Finite Group Databasebeta

← #27153603

Subgroups — #27153603

Order 12416.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×D388C211yesmaximal, characteristic, complemented
2#20261390C211yesmaximal, characteristic
3#20261086C211yesmaximal, characteristic, complemented
4#20261067C212yesmaximal, complemented
5#20261020C211yesmaximal, characteristic, complemented
6#20260963C212yesmaximal, complemented
7#20260940C211yesmaximal, characteristic, complemented
8#20260872C212yesmaximal, complemented
9C8×D388C211yesmaximal, characteristic, complemented
10#103×D97C212yesmaximal, complemented
11C4×SD16×C97C211yesmaximal, characteristic, complemented
12C2×#13095511—21no—
13C2×Q8×D97C2212yes—
14C2×#13095499C2212yes—
15C2×C8×D97C2212yes—
16#15690810—21no—
17#15690810C2211yescharacteristic
18#15690810C2211yescharacteristic
19#15690792—22no—
20#15690790—22no—
21#15690790C2212yes—
22#15690789—21no—
23#15690781C2212yes—
24C4×D388—21no—
25C4×D388C2211yescharacteristic
26C4⋊C4×D97C2212yes—
27#15690741C2212yes—
28C4⋊#11715895C2211yescharacteristic
29#15690735C2212yes—
30#15690735C2212yes—
31#15690734C2212yes—
32#15690733C2212yes—
33#15690722C2212yes—
34C2×SD16×C97C2211yescharacteristic
35C4×Q8×C97C2211yescharacteristic
36C4×D4×C97C2211yescharacteristic
37#106×C97C2211yescharacteristic
38#103×C97C2211yescharacteristic
39#102×C97C2211yescharacteristic
40C4×C8×C97C2211yescharacteristic
41#13095511—44no—
42C2×Dic194—21no—
43C2×Dic194—22no—
44C2×Dic194C2312yes—
45Q8×D97—24no—
46Q8×D97—42no—
47C2×D388D411yescharacteristic
48C2×C22⋊D97—22no—
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97C2312yes—
52#13095499—42no—
53C2×#11715895C2312yes—
54C8×D97—42no—
55C4⋊Dic97—21no—
56C4⋊Dic97—21no—
57C4⋊Dic97—21no—
58C4⋊Dic97—22no—
59C4⋊Dic97D411yescharacteristic
60C4⋊Dic97C2312yes—
61#13095490—22no—
62C22⋊Dic97—22no—
63#13095488—22no—
64#13095488D412yes—
65C2×Q8×C97C2311yescharacteristic
66C2×D4×C97C2311yescharacteristic
67C22×C4×C97—21no—
68SD16×C97—22no—
69C2×C8×C97C2311yescharacteristic
70C2×C8×C97C2311yescharacteristic
71C4⋊C4×C97—21no—
72C4⋊C4×C97C2311yescharacteristic
73C4⋊C4×C97C2311yescharacteristic
74C22⋊C4×C97—21no—
75C42×C97—21no—
76C42×C97C2311yescharacteristic
77C23×C97—21no—
78C22×D97C2×D412yes—
79Q8×C97—21no—
80Q8×C97#4112yescomplemented
81D4×C97—21no—
82D4×C97—21no—
83C2×C4×C97—21no—
84C2×C4×C97—21no—
85C2×C4×C97—21no—
86C2×C4×C97—42no—
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2411yescharacteristic
90C22⋊D97—44no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97C2×D412yes—
94D388SD1614yescomplemented
95C4×D97—22no—
96C4×D97—22no—
97C4×D97—44no—
98C4×D97—44no—
99Dic194—22no—
100Dic194—22no—
101Dic194—44no—
102Dic194—42no—
103C8×C97—21no—
104C8×C97—21no—
105#11715895—22no—
106C22×C97—41no—
107C22×C97—41no—
108C22×C97—41no—
109C22×C97C22×D411yescharacteristic
110C2×D97—22no—
111C2×D97—22no—
112C2×D97C2×SD1614yes—
113C4×C97—22no—
114C4×C97—21no—
115C4×C97—41no—
116C4×C97—41no—
117C4×C97C2×SD1612yes—
118C4×C97C2×#4111yescharacteristic
119C4×C97H(5,𝔽2)11yescharacteristic
120Dic97—22no—
121Dic97—42no—
122Dic97—42no—
123C2×C97—81no—
124C2×C97#54611yescharacteristic
125C2×C97C22×SD1611yescharacteristic
126C2×C97#57811yescharacteristic
127D97—24no—
128#3298—971nomaximal, Sylow
129C97#329811yescharacteristic, complemented, Sylow
130D4×Q8—971no—
131#547—971no—
132#487—971no—
133#483—972no—
134#475—971no—
135#465—972no—
136#461—971no—
137#448—972no—
138C4×SD16D9711yescharacteristic, complemented
139C8×D4—971no—
140C2×#103—972no—
141C2×#41—1941no—
142C22×Q8—972no—
143C2×SD16—972no—
144C2×SD16C2×D9711yescharacteristic
145C22×C8—972no—
146C4⋊Q8—971no—
147C4⋊Q8—971no—
148C4⋊Q8—1941no—
149#123—1942no—
150C22⋊Q8—972no—
151C22⋊Q8—1942no—
152C22⋊Q8—1941no—
153#121—972no—
154C4×Q8C2×D9711yescharacteristic
155C4×D4—971no—
156C4×D4—1941no—
157C4×D4C2×D9711yescharacteristic
158C2×C4⋊C4—972no—
159#106—972no—
160#106C2×D9711yescharacteristic
161C4⋊C8—971no—
162#103—972no—
163#103—972no—
164#103—972no—
165#103—972no—
166#103C2×D9711yescharacteristic
167#102C2×D9711yescharacteristic
168C22⋊C8—972no—
169C4×C8C2×D9711yescharacteristic
170#41—3884no—
171C2×Q8—972no—
172C2×Q8—3882no—
173C2×Q8—1942no—
174C2×Q8—1944no—
175C2×Q8—1941no—
176C2×Q8C22×D9711yescharacteristic
177C2×D4—971no—
178C2×D4—1942no—
179C2×D4C22×D9711yescharacteristic
180C22×C4—21no—
181C22×C4—972no—
182C22×C4—1942no—
183C22×C4—1942no—
184SD16—22no—
185SD16—3882no—
186C2×C8—972no—
187C2×C8—3882no—
188C2×C8C22×D9711yescharacteristic
189C2×C8C22×D9711yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—971no—
192C4⋊C4—972no—
193C4⋊C4—1942no—
194C4⋊C4—1942no—
195C4⋊C4—1941no—
196C4⋊C4—1941no—
197C4⋊C4—1941no—
198C4⋊C4C22×D9711yescharacteristic
199C4⋊C4C22×D9711yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—972no—
202C22⋊C4—1942no—
203C22⋊C4—1942no—
204C42—21no—
205C42C22×D9711yescharacteristic
206C23—21no—
207C23—972no—
208Q8—21no—
209Q8—3884no—
210Q8—3882no—
211Q8—1942no—
212Q8—1942no—
213Q8#1309551112yescomplemented
214D4—21no—
215D4—21no—
216D4—974no—
217D4—3884no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—972no—
223C2×C4—3884no—
224C2×C4—3884no—
225C2×C4—1942no—
226C2×C4—1942no—
227C2×C4—1942no—
228C2×C4—1942no—
229C2×C4D4×D9711yescharacteristic
230C2×C4D4×D9711yescharacteristic
231C2×C4C23×D9711yescharacteristic
232C8—21no—
233C8—21no—
234C8—1942no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—974no—
239C22—1942no—
240C22—1942no—
241C22C2×D4×D9711yescharacteristic
242C4—22no—
243C4—21no—
244C4—41no—
245C4—41no—
246C4—3882no—
247C4—3882no—
248C4—1942no—
249C4SD16×D9712yes—
250C4C2×#1309551111yescharacteristic
251C4#1569085611yescharacteristic
252C2—81no—
253C2—1944no—
254C2#2026138111yescharacteristic
255C2C2×SD16×D9711yescharacteristic
256C2#2026154911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×SD16×C97C2
socleC22×C97C22×D4
prime radical#20261390C2
hypercenterC4×SD16D97
hypocenterC97#3298
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×D388C2semidirect
3. #20261086C2semidirect
4. #20261067C2semidirect
5. #20261020C2semidirect
6. #20260963C2semidirect
7. #20260940C2semidirect
8. #20260872C2semidirect
9. C8×D388C2semidirect
10. #103×D97C2semidirect
11. C4×SD16×C97C2semidirect
80. Q8×C97#41semidirect
94. D388SD16semidirect
129. C97#3298semidirect
138. C4×SD16D97semidirect
213. Q8#13095511semidirect

Maximal subgroups

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

subgroupindex
Q8×D3882
#202613902
#202610862
#202610672
#202610202
#202609632
#202609402
#202608722
C8×D3882
#103×D972
C4×SD16×C972
#329897