Finite Group Databasebeta

← #30608362

Subgroups — #30608362

Order 15616.

406 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 56 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#21462093C212yesmaximal
2#21460953C212yesmaximal, complemented
3#21460952C212yesmaximal, complemented
4#21456277C211yesmaximal, characteristic
5#16840743—22no—
6#16840668—22no—
7#16840667C2214yes—
8C2×#14519809—22no—
9#16840313C2211yescharacteristic
10C4⋊Q8×D61C411yescharacteristic, complemented
11#16840155—22no—
12#16840141C411yescharacteristic, complemented
13#16840136C2211yescharacteristic
14#16840135C2211yescharacteristic
15C4×#11758189C411yescharacteristic, complemented
16C61×#494C411yescharacteristic, complemented
17#14519856—42no—
18#14519836—24no—
19#14519836—42no—
20C2×#11758226D412yes—
21C2×Q8×D61—21no—
22#14519809—44no—
23#14519809—44no—
24C2×#11758202—22no—
25C2×#11758190—22no—
26C2×#11758201—22no—
27C2×#11758189—21no—
28#14519777C2×C411yescharacteristic
29#14519776—41no—
30#14519775—21no—
31#14519774C2311yescharacteristic
32#14519766D412yes—
33#14519752—41no—
34#14519739—41no—
35C4⋊C4×D61—41no—
36C42×D61C2×C411yescharacteristic
37#14519687—21no—
38C4×#92418C2×C411yescharacteristic
39C2×Q16×C61—22no—
40C4⋊Q8×C61C2×C411yescharacteristic
41C4⋊Q8×C61C2×C411yescharacteristic
42C4×C8×C61C2×C411yescharacteristic
43C2×Q8×C61—21no—
44C2×Q8×C61C22⋊C412yes—
45#11758226—42no—
46#11758226—42no—
47Q8×D61—42no—
48Q8×D61—42no—
49C2×D244—22no—
50C2×D244—41no—
51C2×D244C2×D412yes—
52C2×C4×D61—21no—
53C2×C4×D61—22no—
54C2×C4×D61—41no—
55C2×C4×D61C22⋊C411yescharacteristic
56C2×Dic122—21no—
57C2×Dic122—41no—
58C22⋊#6923—44no—
59C4⋊#6923—82no—
60C4×#6923—82no—
61Q16×C61—44no—
62C2×C8×C61—21no—
63C4⋊C4×C61—41no—
64C4⋊C4×C61—41no—
65C42×C61C22×C411yescharacteristic
66#11758202—44no—
67#11758201—44no—
68#11758198—42no—
69C4⋊Dic61—41no—
70#11758196—42no—
71C4×Dic61—21no—
72C4×Dic61C22⋊C411yescharacteristic
73C2×#92418—21no—
74#11758190—44no—
75#11758189—24no—
76C22×D61—22no—
77C22×D61#9911yescharacteristic
78C2×#6923—84no—
79Q8×C61—42no—
80Q8×C61—42no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—41no—
84C2×C4×C61C2×C22⋊C411yescharacteristic
85C2×Dic61—21no—
86C2×Dic61—41no—
87C2×Dic61#9911yescharacteristic
88D244—42no—
89D244—82no—
90D244—82no—
91C4×D61—22no—
92C4×D61—42no—
93C4×D61—42no—
94C4×D61—42no—
95C4×D61—81no—
96C4×D61—81no—
97Dic122—42no—
98Dic122—82no—
99C8×C61—22no—
100#92418—22no—
101#13586—611nomaximal, Sylow
102C22×C61C2×#9911yescharacteristic
103C2×D61—21no—
104C2×D61—41no—
105C2×D61—42no—
106C2×D61—82no—
107#6923—164no—
108C4×C61—22no—
109C4×C61—21no—
110C4×C61—42no—
111C4×C61—81no—
112Dic61—41no—
113Dic61—81no—
114Dic61#35312yes—
115#3152—611no—
116#2126—612no—
117#1409—612no—
118#1409—612no—
119C2×C61—21no—
120C2×C61C2×#35311yescharacteristic
121D61—41no—
122D61—82no—
123C2×#137—1222no—
124#532—611no—
125C2×C4⋊Q8—611no—
126#497—1222no—
127#494—611no—
128#494#692311yescharacteristic, complemented
129#492—611no—
130#492—611no—
131#410—1222no—
132C4×M16—611no—
133#354—1222no—
134#353—614no—
135C61#1358611yescharacteristic, complemented, Sylow
136C2×#41—612no—
137C22×Q8—1221no—
138#137—2444no—
139#137—2444no—
140C2×Q16—22no—
141C2×Q16—1222no—
142C2×SD16—1222no—
143C2×SD16—1222no—
144C2×M16—1221no—
145C4⋊Q8—611no—
146C4⋊Q8—1221no—
147C4⋊Q8—2441no—
148C4⋊Q8C2×#692311yescharacteristic
149C4⋊Q8C2×#692311yescharacteristic
150#127—611no—
151#124—612no—
152#121—2441no—
153C4×D4—2441no—
154#117—2442no—
155C2×C4⋊C4—2441no—
156C2×C42—611no—
157#99—1224no—
158#99—2442no—
159#97—1221no—
160C4×C8—611no—
161C4×C8C2×#692311yescharacteristic
162#41—2442no—
163#41—2442no—
164C2×Q8—21no—
165C2×Q8—1221no—
166C2×Q8—2441no—
167C2×Q8—2442no—
168C2×Q8—2442no—
169C2×Q8C22⋊#692312yes—
170C2×D4—612no—
171C2×D4—1222no—
172C2×D4—2441no—
173C22×C4—611no—
174C22×C4—1222no—
175C22×C4—1221no—
176C22×C4—2441no—
177Q16—44no—
178Q16—2444no—
179SD16—2444no—
180SD16—2444no—
181M16—1224no—
182C2×C8—21no—
183C2×C8—1221no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—4882no—
187C4⋊C4—2441no—
188C4⋊C4—2442no—
189C22⋊C4—2442no—
190C22⋊C4—2444no—
191C42—611no—
192C42—1221no—
193C42—4882no—
194C42C22×#692311yescharacteristic
195C23—611no—
196C23—1222no—
197Q8—42no—
198Q8—42no—
199Q8—4882no—
200Q8—2442no—
201D4—4882no—
202D4—4882no—
203D4—2442no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—611no—
208C2×C4—1222no—
209C2×C4—1221no—
210C2×C4—4881no—
211C2×C4—4881no—
212C2×C4—4884no—
213C2×C4—2442no—
214C2×C4—2442no—
215C2×C4—2442no—
216C2×C4—2441no—
217C2×C4C2×C22⋊#692311yescharacteristic
218C8—22no—
219C8—1222no—
220C22—1221no—
221C22—4882no—
222C22—2441no—
223C22—2442no—
224C22D61⋊#9911yescharacteristic
225C4—22no—
226C4—21no—
227C4—42no—
228C4—81no—
229C4—612no—
230C4—9764no—
231C4—4881no—
232C4—2441no—
233C2—21no—
234C2—4882no—
235C2—2441no—
236C2D61⋊#35311yescharacteristic
subgroupG/H
centerC2D61⋊#353
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC2×C61C2×#353
prime radical#16840313C22
cosocle#14519774C23
hypercenter#494#6923
hypocenterC61#13586
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #21460953C2semidirect
3. #21460952C2semidirect
10. C4⋊Q8×D61C4semidirect
12. #16840141C4semidirect
15. C4×#11758189C4semidirect
16. C61×#494C4semidirect
128. #494#6923semidirect
135. C61#13586semidirect

Maximal subgroups

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

subgroupindex
#214620932
#214609532
#214609522
#214562772
#1358661