Finite Group Databasebeta

← #46820363

Subgroups — #46820363

Order 34432.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#22575478C211yesmaximal, characteristic, complemented
2C2×#22575449C211yesmaximal, characteristic, complemented
3#41×Dic269C211yesmaximal, characteristic, complemented
4#31770803C214yesmaximal, complemented
5#31770798C214yesmaximal, complemented
6#31770628C211yesmaximal, characteristic, complemented
7#31770611C211yesmaximal, characteristic, complemented
8#31770610C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
10C22×D1076C2211yescharacteristic
11C22×C22⋊D269—22no—
12#22575478—24no—
13C2×#16864389C2212yescomplemented
14C2×#16864387C2212yescomplemented
15C2×#16864386C2211yescharacteristic
16#22575449—21no—
17#22575449—21no—
18#22575449C2214yes—
19Q8×Dic269C2212yescomplemented
20D4×Dic269—22no—
21D4×Dic269C2212yescomplemented
22#22575412—21no—
23#22575412C2211yescharacteristic
24C2×C4×Dic269—21no—
25C2×C4×Dic269C2211yescharacteristic
26#22575382C2214yescomplemented
27#22575375C2214yescomplemented
28#22575374C2214yescomplemented
29C2×#41×C269C2211yescharacteristic
30C22×D4×C269C2211yescharacteristic
31AGL(1,ℤ/8)×C269—24no—
32C2×SD16×C269C2212yescomplemented
33C2×D8×C269C2212yescomplemented
34C2×M16×C269C2211yescharacteristic
35C23×D269—21no—
36C2×D1076—22no—
37C2×D1076—21no—
38C2×D1076—22no—
39C2×D1076C2312yes—
40C2×C22⋊D269—24no—
41C2×C22⋊D269—24no—
42C2×C22⋊D269—48no—
43C22×Dic269—22no—
44C22×Dic269D412yes—
45#16864389—24no—
46#16864387—24no—
47#16864386—22no—
48C2×#14524130C2312yes—
49C4⋊Dic269—22no—
50C4⋊Dic269C2312yes—
51C22⋊Dic269—22no—
52C22⋊Dic269—21no—
53C4×Dic269—22no—
54C4×Dic269—21no—
55C4×Dic269D411yescharacteristic
56C4×Dic269D411yescharacteristic
57C4×Dic269C2312yes—
58C24×C269—21no—
59#41×C269—22no—
60#41×C269D414yescomplemented
61C2×Q8×C269C2311yescharacteristic
62C2×D4×C269—21no—
63C2×D4×C269—22no—
64C2×D4×C269—21no—
65C2×D4×C269—22no—
66C2×D4×C269C2312yes—
67C2×D4×C269C2311yescharacteristic
68C22×C4×C269—21no—
69C22×C4×C269C2311yescharacteristic
70SD16×C269—24no—
71D8×C269—24no—
72M16×C269—22no—
73C2×C8×C269C2312yes—
74C22×D269—22no—
75C22×D269—44no—
76C22×D269—42no—
77D1076—22no—
78D1076—22no—
79D1076—42no—
80C22⋊D269—48no—
81C22⋊D269—48no—
82C2×Dic269—24no—
83C2×Dic269—22no—
84C2×Dic269—44no—
85C2×Dic269C2×D412yes—
86C2×Dic269C2×D412yes—
87#14524130—22no—
88C23×C269—21no—
89C23×C269—22no—
90C23×C269—44no—
91C23×C269—42no—
92C23×C269C2×D411yescharacteristic
93Q8×C269—21no—
94Q8×C269C2×D412yescomplemented
95D4×C269—24no—
96D4×C269—22no—
97D4×C269—21no—
98D4×C269—22no—
99D4×C269—42no—
100D4×C269C2×D412yescomplemented
101C2×C4×C269—21no—
102C2×C4×C269—24no—
103C2×C4×C269C2×D411yescharacteristic
104C2×C4×C269C2×D412yes—
105C2×C4×C269#4112yes—
106C2×C4×C269C2411yescharacteristic
107C8×C269—22no—
108C22×C269—22no—
109C22×C269—22no—
110C22×C269—42no—
111C22×C269—42no—
112C22×C269—44no—
113C22×C269—84no—
114C22×C269#12112yes—
115C22×C269C22×D411yescharacteristic
116C2×D269—42no—
117C2×D269—44no—
118C2×D269—84no—
119C4×C269—22no—
120C4×C269#12112yes—
121C4×C269C22×D411yescharacteristic
122C4×C269C2×#4111yescharacteristic
123Dic269—22no—
124Dic269—42no—
125Dic269AGL(1,ℤ/8)14yescomplemented
126C2×C269—22no—
127C2×C269—42no—
128C2×C269—82no—
129C2×C269C2×#12111yescharacteristic
130C2×C269C2×AGL(1,ℤ/8)12yes—
131D269—82no—
132C269#303811yescharacteristic, complemented, Sylow
133#3038—2691nomaximal, Sylow
134C2×AGL(1,ℤ/8)—2691no—
135C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
136C2×#127—2691no—
137C4×#41—2691no—
138#460—2694no—
139#459—2694no—
140#423—2691no—
141#418—2691no—
142#418—2691no—
143C2×#41C2×D26911yescharacteristic
144C22×D4—2691no—
145C22×D4—5382no—
146C22×D4C2×D26911yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—5384no—
149C2×SD16—2692no—
150C2×SD16C2×D26912yescomplemented
151C2×D8—2692no—
152C2×D8C2×D26912yescomplemented
153C2×M16—2691no—
154C2×M16C2×D26911yescharacteristic
155#127—2694no—
156#127—5381no—
157#127—5381no—
158C4×Q8—2692no—
159C4×D4—2692no—
160C4×D4—5382no—
161#117—2691no—
162#117—5381no—
163C2×C42—2691no—
164C2×C42—5381no—
165C4⋊C8—2694no—
166#102—2694no—
167#102—2694no—
168C24—21no—
169C24—5381no—
170#41—22no—
171#41C22⋊D26914yescomplemented
172C2×Q8C22×D26911yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—2692no—
178C2×D4—5384no—
179C2×D4—5384no—
180C2×D4—5381no—
181C2×D4—5382no—
182C2×D4—5382no—
183C2×D4—10768no—
184C2×D4C22×D26912yes—
185C2×D4C22×D26911yescharacteristic
186C22×C4—21no—
187C22×C4—2692no—
188C22×C4—5382no—
189C22×C4C22×D26911yescharacteristic
190SD16—24no—
191SD16—5384no—
192D8—24no—
193D8—5384no—
194M16—22no—
195M16—5382no—
196C2×C8—2692no—
197C2×C8C22×D26912yes—
198C4⋊C4—2692no—
199C4⋊C4—5382no—
200C22⋊C4—5382no—
201C22⋊C4—5381no—
202C42—2691no—
203C42—2691no—
204C42—2692no—
205C42—5382no—
206C42—5381no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—5382no—
212C23—10764no—
213C23—10762no—
214C23D4×D26911yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D26912yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—5382no—
223D4—5382no—
224D4—10768no—
225D4—10768no—
226D4—10762no—
227D4C2×C22⋊D26912yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—2692no—
231C2×C4—2692no—
232C2×C4—5384no—
233C2×C4—5382no—
234C2×C4—10764no—
235C2×C4D4×D26911yescharacteristic
236C2×C4C2×C22⋊D26912yes—
237C2×C4#1686440112yes—
238C2×C4C23×D26911yescharacteristic
239C8—22no—
240C8—5382no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—10762no—
248C22—10764no—
249C22—21524no—
250C22#2257542512yes—
251C22C2×D4×D26911yescharacteristic
252C4—22no—
253C4—2694no—
254C4—5382no—
255C4—10762no—
256C4#2257542512yes—
257C4C22×C22⋊D26911yescharacteristic
258C4C2×#1686440111yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—21522no—
263C2C2×#2257542511yescharacteristic
264C2AGL(1,ℤ/8)×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×AGL(1,ℤ/8)×C269C2
socleC22×C269C22×D4
hypercenterC2×AGL(1,ℤ/8)D269
hypocenterC269#3038
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22575478C2semidirect
2. C2×#22575449C2semidirect
3. #41×Dic269C2semidirect
4. #31770803C2semidirect
5. #31770798C2semidirect
6. #31770628C2semidirect
7. #31770611C2semidirect
8. #31770610C2semidirect
9. C2×AGL(1,ℤ/8)×C269C2semidirect
13. C2×#16864389C22semidirect
14. C2×#16864387C22semidirect
19. Q8×Dic269C22semidirect
21. D4×Dic269C22semidirect
26. #22575382C22semidirect
27. #22575375C22semidirect
28. #22575374C22semidirect
32. C2×SD16×C269C22semidirect
33. C2×D8×C269C22semidirect
60. #41×C269D4semidirect
94. Q8×C269C2×D4semidirect
100. D4×C269C2×D4semidirect
125. Dic269AGL(1,ℤ/8)semidirect
132. C269#3038semidirect
135. C2×AGL(1,ℤ/8)D269semidirect
150. C2×SD16C2×D269semidirect
152. C2×D8C2×D269semidirect
171. #41C22⋊D269semidirect
216. Q8C2×C22⋊D269semidirect
227. D4C2×C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754782
C2×#225754492
#41×Dic2692
#317708032
#317707982
#317706282
#317706112
#317706102
C2×AGL(1,ℤ/8)×C2692
#3038269