Finite Group Databasebeta

← #46821057

Subgroups — #46821057

Order 34432.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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×Q8×Dic269C211yesmaximal, characteristic, complemented
3#31770938C211yesmaximal, characteristic, complemented
4#31770924C211yesmaximal, characteristic
5#31770910C212yesmaximal, complemented
6#31770829C212yesmaximal, complemented
7#31770744C211yesmaximal, characteristic
8#31770712C212yesmaximal, complemented
9#31770708C212yesmaximal, complemented
10#31770659C211yesmaximal, characteristic, complemented
11#451×C269C211yesmaximal, characteristic, complemented
12C2×#16864389C2212yes—
13#22575448—21no—
14#22575444C2211yescharacteristic
15#22575442C2211yescharacteristic
16#22575437—21no—
17#22575427C2211yescharacteristic
18Q8×Dic269—21no—
19Q8×Dic269—21no—
20Q8×Dic269C2211yescharacteristic
21Q8×Dic269C2212yes—
22Q8×Dic269C2212yes—
23#22575415C2211yescharacteristic
24#22575413—21no—
25#22575411—21no—
26C2×C4⋊Dic269—21no—
27C2×C4⋊Dic269C2211yescharacteristic
28C2×C4×Dic269—21no—
29C2×C4×Dic269C2211yescharacteristic
30#22575386C2212yes—
31#22575385C2212yes—
32#22575382C2212yes—
33#22575377C2212yes—
34#22575377C2212yes—
35#22575375C2212yes—
36C22⋊#14524130C2211yescharacteristic
37#22575363C2212yes—
38C22×Q8×C269C2211yescharacteristic
39C2×Q16×C269C2212yes—
40C22⋊Q8×C269C2211yescharacteristic
41#103×C269C2212yes—
42C22⋊C8×C269C2211yescharacteristic
43C2×D1076C2311yescharacteristic
44C2×C22⋊D269—21no—
45C22×Dic269—21no—
46C22×Dic269D411yescharacteristic
47C22×Dic269D411yescharacteristic
48C2×C4×D269—21no—
49#16864389—24no—
50C2×#14524130C2312yes—
51C4⋊Dic269—22no—
52C4⋊Dic269—22no—
53C4⋊Dic269—21no—
54C4⋊Dic269—21no—
55C4⋊Dic269—21no—
56C4⋊Dic269C2312yes—
57C4⋊Dic269C2311yescharacteristic
58#16864381—21no—
59#16864379—21no—
60#16864379—21no—
61#16864379—21no—
62#16864379—21no—
63C4×Dic269—22no—
64C4×Dic269—22no—
65C4×Dic269—21no—
66C4×Dic269D411yescharacteristic
67C4×Dic269D411yescharacteristic
68C4×Dic269C2311yescharacteristic
69C4×Dic269C2311yescharacteristic
70C2×Q8×C269—24no—
71C2×Q8×C269—21no—
72C2×Q8×C269C2311yescharacteristic
73C2×Q8×C269C2312yes—
74C22×C4×C269—21no—
75C22×C4×C269C2311yescharacteristic
76Q16×C269—24no—
77C2×C8×C269C2312yes—
78C4⋊C4×C269—21no—
79C4⋊C4×C269C2311yescharacteristic
80C22⋊C4×C269—21no—
81C22×D269—21no—
82D1076—22no—
83C22⋊D269—42no—
84C2×Dic269—22no—
85C2×Dic269—22no—
86C2×Dic269—21no—
87C2×Dic269—41no—
88C2×Dic269—42no—
89C2×Dic269C2×D411yescharacteristic
90C2×Dic269C2×D411yescharacteristic
91C2×Dic269C2×D411yescharacteristic
92C2×Dic269C2×D411yescharacteristic
93C4×D269—42no—
94#14524130—22no—
95C23×C269C2×D411yescharacteristic
96Q8×C269—24no—
97Q8×C269—22no—
98Q8×C269—22no—
99Q8×C269#4114yes—
100C2×C4×C269—24no—
101C2×C4×C269—22no—
102C2×C4×C269—22no—
103C2×C4×C269—21no—
104C2×C4×C269—21no—
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—24no—
116C4×C269—21no—
117C4×C269—41no—
118C4×C269—41no—
119C4×C269C2×#4112yes—
120Dic269—21no—
121Dic269—21no—
122Dic269—21no—
123Dic269—21no—
124Dic269—41no—
125Dic269—42no—
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#484—2691no—
136#481—2691no—
137#479—2692no—
138#464—2692no—
139#451D26911yescharacteristic, complemented
140#448—2691no—
141#441—2692no—
142#440—2692no—
143#432—2691no—
144C22×Q8C2×D26911yescharacteristic
145C2×Q16C2×D26912yes—
146C2×SD16—2692no—
147#126—5381no—
148#125—2691no—
149#124—2691no—
150#123—5381no—
151C22⋊Q8C2×D26911yescharacteristic
152#121—2691no—
153C4×Q8—2692no—
154C4×Q8—2692no—
155C4×Q8—2691no—
156C4×Q8—5381no—
157C4×Q8—5381no—
158C4×D4—2691no—
159C4×D4—5381no—
160#117—5381no—
161C2×C4⋊C4—2691no—
162C2×C4⋊C4—5381no—
163C2×C42—2691no—
164C2×C42—5381no—
165#107—2692no—
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—24no—
176C2×Q8—21no—
177C2×Q8C22×D26911yescharacteristic
178C2×Q8C22×D26912yes—
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—24no—
188SD16—5384no—
189C2×C8—2692no—
190C2×C8C22×D26912yes—
191C4⋊C4—21no—
192C4⋊C4—2692no—
193C4⋊C4—2691no—
194C4⋊C4—5382no—
195C4⋊C4—5382no—
196C4⋊C4—5381no—
197C4⋊C4—5381no—
198C4⋊C4—5381no—
199C4⋊C4—5381no—
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—24no—
216Q8—22no—
217Q8—22no—
218Q8#1686440114yes—
219D4—5382no—
220D4—10762no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—2691no—
227C2×C4—2691no—
228C2×C4—2691no—
229C2×C4—2691no—
230C2×C4—5382no—
231C2×C4—5382no—
232C2×C4—5381no—
233C2×C4—10762no—
234C2×C4—10761no—
235C2×C4—10762no—
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—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—5381no—
252C4—5381no—
253C4—5381no—
254C4—5381no—
255C4—10761no—
256C4—10762no—
257C4C2×#1686440112yes—
258C2—22no—
259C2—21521no—
260C2#3177111111yescharacteristic
261C2C2×#2257548511yescharacteristic
262C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#451×C269C2
socleC22×C269C22×D4
prime radical#22575427C22
hypercenter#451D269
hypocenterC269#3117
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic269C2semidirect
3. #31770938C2semidirect
5. #31770910C2semidirect
6. #31770829C2semidirect
8. #31770712C2semidirect
9. #31770708C2semidirect
10. #31770659C2semidirect
11. #451×C269C2semidirect
131. C269#3117semidirect
139. #451D269semidirect

Maximal subgroups

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

subgroupindex
#317711412
C2×Q8×Dic2692
#317709382
#317709242
#317709102
#317708292
#317707442
#317707122
#317707082
#317706592
#451×C2692
#3117269