Finite Group Databasebeta

← #46820364

Subgroups — #46820364

Order 34432.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22575481C211yesmaximal, characteristic, complemented
2C2×#22575479C211yesmaximal, characteristic, complemented
3C2×#22575449C211yesmaximal, characteristic, complemented
4#31771073C211yesmaximal, characteristic, complemented
5#31770804C214yesmaximal, complemented
6#31770799C214yesmaximal, complemented
7#31770628C211yesmaximal, characteristic, complemented
8#31770609C211yesmaximal, characteristic, complemented
9#418×C269C211yesmaximal, characteristic, complemented
10C2×#16864402C2211yescharacteristic
11C22×D1076C2211yescharacteristic
12C22×C22⋊D269—22no—
13#22575481—24no—
14#22575479—24no—
15C2×#16864391C2212yescomplemented
16C2×#16864390C2212yescomplemented
17C2×D2152C2212yescomplemented
18C2×#16864387C2212yescomplemented
19C2×#16864386C2211yescharacteristic
20#22575449—21no—
21#22575449—21no—
22#22575449C2214yes—
23#22575418C2212yescomplemented
24#22575415C2212yescomplemented
25#22575413—22no—
26#22575410—21no—
27C2×C4×Dic269C2211yescharacteristic
28C42×D269—21no—
29#22575382C2214yescomplemented
30#22575373C2214yescomplemented
31C22×D4×C269C2211yescharacteristic
32C2×M16×C269C2211yescharacteristic
33#117×C269C2211yescharacteristic
34#102×C269C2214yescomplemented
35C23×D269—21no—
36#16864402—22no—
37#16864402D414yescomplemented
38C2×Dic538C2311yescharacteristic
39C2×D1076—22no—
40C2×D1076—21no—
41C2×D1076—22no—
42C2×D1076C2312yes—
43C2×D1076C2311yescharacteristic
44C2×C22⋊D269—24no—
45C2×C22⋊D269—24no—
46C2×C22⋊D269—21no—
47C2×C22⋊D269—48no—
48C22×Dic269D412yes—
49C2×C4×D269—22no—
50C2×C4×D269—21no—
51#16864391—24no—
52#16864390—24no—
53D2152—24no—
54#16864387—24no—
55#16864386—22no—
56C2×#14524130C2312yes—
57#16864381—22no—
58#16864379—22no—
59C4×Dic269—22no—
60C4×Dic269D411yescharacteristic
61C4×Dic269D411yescharacteristic
62C4×Dic269C2312yes—
63C24×C269—21no—
64C2×D4×C269—22no—
65C2×D4×C269—21no—
66C2×D4×C269—22no—
67C2×D4×C269C2312yes—
68C22×C4×C269C2311yescharacteristic
69M16×C269—22no—
70C2×C8×C269C2312yes—
71C4⋊C4×C269C2312yes—
72C22⋊C4×C269—21no—
73C42×C269—21no—
74C22×D269—21no—
75C22×D269—22no—
76C22×D269—44no—
77C22×D269—42no—
78Dic538—21no—
79Dic538C2×D412yescomplemented
80D1076—22no—
81D1076—21no—
82D1076—22no—
83D1076—42no—
84D1076C2×D412yescomplemented
85C22⋊D269—24no—
86C22⋊D269—48no—
87C22⋊D269—48no—
88C2×Dic269—21no—
89C2×Dic269—24no—
90C2×Dic269C2×D412yes—
91C2×Dic269C2×D412yes—
92C4×D269—24no—
93C4×D269—44no—
94#14524130—22no—
95C23×C269—22no—
96C23×C269—44no—
97C23×C269—42no—
98C23×C269C2×D411yescharacteristic
99D4×C269—22no—
100D4×C269—22no—
101D4×C269—42no—
102C2×C4×C269—22no—
103C2×C4×C269C2×D411yescharacteristic
104C2×C4×C269C2×D412yes—
105C2×C4×C269#4112yes—
106C2×C4×C269C2411yescharacteristic
107C8×C269—22no—
108C22×C269—22no—
109C22×C269—42no—
110C22×C269—44no—
111C22×C269—84no—
112C22×C269#12112yes—
113C22×C269C22×D411yescharacteristic
114C2×D269—22no—
115C2×D269—42no—
116C2×D269—42no—
117C2×D269—44no—
118C2×D269—84no—
119C4×C269—42no—
120C4×C269#12112yes—
121C4×C269C22×D411yescharacteristic
122C4×C269C2×#4111yescharacteristic
123Dic269—22no—
124Dic269—22no—
125Dic269AGL(1,ℤ/8)14yescomplemented
126C2×C269—22no—
127C2×C269—82no—
128C2×C269C2×#12111yescharacteristic
129C2×C269C2×AGL(1,ℤ/8)12yes—
130D269—42no—
131D269—82no—
132C269#303811yescharacteristic, complemented, Sylow
133#3038—2691nomaximal, Sylow
134C2×AGL(1,ℤ/8)—2691no—
135C2×AGL(1,ℤ/8)—2691no—
136C2×#127—2691no—
137C4×#41—2691no—
138#460—2694no—
139#459—2694no—
140#423—2691no—
141#418—2691no—
142#418D26911yescharacteristic, complemented
143C2×#41—2691no—
144C22×D4—2691no—
145C22×D4—5382no—
146C22×D4C2×D26911yescharacteristic
147AGL(1,ℤ/8)—5384no—
148AGL(1,ℤ/8)—5384no—
149C2×SD16—2692no—
150C2×SD16—2692no—
151C2×D8—2692no—
152C2×D8—2692no—
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—5381no—
162#117C2×D26911yescharacteristic
163C2×C42—2691no—
164C2×C42—5381no—
165C4⋊C8—2694no—
166#102—2694no—
167#102C2×D26914yescomplemented
168C24—21no—
169C24—5381no—
170#41—2694no—
171#41—5382no—
172C2×Q8—2691no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—2691no—
177C2×D4—2692no—
178C2×D4—5384no—
179C2×D4—5384no—
180C2×D4—5381no—
181C2×D4—5382no—
182C2×D4—5381no—
183C2×D4—5382no—
184C2×D4—10768no—
185C2×D4C22×D26912yes—
186C22×C4—2692no—
187C22×C4—5382no—
188C22×C4—5381no—
189C22×C4C22×D26911yescharacteristic
190SD16—5384no—
191SD16—5384no—
192D8—5384no—
193D8—5384no—
194M16—22no—
195M16—5382no—
196C2×C8—2692no—
197C2×C8C22×D26912yes—
198C4⋊C4—5382no—
199C4⋊C4C22×D26912yes—
200C22⋊C4—21no—
201C22⋊C4—5382no—
202C42—21no—
203C42—2691no—
204C42—2691no—
205C42—2692no—
206C42—5382no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—5381no—
211C23—5382no—
212C23—10764no—
213C23—10762no—
214C23D4×D26911yescharacteristic
215Q8—2692no—
216Q8—5381no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—2692no—
221D4—5384no—
222D4—5382no—
223D4—5381no—
224D4—5382no—
225D4—10768no—
226D4—10768no—
227D4—10762no—
228C2×C4—22no—
229C2×C4—2692no—
230C2×C4—2692no—
231C2×C4—5384no—
232C2×C4—5381no—
233C2×C4—5384no—
234C2×C4—10764no—
235C2×C4D4×D26911yescharacteristic
236C2×C4D4×D26912yes—
237C2×C4#1686440212yes—
238C2×C4C23×D26911yescharacteristic
239C8—22no—
240C8—5382no—
241C22—22no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—5382no—
246C22—10762no—
247C22—10762no—
248C22—10764no—
249C22—21524no—
250C22#2257542612yes—
251C22C2×D4×D26911yescharacteristic
252C4—42no—
253C4—2694no—
254C4—5382no—
255C4—5382no—
256C4#2257542612yes—
257C4C2×D4×D26911yescharacteristic
258C4C2×#1686440211yescharacteristic
259C2—22no—
260C2—82no—
261C2—10762no—
262C2—21522no—
263C2C2×#2257542611yescharacteristic
264C2AGL(1,ℤ/8)×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#418×C269C2
socleC22×C269C22×D4
hypercenter#418D269
hypocenterC269#3038
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#22575481C2semidirect
2. C2×#22575479C2semidirect
3. C2×#22575449C2semidirect
4. #31771073C2semidirect
5. #31770804C2semidirect
6. #31770799C2semidirect
7. #31770628C2semidirect
8. #31770609C2semidirect
9. #418×C269C2semidirect
15. C2×#16864391C22semidirect
16. C2×#16864390C22semidirect
17. C2×D2152C22semidirect
18. C2×#16864387C22semidirect
23. #22575418C22semidirect
24. #22575415C22semidirect
29. #22575382C22semidirect
30. #22575373C22semidirect
34. #102×C269C22semidirect
37. #16864402D4semidirect
79. Dic538C2×D4semidirect
84. D1076C2×D4semidirect
125. Dic269AGL(1,ℤ/8)semidirect
132. C269#3038semidirect
142. #418D269semidirect
167. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×#225754792
C2×#225754492
#317710732
#317708042
#317707992
#317706282
#317706092
#418×C2692
#3038269