Finite Group Databasebeta

← #30604213

Subgroups — #30604213

Order 15616.

334 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 70 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#21462192C212yesmaximal, complemented
2#21460753C212yesmaximal, complemented
3#21460751C212yesmaximal, complemented
4#21454523C211yesmaximal, characteristic
5#16840685—22no—
6#16840658C2214yes—
7#16840654—22no—
8C2×#14519784C411yescharacteristic
9C22×#11758194C411yescharacteristic
10#16839840—22no—
11#16839834—22no—
12#16839789C411yescharacteristic
13#16839729C2211yescharacteristic
14#16839726C2211yescharacteristic
15#16839718C2211yescharacteristic
16C61×#525C411yescharacteristic
17#14519851—24no—
18#14519784—42no—
19C2×#11758194—21no—
20C2×#11758194—21no—
21C2×#11758194—21no—
22C2×#11758194—42no—
23C2×#11758194C2×C411yescharacteristic
24C2×#11758194C2311yescharacteristic
25C22×#92418—21no—
26C22×#92418C2×C411yescharacteristic
27C22×#92418C2×C411yescharacteristic
28C4⋊#92418—22no—
29C22⋊#92418—22no—
30C22⋊#92418—22no—
31#14519688D412yes—
32C4×#92418D412yes—
33C2×#41×C61C2×C411yescharacteristic
34C23×C4×C61C2×C411yescharacteristic
35#123×C61—21no—
36C22⋊Q8×C61—21no—
37#121×C61—21no—
38#120×C61—21no—
39C4×D4×C61—22no—
40#117×C61C2×C411yescharacteristic
41C24×C61M1611yescharacteristic
42#41×C61—41no—
43#41×C61—41no—
44C2×Q8×C61M1611yescharacteristic
45C2×D4×C61—21no—
46C2×D4×C61—22no—
47C2×D4×C61M1611yescharacteristic
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61—41no—
52C22×C4×C61—41no—
53C22×C4×C61M1611yescharacteristic
54C22×C4×C61C22×C411yescharacteristic
55C4⋊C4×C61—22no—
56C4⋊C4×C61—21no—
57C22⋊C4×C61—22no—
58C22⋊C4×C61—22no—
59C22⋊C4×C61—21no—
60C42×C61C22⋊C412yes—
61#11758194—41no—
62#11758194—42no—
63#11758194—42no—
64#11758194—42no—
65#11758194—41no—
66C2×#92418—22no—
67C2×#92418—21no—
68C2×#92418—21no—
69C2×#92418—42no—
70C2×#92418—41no—
71C2×#92418C22⋊C411yescharacteristic
72C2×#92418C22⋊C411yescharacteristic
73C2×#92418C2×D412yes—
74#11758187—44no—
75C23×C61—21no—
76C23×C61—21no—
77C23×C61—41no—
78C23×C61—81no—
79C23×C61C2×M1611yescharacteristic
80Q8×C61—41no—
81D4×C61—41no—
82D4×C61—42no—
83D4×C61—82no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—22no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—42no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61—41no—
95C2×C4×C61C2×C22⋊C411yescharacteristic
96C2×C4×C61C2×M1611yescharacteristic
97#92418—22no—
98#92418—21no—
99#92418—41no—
100#92418—41no—
101#92418#10012yes—
102#11538—611nomaximal, Sylow
103C22×C61—21no—
104C22×C61—41no—
105C22×C61—41no—
106C22×C61—42no—
107C22×C61—41no—
108C22×C61—41no—
109C22×C61—41no—
110C22×C61—82no—
111C22×C61#40711yescharacteristic
112C4×C61—21no—
113C4×C61—42no—
114C4×C61—41no—
115C4×C61—41no—
116C4×C61#41011yescharacteristic
117C4×C61C2×#10011yescharacteristic
118#2930—611no—
119#2166—612no—
120#1321—612no—
121#1320—612no—
122C2×C61—21no—
123C2×C61—41no—
124C2×C61—81no—
125C2×C61—81no—
126C2×C61#146611yescharacteristic
127C2×#131—611no—
128C22×M16—611no—
129#525#692311yescharacteristic
130#436—1222no—
131#435—1222no—
132#424—611no—
133#408—611no—
134#407—611no—
135#405—611no—
136#364—1222no—
137#349—614no—
138#347—1222no—
139C61#1153811yescharacteristic, complemented, Sylow
140C2×#41C2×#692311yescharacteristic
141C23×C4C2×#692311yescharacteristic
142#131—2442no—
143C2×M16—611no—
144C2×M16—611no—
145C2×M16—1221no—
146C2×M16—1221no—
147C2×M16—1221no—
148C2×M16—2442no—
149C22×C8—611no—
150C22×C8—611no—
151C22×C8—1221no—
152#123—21no—
153C22⋊Q8—21no—
154#121—21no—
155#120—21no—
156C4×D4—22no—
157#117C2×#692311yescharacteristic
158M32—1224no—
159C4⋊C8—1222no—
160C22⋊C8—1222no—
161C22⋊C8—1222no—
162#97—612no—
163C4×C8—612no—
164C24#1175821711yescharacteristic
165#41—41no—
166#41—41no—
167C2×Q8#1175821711yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4#1175821711yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4#1175821711yescharacteristic
177C22×C4C22×#692311yescharacteristic
178M16—2442no—
179M16—2442no—
180M16—2441no—
181M16—2442no—
182M16—2441no—
183C2×C8—611no—
184C2×C8—612no—
185C2×C8—611no—
186C2×C8—1222no—
187C2×C8—1221no—
188C2×C8—1221no—
189C2×C8—2442no—
190C2×C8—2441no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—21no—
196C42C22⋊#692312yes—
197C16—2444no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—81no—
202C23C2×#1175821711yescharacteristic
203Q8—41no—
204D4—41no—
205D4—42no—
206D4—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4C2×C22⋊#692311yescharacteristic
219C2×C4C2×#1175821711yescharacteristic
220C8—612no—
221C8—1222no—
222C8—1221no—
223C8—2441no—
224C8—2441no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—82no—
233C22#1684073211yescharacteristic
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4#1684074211yescharacteristic
239C4D61⋊#10011yescharacteristic
240C2—21no—
241C2—41no—
242C2—81no—
243C2—81no—
244C2#2146107911yescharacteristic
subgroupG/H
centerC4#16840742
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC61×#525C4
socleC2×C61#1466
cosocleC2×#11758194C23
hypercenter#525#6923
hypocenterC61#11538
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ #117 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #21462192C2semidirect
2. #21460753C2semidirect
3. #21460751C2semidirect
139. C61#11538semidirect

Maximal subgroups

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

subgroupindex
#214621922
#214607532
#214607512
#214545232
#1153861