Finite Group Databasebeta

← #46821059

Subgroups — #46821059

Order 34432.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#31771141C211yesmaximal, characteristic
2C2×#22575418C211yesmaximal, characteristic, complemented
3#31770923C211yesmaximal, characteristic
4#31770911C212yesmaximal, complemented
5#31770830C212yesmaximal, complemented
6#31770760C211yesmaximal, characteristic, complemented
7#31770746C211yesmaximal, characteristic
8#31770709C212yesmaximal, complemented
9#31770704C212yesmaximal, complemented
10#31770659C211yesmaximal, characteristic, complemented
11#484×C269C211yesmaximal, characteristic, complemented
12C22×Dic538C2211yescharacteristic
13C2×#16864392C2212yes—
14C2×#16864391C2212yes—
15#22575448—21no—
16#22575444C2211yescharacteristic
17#22575442C2211yescharacteristic
18#22575437—21no—
19#22575427C2211yescharacteristic
20Q8×Dic269C2211yescharacteristic
21#22575418—21no—
22#22575418—21no—
23#22575418C2212yes—
24#22575418C2212yes—
25#22575415C2211yescharacteristic
26#22575413—21no—
27#22575411—21no—
28C2×#16864381—21no—
29C2×C4×Dic269—21no—
30C2×C4×Dic269C2211yescharacteristic
31#22575384C2212yes—
32#22575382C2212yes—
33#22575378C2212yes—
34#22575376C2212yes—
35#22575373C2212yes—
36C22⋊#14524130C2211yescharacteristic
37#22575363C2212yes—
38C22⋊Q8×C269C2211yescharacteristic
39C2×C4⋊C4×C269C2211yescharacteristic
40#107×C269C2212yes—
41#103×C269C2212yes—
42C22⋊C8×C269C2211yescharacteristic
43C2×Dic538—24no—
44C2×Dic538—21no—
45C2×Dic538C2312yes—
46C2×D1076C2311yescharacteristic
47C2×C22⋊D269—21no—
48C22×Dic269—21no—
49C22×Dic269D411yescharacteristic
50C22×Dic269D411yescharacteristic
51C2×C4×D269—21no—
52#16864392—24no—
53#16864391—24no—
54C2×#14524130C2312yes—
55C4⋊Dic269—21no—
56C4⋊Dic269—21no—
57C4⋊Dic269C2311yescharacteristic
58#16864381—22no—
59#16864381—22no—
60#16864381—21no—
61#16864379—21no—
62#16864379—21no—
63#16864379—21no—
64#16864379—21no—
65C4×Dic269—22no—
66C4×Dic269—22no—
67C4×Dic269—21no—
68C4×Dic269D411yescharacteristic
69C4×Dic269D411yescharacteristic
70C4×Dic269C2311yescharacteristic
71C4×Dic269C2311yescharacteristic
72C2×Q8×C269C2311yescharacteristic
73C22×C4×C269—21no—
74C22×C4×C269C2311yescharacteristic
75C2×C8×C269C2312yes—
76C4⋊C4×C269—21no—
77C4⋊C4×C269—21no—
78C4⋊C4×C269C2312yes—
79C4⋊C4×C269C2311yescharacteristic
80C22⋊C4×C269—21no—
81C22×D269—21no—
82Dic538—24no—
83Dic538—22no—
84Dic538#4114yes—
85D1076—22no—
86C22⋊D269—42no—
87C2×Dic269—24no—
88C2×Dic269—22no—
89C2×Dic269—22no—
90C2×Dic269—21no—
91C2×Dic269—41no—
92C2×Dic269C2×D411yescharacteristic
93C2×Dic269C2×D411yescharacteristic
94C2×Dic269C2×D411yescharacteristic
95C2×Dic269C2×D411yescharacteristic
96C4×D269—42no—
97#14524130—22no—
98C23×C269C2×D411yescharacteristic
99Q8×C269—22no—
100C2×C4×C269—22no—
101C2×C4×C269—21no—
102C2×C4×C269—21no—
103C2×C4×C269—22no—
104C2×C4×C269—42no—
105C2×C4×C269C2×D411yescharacteristic
106C2×C4×C269C2411yescharacteristic
107C8×C269—22no—
108C22×C269—21no—
109C22×C269—21no—
110C22×C269#13712yes—
111C22×C269C22×D411yescharacteristic
112C2×D269—41no—
113C2×D269—41no—
114C2×D269—41no—
115C4×C269—21no—
116C4×C269—41no—
117C4×C269—42no—
118C4×C269—41no—
119C4×C269C2×#4112yes—
120Dic269—24no—
121Dic269—21no—
122Dic269—21no—
123Dic269—21no—
124Dic269—21no—
125Dic269—41no—
126C2×C269—22no—
127C2×C269#52511yescharacteristic
128C2×C269C2×#13711yescharacteristic
129C2×C269#57511yescharacteristic
130D269—81no—
131C269#311711yescharacteristic, complemented, Sylow
132#3117—2691nomaximal, Sylow
133#529—2691no—
134C2×C4×Q8—2691no—
135#484D26911yescharacteristic, complemented
136#481—2691no—
137#479—2692no—
138#464—2692no—
139#451—2691no—
140#448—2691no—
141#441—2692no—
142#440—2692no—
143#432—2691no—
144C22×Q8—2691no—
145C2×Q16—2692no—
146C2×SD16—2692no—
147#126—5381no—
148#125—2691no—
149#124—2691no—
150#123—5381no—
151C22⋊Q8C2×D26911yescharacteristic
152#121—2691no—
153C4×Q8—2691no—
154C4×Q8—2692no—
155C4×Q8—2692no—
156C4×Q8—5381no—
157C4×Q8—5381no—
158C4×D4—2691no—
159C4×D4—5381no—
160#117—5381no—
161C2×C4⋊C4—5381no—
162C2×C4⋊C4C2×D26911yescharacteristic
163C2×C42—2691no—
164C2×C42—5381no—
165#107C2×D26912yes—
166#106—2692no—
167C4⋊C8—2692no—
168#103—2692no—
169#103—2692no—
170#103C2×D26912yes—
171#102—2692no—
172C22⋊C8—2691no—
173C22⋊C8C2×D26911yescharacteristic
174#97—2692no—
175C2×Q8—2692no—
176C2×Q8—5384no—
177C2×Q8—5381no—
178C2×Q8C22×D26911yescharacteristic
179C2×D4—2691no—
180C2×D4—5381no—
181C22×C4—21no—
182C22×C4—2691no—
183C22×C4—2691no—
184C22×C4—5381no—
185C22×C4—5381no—
186C22×C4C22×D26911yescharacteristic
187Q16—5384no—
188SD16—5384no—
189C2×C8—2692no—
190C2×C8C22×D26912yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—2691no—
194C4⋊C4—5382no—
195C4⋊C4—5382no—
196C4⋊C4—5381no—
197C4⋊C4—5381no—
198C4⋊C4—5381no—
199C4⋊C4C22×D26912yes—
200C4⋊C4C22×D26911yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—5381no—
203C22⋊C4—5381no—
204C22⋊C4—5381no—
205C22⋊C4—5381no—
206C42—2691no—
207C42—2691no—
208C42—2691no—
209C42—2691no—
210C42—5382no—
211C42—5382no—
212C42—5381no—
213C23—5381no—
214C23D4×D26911yescharacteristic
215Q8—22no—
216Q8—2694no—
217Q8—5384no—
218Q8—5382no—
219D4—5382no—
220D4—10762no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—2691no—
227C2×C4—2691no—
228C2×C4—2691no—
229C2×C4—2691no—
230C2×C4—5384no—
231C2×C4—5382no—
232C2×C4—5382no—
233C2×C4—5381no—
234C2×C4—10762no—
235C2×C4—10761no—
236C2×C4D4×D26911yescharacteristic
237C2×C4C23×D26911yescharacteristic
238C8—22no—
239C8—5382no—
240C22—21no—
241C22—21no—
242C22—10761no—
243C22—10761no—
244C22—10761no—
245C22#2257548512yes—
246C22C2×D4×D26911yescharacteristic
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—5381no—
252C4—5381no—
253C4—5381no—
254C4—5381no—
255C4—5384no—
256C4—10761no—
257C4#41×D26912yes—
258C2—22no—
259C2—21521no—
260C2#3177111311yescharacteristic
261C2C2×#2257548511yescharacteristic
262C2#3177140111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#484×C269C2
socleC22×C269C22×D4
prime radical#22575427C22
hypercenter#484D269
hypocenterC269#3117
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#22575418C2semidirect
4. #31770911C2semidirect
5. #31770830C2semidirect
6. #31770760C2semidirect
8. #31770709C2semidirect
9. #31770704C2semidirect
10. #31770659C2semidirect
11. #484×C269C2semidirect
131. C269#3117semidirect
135. #484D269semidirect

Maximal subgroups

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

subgroupindex
#317711412
C2×#225754182
#317709232
#317709112
#317708302
#317707602
#317707462
#317707092
#317707042
#317706592
#484×C2692
#3117269