
206M16—5024no—
207M16—2512no—
208M16—2512no—
209M16C22×D25112yes—
210C2×C8—21no—
211C2×C8—5021no—
212C2×C8—5021no—
213C2×C8—5024no—
214C2×C8—2511no—
215C2×C8—2511no—
216C2×C8—2512no—
217C2×C8—2512no—
218C23—21no—
219C23—20082no—
220C23—10041no—
221C23—10041no—
222C23—5022no—
223C23#1686080911yescharacteristic
224Q8—21no—
225Q8—22no—
226Q8—22no—
227Q8—41no—
228Q8—41no—
229Q8C2×C22⋊D25112yes—
230D4—22no—
231D4—22no—
232D4—21no—
233D4—10041no—
234D4—10041no—
235D4—5022no—
236D4—5022no—
237D4C2×C22⋊D25112yes—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4D4×D25111yescharacteristic
245C2×C4D4×D25111yescharacteristic
246C2×C4C2×C22⋊D25111yescharacteristic
247C2×C4C2×C22⋊D25111yescharacteristic
248C2×C4#1686080911yescharacteristic
249C2×C4C23×D25111yescharacteristic
250C8—22no—
251C8—5022no—
252C8—5022no—
253C8—2512no—
254C8—2512no—
255C22—22no—
256C22—22no—
257C22—42no—
258C22—20082no—
259C22—20082no—
260C22—20082no—
261C22—10042no—
262C22#2254868711yescharacteristic
263C22#2254868711yescharacteristic
264C22C2×#1686080911yescharacteristic
265C4—22no—
266C4—42no—
267C4#2254868711yescharacteristic
268C4#2254868711yescharacteristic
269C4C2×D4×D25111yescharacteristic
270C4C22×C22⋊D25111yescharacteristic
271C2—21no—
272C2—21no—
273C2—42no—
274C2—20082no—
275C2#3063195212yesdirect product factor
276C2C2×#2254868711yescharacteristic
subgroupG/H
centerC22#22548687
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251#121
prime radicalC2×#22548740C2
hypercenterC2×#137D251
hypocenterC251C2×#471
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#16860797C2semidirect
3. C2×#22548722C2semidirect
4. #30631952C2direct (N × H)
5. C2×#22548650C2semidirect
6. C2×#22548634C2semidirect
7. C2×#22548631C2semidirect
8. C2×#137×C251C2semidirect
16. C2×#16860797C22semidirect
19. #22548722C22semidirect
25. #22548650C22semidirect
26. #22548634C22semidirect
27. #22548631C22semidirect
31. #137×C251C22semidirect
135. C2×C251#471semidirect
138. C251C2×#471semidirect
140. C2×#137D251semidirect
152. #137C2×D251semidirect
275. C2#30631952direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487402
C22×#168607972
C2×#225487222
#306319522
C2×#225486502
C2×#225486342
C2×#225486312
C2×#137×C2512
C2×#471251