Finite Group Databasebeta

← #46809231

Subgroups — #46809231

Order 34432.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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×#22575436C211yesmaximal, characteristic
2#31770569C211yesmaximal, characteristic, complemented
3#31770330C212yesmaximal, complemented
4#31770328C212yesmaximal, complemented
5C2×C22⋊C8×C269C211yesmaximal, characteristic, complemented
6C22×C22⋊D269C411yescharacteristic
7C2×#16864386—21no—
8#22575436—21no—
9#22575436—21no—
10#22575436—21no—
11#22575436—21no—
12C2×#16864381C411yescharacteristic
13C2×#16864381C2211yescharacteristic
14C2×C22⋊Dic269C411yescharacteristic
15C2×#16864379C411yescharacteristic
16C2×#16864379C2211yescharacteristic
17C22⋊#14524130—21no—
18C22⋊#14524130C2212yes—
19C23×C4×C269C2211yescharacteristic
20C22×C8×C269—21no—
21C22⋊C8×C269—21no—
22C22⋊C8×C269C2212yes—
23C23×D269C2×C411yescharacteristic
24C2×C22⋊D269—21no—
25C2×C22⋊D269—21no—
26C2×C22⋊D269—42no—
27C22×Dic269C2×C411yescharacteristic
28C22×Dic269C2×C411yescharacteristic
29C22×Dic269C2×C411yescharacteristic
30#16864386—42no—
31C2×#14524130—22no—
32#16864381—21no—
33#16864381—21no—
34#16864381—21no—
35#16864381—21no—
36C22⋊Dic269—21no—
37C22⋊Dic269—21no—
38#16864379—21no—
39#16864379—21no—
40#16864379—21no—
41#16864379—21no—
42C24×C269C2×C411yescharacteristic
43C22×C4×C269—42no—
44C22×C4×C269C2×C411yescharacteristic
45C22×C4×C269D411yescharacteristic
46C22×C4×C269D411yescharacteristic
47C22×C4×C269D411yescharacteristic
48C22×C4×C269D411yescharacteristic
49C22×C4×C269C2311yescharacteristic
50C2×C8×C269—22no—
51C2×C8×C269—42no—
52C22×D269—21no—
53C22×D269—22no—
54C22×D269—41no—
55C22×D269—41no—
56C22⋊D269—42no—
57C22⋊D269—42no—
58C2×Dic269—21no—
59C2×Dic269—21no—
60C2×Dic269—21no—
61C2×Dic269—22no—
62C2×Dic269—41no—
63C2×Dic269—41no—
64#14524130—42no—
65C23×C269—22no—
66C23×C269—22no—
67C23×C269—41no—
68C23×C269C22⋊C411yescharacteristic
69C23×C269C22⋊C411yescharacteristic
70C23×C269C22×C411yescharacteristic
71C2×C4×C269—22no—
72C2×C4×C269—22no—
73C2×C4×C269—41no—
74C2×C4×C269—41no—
75C2×C4×C269—41no—
76C2×C4×C269—41no—
77C2×C4×C269C22⋊C411yescharacteristic
78C2×C4×C269C22⋊C411yescharacteristic
79C2×C4×C269C2×D411yescharacteristic
80C2×C4×C269C2×D411yescharacteristic
81C8×C269—42no—
82C22×C269—21no—
83C22×C269—21no—
84C22×C269—21no—
85C22×C269—21no—
86C22×C269—21no—
87C22×C269—21no—
88C22×C269—21no—
89C22×C269—44no—
90C22×C269#9912yes—
91C22×C269C2×C22⋊C411yescharacteristic
92C22×C269#13112yes—
93C2×D269—41no—
94C2×D269—41no—
95C2×D269—41no—
96C2×D269—82no—
97C4×C269—21no—
98C4×C269—21no—
99C4×C269—21no—
100C4×C269—21no—
101Dic269—41no—
102Dic269—41no—
103Dic269—41no—
104C2×C269—22no—
105C2×C269—22no—
106C2×C269—41no—
107C2×C269#40811yescharacteristic
108C2×C269C2×#9911yescharacteristic
109C2×C269#41311yescharacteristic
110D269—81no—
111C269#146811yescharacteristic, complemented, Sylow
112#1468—2691nomaximal, Sylow
113C2×#123—2691no—
114#407—2691no—
115C2×C22⋊C8D26911yescharacteristic, complemented
116#324—2692no—
117#323—2692no—
118C22×D4—2691no—
119C23×C4C2×D26911yescharacteristic
120C2×M16—5381no—
121C22×C8—21no—
122#123—5381no—
123#123—5381no—
124#123—5381no—
125#123—5381no—
126C2×C4⋊C4—2691no—
127C2×C4⋊C4—2691no—
128C2×C22⋊C4—2691no—
129C2×C22⋊C4—2691no—
130C2×C22⋊C4—2691no—
131C22⋊C8—21no—
132C22⋊C8—2692no—
133C22⋊C8—5381no—
134C22⋊C8C2×D26912yes—
135C24—2691no—
136C24C4×D26911yescharacteristic
137C2×D4—5381no—
138C2×D4—5381no—
139C2×D4—10762no—
140C22×C4—42no—
141C22×C4—2691no—
142C22×C4—2691no—
143C22×C4—2691no—
144C22×C4C4×D26911yescharacteristic
145C22×C4C22⋊D26911yescharacteristic
146C22×C4C22⋊D26911yescharacteristic
147C22×C4D107611yescharacteristic
148C22×C4D107611yescharacteristic
149C22×C4C22×D26911yescharacteristic
150M16—10762no—
151C2×C8—22no—
152C2×C8—42no—
153C2×C8—5382no—
154C4⋊C4—5381no—
155C4⋊C4—5381no—
156C4⋊C4—5381no—
157C4⋊C4—5381no—
158C22⋊C4—5381no—
159C22⋊C4—5381no—
160C22⋊C4—5381no—
161C22⋊C4—5381no—
162C22⋊C4—5381no—
163C22⋊C4—5381no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—5381no—
168C23—5382no—
169C23—10761no—
170C23—10761no—
171C23#1686437911yescharacteristic
172C23#1686437911yescharacteristic
173C23C2×C4×D26911yescharacteristic
174D4—10762no—
175D4—10762no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—5381no—
183C2×C4—5381no—
184C2×C4—5381no—
185C2×C4—5382no—
186C2×C4—10761no—
187C2×C4—10761no—
188C2×C4#1686437911yescharacteristic
189C2×C4#1686437911yescharacteristic
190C2×C4C2×C22⋊D26911yescharacteristic
191C2×C4C2×D107611yescharacteristic
192C8—42no—
193C8—10762no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—10761no—
203C22—10761no—
204C22—10761no—
205C22—21522no—
206C22#2257536712yes—
207C22C2×#1686437911yescharacteristic
208C22#2257545912yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—10761no—
214C4—10761no—
215C4—10761no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—21521no—
220C2#3177057111yescharacteristic
221C2C2×#2257536711yescharacteristic
222C2#3177058711yescharacteristic
subgroupG/H
centerC22C2×#16864379
commutatorC23×C269C22×C4
FrattiniC22×C4C22×D269
FittingC2×C22⋊C8×C269C2
socleC22×C269C2×C22⋊C4
prime radicalC22×C22⋊D269C4
cosocleC22×C4×C269C23
hypercenterC2×C22⋊C8D269
hypocenterC269#1468
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #31770569C2semidirect
3. #31770330C2semidirect
4. #31770328C2semidirect
5. C2×C22⋊C8×C269C2semidirect
111. C269#1468semidirect
115. C2×C22⋊C8D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754362
#317705692
#317703302
#317703282
C2×C22⋊C8×C2692
#1468269