Finite Group Databasebeta

← #27151388

Subgroups — #27151388

Order 12416.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13095501C211yesmaximal, characteristic, complemented
2C2×#15690817C211yesmaximal, characteristic
3#20261306C211yesmaximal, characteristic
4#20260991C214yesmaximal, complemented
5#20260974C214yesmaximal, complemented
6#20260771C211yesmaximal, characteristic, complemented
7#20260744C211yesmaximal, characteristic, complemented
8#20260743C211yesmaximal, characteristic, complemented
9#536×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C2×#13095501—24no—
12C2×#13095501C2212yes—
13C2×#13095501C2212yes—
14#15690817—24no—
15C2×#13095495C2212yes—
16C2×#13095495C2211yescharacteristic
17C22×#11715895C2212yes—
18C22×#11715895C2211yescharacteristic
19#15690808—21no—
20#15690797—21no—
21#15690791—22no—
22#15690789C2214yes—
23#15690769C2211yescharacteristic
24#15690743C2212yes—
25#15690741C2212yes—
26#15690734C2214yes—
27#15690733C2214yes—
28C2×#41×C97C2211yescharacteristic
29C22×Q8×C97C2211yescharacteristic
30C4⋊Q8×C97—21no—
31#124×C97—21no—
32#123×C97—22no—
33C22⋊Q8×C97C2214yes—
34#117×C97C2211yescharacteristic
35C2×Dic194—22no—
36C2×Dic194—22no—
37C2×Dic194—21no—
38C2×Dic194C2312yes—
39C22×Dic97—21no—
40#13095501—24no—
41#13095501—24no—
42#13095495—22no—
43#13095495—24no—
44C2×#11715895—24no—
45C2×#11715895D414yes—
46C2×#11715895C2312yes—
47C2×#11715895C2311yescharacteristic
48C2×#11715895C2311yescharacteristic
49C4⋊Dic97C2312yes—
50#13095490—24no—
51C22⋊Dic97—21no—
52C22⋊Dic97—24no—
53C4×Dic97—21no—
54#41×C97—22no—
55#41×C97—22no—
56C2×Q8×C97—21no—
57C2×Q8×C97—22no—
58C2×Q8×C97—22no—
59C2×Q8×C97D411yescharacteristic
60C2×Q8×C97C2312yes—
61C2×D4×C97D412yes—
62C2×D4×C97D411yescharacteristic
63C22×C4×C97—21no—
64C22×C4×C97C2312yes—
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—24no—
67C4⋊C4×C97C2312yes—
68C22⋊C4×C97—21no—
69C22⋊C4×C97—24no—
70C42×C97—21no—
71C23×C97C2×D412yes—
72C23×C97C2×D411yescharacteristic
73Q8×C97—22no—
74Q8×C97—22no—
75Q8×C97—22no—
76Q8×C97—42no—
77D4×C97—24no—
78D4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—42no—
84C2×C4×C97C2×D412yes—
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97#4112yes—
88C2×C4×C97C2411yescharacteristic
89C2×Dic97—22no—
90C2×Dic97—22no—
91C2×Dic97—42no—
92Dic194—22no—
93Dic194—22no—
94Dic194—42no—
95#11715895—22no—
96#11715895C2×D412yes—
97#11715895C2×D412yes—
98C22×C97—22no—
99C22×C97—22no—
100C22×C97—24no—
101C22×C97#12112yes—
102C22×C97C22×D411yescharacteristic
103C4×C97—22no—
104C4×C97—42no—
105C4×C97—42no—
106C4×C97#12112yes—
107C4×C97C22×D411yescharacteristic
108C4×C97C2×#4111yescharacteristic
109Dic97—42no—
110Dic97—42no—
111C2×C97—22no—
112C2×C97—42no—
113C2×C97C2×#12111yescharacteristic
114C2×C97#57812yes—
115#3059—971nomaximal, Sylow
116C97#305911yescharacteristic, complemented, Sylow
117C22×Q16—971no—
118C2×#131—971no—
119#536—971no—
120#536D9711yescharacteristic
121#470—974no—
122#467—974no—
123#427—971no—
124#419—971no—
125#419—971no—
126C2×#41C2×D9711yescharacteristic
127C22×Q8—971no—
128C22×Q8C2×D9711yescharacteristic
129C2×Q16—972no—
130C2×Q16—972no—
131C2×Q16—1944no—
132#131—1944no—
133C2×M16—971no—
134C2×M16—972no—
135C22×C8—971no—
136C22×C8—972no—
137C4⋊Q8—21no—
138C4⋊Q8—1941no—
139#124—21no—
140#124—1941no—
141#123—22no—
142#123—1942no—
143C22⋊Q8—974no—
144C22⋊Q8C2×D9714yes—
145#117—971no—
146#117C2×D9711yescharacteristic
147#107—972no—
148#106—972no—
149#103—974no—
150#103—974no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—972no—
157C2×Q8—1942no—
158C2×Q8—1942no—
159C2×Q8—1941no—
160C2×Q8C22⋊D9711yescharacteristic
161C2×Q8C22×D9712yes—
162C2×D4C22⋊D9712yes—
163C2×D4C22⋊D9711yescharacteristic
164C22×C4—21no—
165C22×C4—1941no—
166C22×C4C22×D9712yes—
167C22×C4C22×D9711yescharacteristic
168Q16—1944no—
169Q16—1944no—
170M16—1944no—
171M16—1942no—
172C2×C8—974no—
173C2×C8—971no—
174C2×C8—971no—
175C2×C8—972no—
176C2×C8—1944no—
177C4⋊C4—24no—
178C4⋊C4—972no—
179C4⋊C4—1944no—
180C4⋊C4C22×D9712yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—1941no—
184C22⋊C4—1944no—
185C42—21no—
186C42—1941no—
187C23C2×C22⋊D9712yes—
188C23C2×C22⋊D9711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—3882no—
194Q8—1942no—
195Q8—1942no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—3882no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4D4×D9712yes—
207C2×C4C2×C22⋊D9712yes—
208C2×C4C2×C22⋊D9711yescharacteristic
209C2×C4#1309551012yes—
210C2×C4C23×D9711yescharacteristic
211C8—972no—
212C8—972no—
213C8—1942no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22#1569078212yes—
218C22C22×C22⋊D9711yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—3882no—
223C4—3882no—
224C4#1569078212yes—
225C4C2×D4×D9711yescharacteristic
226C4C2×#1309551011yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1569078211yescharacteristic
230C2#2026155012yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#536×C97C2
socleC22×C97C22×D4
prime radicalC2×#41×C97C22
hypercenter#536D97
hypocenterC97#3059
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#13095501C2semidirect
4. #20260991C2semidirect
5. #20260974C2semidirect
6. #20260771C2semidirect
7. #20260744C2semidirect
8. #20260743C2semidirect
116. C97#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
C2×#156908172
#202613062
#202609912
#202609742
#202607712
#202607442
#202607432
#536×C972
#305997