Finite Group Databasebeta

← #342×D269 #46810613

Subgroups — #342×D269 #46810613

Order 34432.

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

normal subgroups only

This group has 680 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×C22⋊C4×D269C211yesmaximal, characteristic
2C2×C22⋊C4×D269C212yesmaximal
3#31770372C211yesmaximal, characteristic, complemented
4#31770371C212yesmaximal, complemented
5#342×C269C211yesmaximal, characteristic, complemented
6C24×D269C2211yescharacteristic
7C22×C4×D269—22no—
8C22×C4×D269C412yescomplemented
9C2×C22⋊Dic269C2211yescharacteristic
10C2×C22⋊Dic269C2212yes—
11C2×#16864379—22no—
12C2×#16864379C412yescomplemented
13C22⋊C4×D269—22no—
14C22⋊C4×D269—22no—
15C22⋊C4×D269—24no—
16C22⋊C4×D269C418yescomplemented
17C2×C22⋊C4×C269C2211yescharacteristic
18C2×C22⋊C4×C269C2212yes—
19C23×D269—44no—
20C23×D269C2×C411yescharacteristic
21C23×D269C2×C414yes—
22C23×D269C2×C411yescharacteristic
23C23×D269D412yes—
24C23×D269D412yes—
25C23×D269D412yes—
26C23×D269Q812yes—
27C22×Dic269—22no—
28C22×Dic269C2×C412yescomplemented
29C2×C4×D269—22no—
30C2×C4×D269—22no—
31C2×C4×D269—28no—
32C2×C4×D269—48no—
33C2×C4×D269—44no—
34C22⋊Dic269—22no—
35C22⋊Dic269—22no—
36C22⋊Dic269C2×C414yescomplemented
37#16864379—24no—
38#16864379—28no—
39C24×C269C2311yescharacteristic
40C22×C4×C269—22no—
41C22×C4×C269C2×C412yescomplemented
42C22⋊C4×C269—22no—
43C22⋊C4×C269—22no—
44C22⋊C4×C269C2×C414yescomplemented
45C22×D269—24no—
46C22×D269—28no—
47C22×D269—48no—
48C22×D269—44no—
49C22×D269—44no—
50C22×D269—42no—
51C22×D269—44no—
52C22×D269C4212yes—
53C22×D269C4212yes—
54C22×D269C22⋊C412yes—
55C22×D269C22⋊C414yes—
56C22×D269C22⋊C411yescharacteristic
57C22×D269C22⋊C414yes—
58C22×D269C22⋊C411yescharacteristic
59C22×D269C4⋊C412yes—
60C22×D269C4⋊C414yes—
61C22×D269C4⋊C414yes—
62C22×D269C4⋊C412yes—
63C2×Dic269—22no—
64C2×Dic269—24no—
65C2×Dic269—44no—
66C2×Dic269—42no—
67C4×D269—416no—
68C4×D269—44no—
69C4×D269—44no—
70C23×C269—42no—
71C23×C269C22×C411yescharacteristic
72C23×C269C22×C412yes—
73C23×C269C2×D412yes—
74C23×C269C2×D411yescharacteristic
75C23×C269C2×Q811yescharacteristic
76C2×C4×C269—22no—
77C2×C4×C269—24no—
78C2×C4×C269—44no—
79C2×C4×C269—42no—
80C22×C269—22no—
81C22×C269—24no—
82C22×C269—42no—
83C22×C269—41no—
84C22×C269—41no—
85C22×C269C2×C4211yescharacteristic
86C22×C269C2×C22⋊C412yes—
87C22×C269C2×C22⋊C411yescharacteristic
88C22×C269C2×C4⋊C411yescharacteristic
89C22×C269C2×C4⋊C412yes—
90C2×D269—22no—
91C2×D269—24no—
92C2×D269—28no—
93C2×D269—24no—
94C2×D269—24no—
95C2×D269—28no—
96C2×D269—42no—
97C2×D269—44no—
98C2×D269—44no—
99C2×D269—42no—
100C2×D269#9512yes—
101C2×D269#9514yes—
102C2×D269#9512yes—
103C2×D269#9914yes—
104C4×C269—44no—
105C4×C269—42no—
106Dic269—44no—
107Dic269—42no—
108C2×C269—22no—
109C2×C269—21no—
110C2×C269—21no—
111C2×C269—22no—
112C2×C269C2×#9511yescharacteristic
113C2×C269C2×#9912yes—
114D269—22no—
115D269—24no—
116D269#34214yesdirect product factor
117C269C2×#34211yescharacteristic, complemented, Sylow
118C2×#342—2691nomaximal, Sylow
119C22×C22⋊C4—2692no—
120C22×C22⋊C4—2691no—
121#342—2692no—
122#342—2691no—
123#342D26911yescharacteristic, direct product factor
124C25—2691no—
125C23×C4—2692no—
126C23×C4—5382no—
127C2×C22⋊C4—2692no—
128C2×C22⋊C4—2698no—
129C2×C22⋊C4—2691no—
130C2×C22⋊C4—2692no—
131C2×C22⋊C4—5384no—
132C2×C22⋊C4—5382no—
133C2×C22⋊C4—5382no—
134C2×C22⋊C4—5382no—
135C2×C22⋊C4C2×D26911yescharacteristic
136C2×C22⋊C4C2×D26912yes—
137C24—2691no—
138C24—2692no—
139C24—2694no—
140C24—2691no—
141C24—2692no—
142C24—2692no—
143C24—2692no—
144C24—10764no—
145C24C22×D26911yescharacteristic
146C22×C4—22no—
147C22×C4—2692no—
148C22×C4—5388no—
149C22×C4—5382no—
150C22×C4—5382no—
151C22×C4—5382no—
152C22×C4—10768no—
153C22×C4—10764no—
154C22×C4C4×D26912yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—2694no—
158C22⋊C4—5388no—
159C22⋊C4—5384no—
160C22⋊C4—5382no—
161C22⋊C4—5382no—
162C22⋊C4C4×D26914yescomplemented
163C23—42no—
164C23—2691no—
165C23—2692no—
166C23—2694no—
167C23—2692no—
168C23—2691no—
169C23—2692no—
170C23—2694no—
171C23—2692no—
172C23—2694no—
173C23—2694no—
174C23—2692no—
175C23—5384no—
176C23—5388no—
177C23—10762no—
178C23—10764no—
179C23—10768no—
180C23—10764no—
181C23—10764no—
182C23C2×C4×D26911yescharacteristic
183C23C2×C4×D26912yes—
184C23D4×D26912yes—
185C23D4×D26911yescharacteristic
186C23Q8×D26911yescharacteristic
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—5384no—
192C2×C4—5382no—
193C2×C4—10764no—
194C2×C4—107616no—
195C2×C4—10764no—
196C2×C4—10764no—
197C2×C4—10762no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—2692no—
204C22—2694no—
205C22—2692no—
206C22—2694no—
207C22—5388no—
208C22—5384no—
209C22—5384no—
210C22—5388no—
211C22—5382no—
212C22—5384no—
213C22—10764no—
214C22—10762no—
215C22—10762no—
216C22—10764no—
217C22C42×D26911yescharacteristic
218C22C22⋊C4×D26912yes—
219C22C22⋊C4×D26911yescharacteristic
220C22C4⋊C4×D26911yescharacteristic
221C22C4⋊C4×D26912yes—
222C4—44no—
223C4—42no—
224C4—10764no—
225C4—10762no—
226C2—22no—
227C2—21no—
228C2—21no—
229C2—22no—
230C2—2694no—
231C2—5384no—
232C2—5382no—
233C2#95×D26911yescharacteristic
234C2#99×D26912yes—
subgroupG/H
centerC22C22⋊C4×D269
commutatorC22×C269C2×C42
FrattiniC24C22×D269
Fitting#342×C269C2
socleC22×C269C2×C22⋊C4
prime radicalC24×D269C22
cosocleC24×C269C23
hypercenter#342D269
hypocenterC269C2×#342
LCSG ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #31770372C2semidirect
4. #31770371C2semidirect
5. #342×C269C2semidirect
8. C22×C4×D269C4semidirect
12. C2×#16864379C4semidirect
16. C22⋊C4×D269C4semidirect
28. C22×Dic269C2×C4semidirect
36. C22⋊Dic269C2×C4semidirect
41. C22×C4×C269C2×C4semidirect
44. C22⋊C4×C269C2×C4semidirect
116. D269#342direct (N × H)
117. C269C2×#342semidirect
123. #342D269direct (N × H)
154. C22×C4C4×D269semidirect
162. C22⋊C4C4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D2692
#317703722
#317703712
#342×C2692
C2×#342269