Finite Group Databasebeta

← #11736857

Subgroups — #11736857

Order 896.

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

normal subgroups only

This group has 388 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
1#90246C211yesmaximal, characteristic
2#90234C211yesmaximal, characteristic, complemented
3C7×C8×D4C211yesmaximal, characteristic, complemented
4#89886C212yesmaximal, complemented
5C2×#6391C212yesmaximal, complemented
6#89623C211yesmaximal, characteristic, complemented
7#89614C211yesmaximal, characteristic, complemented
8#89507C212yesmaximal, complemented
9#89493C212yesmaximal, complemented
10C4×#1132C211yesmaximal, characteristic, complemented
11#89452C211yesmaximal, characteristic, complemented
12C2×#1159—21no—
13C22×C7×C8C2212yes—
14C4×C7×D4C2211yescharacteristic
15D4×Dic7—21no—
16#6493C2212yes—
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19#6487C2212yes—
20C2×#1132C2211yescharacteristic
21#6460—21no—
22#6453—21no—
23#6449—22no—
24#6441—22no—
25C4×D28C2211yescharacteristic
26#6432C2211yescharacteristic
27#6432C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×C4⋊C8C2211yescharacteristic
30C7×C22⋊C8C2212yes—
31C4×C7×C8C2211yescharacteristic
32#6395C2212yes—
33#6395C2211yescharacteristic
34#6391C2212yes—
35#6391C2211yescharacteristic
36#6391C2212yes—
37#6391C2212yes—
38#6391C2212yes—
39#6390C2212yes—
40#6390C2211yescharacteristic
41#3335—71nomaximal, Sylow
42C2×C7×D4D411yescharacteristic
43C22×C4×C7C2312yes—
44C2×C22⋊D7—22no—
45C22×Dic7—22no—
46C22×Dic7—22no—
47#1159—44no—
48C2×D28C2311yescharacteristic
49C2×C4×D7—21no—
50C2×Dic14—21no—
51C2×Dic14—21no—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8—24no—
54C2×C7×C8C2311yescharacteristic
55C2×C7×C8C2312yes—
56C2×C7×C8C2311yescharacteristic
57C7×C4⋊C4D411yescharacteristic
58C7×C22⋊C4D412yes—
59C42×C7C2311yescharacteristic
60C22⋊Dic7—22no—
61(C2×D7)⋊C4—21no—
62(C2×D7)⋊C4—22no—
63C4⋊Dic7—21no—
64C4⋊Dic7—21no—
65C4⋊Dic7—22no—
66C4⋊Dic7—22no—
67C4⋊Dic7C2311yescharacteristic
68C4⋊Dic7C2312yes—
69C4⋊Dic7C2312yes—
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—21no—
72Dic7⋊C4—22no—
73C4×Dic7—21no—
74#1132—22no—
75#554—71no—
76#547—71no—
77#498—71no—
78#483—72no—
79#481—72no—
80#476—71no—
81#475—71no—
82#465—72no—
83C4×SD16—71no—
84C8×D4D711yescharacteristic, complemented
85C2×#106—72no—
86C23×C7C2×D412yes—
87C22×D7—21no—
88C7×D4SD1614yescomplemented
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7C2×D412yes—
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2411yescharacteristic
95C22⋊D7—44no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—21no—
101C2×Dic7—44no—
102C2×Dic7—82no—
103D28—21no—
104D28—21no—
105C4×D7—42no—
106Dic14—21no—
107Dic14—21no—
108Dic14—42no—
109Dic14—81no—
110C7×C8—22no—
111C7×C8—21no—
112C7×C8#4112yescomplemented
113C2×#41—141no—
114C2×SD16—71no—
115C22×C8C2×D712yes—
116C4⋊Q8—71no—
117C4⋊Q8—71no—
118#125—141no—
119#123—142no—
120C22⋊Q8—72no—
121C22⋊Q8—141no—
122C22⋊Q8—142no—
123#121—72no—
124C4×Q8—71no—
125C4×D4—71no—
126C4×D4—141no—
127C4×D4C2×D711yescharacteristic
128C2×C4⋊C4—72no—
129C2×C4⋊C4—72no—
130#106—71no—
131#106—72no—
132#106—72no—
133#106—72no—
134#106—72no—
135C4⋊C8C2×D711yescharacteristic
136#103—71no—
137#103—72no—
138#102—71no—
139#102—72no—
140C22⋊C8C2×D712yes—
141C4×C8C2×D711yescharacteristic
142C22×C7—22no—
143C22×C7—22no—
144C22×C7C2×SD1614yes—
145C22×C7C22×D411yescharacteristic
146C2×D7—41no—
147C2×D7—41no—
148C2×D7—41no—
149C4×C7—22no—
150C4×C7—21no—
151C4×C7C2×SD1612yes—
152C4×C7C2×#4111yescharacteristic
153C4×C72−1+411yescharacteristic
154Dic7—41no—
155Dic7—42no—
156Dic7—41no—
157Dic7—42no—
158Dic7—41no—
159#41—284no—
160C2×Q8—71no—
161C2×Q8—141no—
162C2×Q8—141no—
163C2×D4—71no—
164C2×D4—142no—
165C2×D4D2811yescharacteristic
166C22×C4—142no—
167C22×C4—142no—
168C22×C4—141no—
169C22×C4C22×D712yes—
170SD16—142no—
171C2×C8—24no—
172C2×C8C22×D711yescharacteristic
173C2×C8C22×D712yes—
174C2×C8C22×D711yescharacteristic
175C4⋊C4—71no—
176C4⋊C4—72no—
177C4⋊C4—71no—
178C4⋊C4—72no—
179C4⋊C4—141no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—142no—
183C4⋊C4—141no—
184C4⋊C4—141no—
185C4⋊C4D2811yescharacteristic
186C22⋊C4—142no—
187C22⋊C4—141no—
188C22⋊C4—142no—
189C22⋊C4D2812yes—
190C42—141no—
191C42C22×D711yescharacteristic
192C2×C7—24no—
193C2×C7#54711yescharacteristic
194C2×C7C22×SD1611yescharacteristic
195C2×C7#57711yescharacteristic
196D7—81no—
197C23—141no—
198C23C2×D2812yes—
199Q8—141no—
200Q8—141no—
201Q8—282no—
202Q8—561no—
203D4—141no—
204D4—141no—
205D4—284no—
206D4#113214yescomplemented
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—141no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—141no—
213C2×C4—142no—
214C2×C4—284no—
215C2×C4—282no—
216C2×C4—562no—
217C2×C4C2×D2811yescharacteristic
218C2×C4C2×D2812yes—
219C2×C4C2×D2811yescharacteristic
220C2×C4C23×D711yescharacteristic
221C8—22no—
222C8—21no—
223C8#115912yescomplemented
224C7#333511yescharacteristic, complemented, Sylow
225C22—22no—
226C22—22no—
227C22—281no—
228C22—281no—
229C22—281no—
230C22C2×#113214yes—
231C22C22×D2811yescharacteristic
232C4—21no—
233C4—22no—
234C4—281no—
235C4—282no—
236C4—281no—
237C4—282no—
238C4—281no—
239C4C2×#113212yes—
240C4C2×#115911yescharacteristic
241C4#655411yescharacteristic
242C2—24no—
243C2—561no—
244C2#9024611yescharacteristic
245C2C22×#113211yescharacteristic
246C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×C8×D4C2
socleC22×C7C22×D4
prime radical#90246C2
hypercenterC8×D4D7
hypocenterC7#3335
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
2. #90234C2semidirect
3. C7×C8×D4C2semidirect
4. #89886C2semidirect
5. C2×#6391C2semidirect
6. #89623C2semidirect
7. #89614C2semidirect
8. #89507C2semidirect
9. #89493C2semidirect
10. C4×#1132C2semidirect
11. #89452C2semidirect
84. C8×D4D7semidirect
88. C7×D4SD16semidirect
112. C7×C8#41semidirect
206. D4#1132semidirect
223. C8#1159semidirect
224. C7#3335semidirect

Maximal subgroups

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

subgroupindex
#902462
#902342
C7×C8×D42
#898862
C2×#63912
#896232
#896142
#895072
#894932
C4×#11322
#894522
#33357