Finite Group Databasebeta

← #49133906

Subgroups — #49133906

Order 36224.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#31827175C211yesmaximal, characteristic, complemented
2#31826819C212yesmaximal, complemented
3#31826815C211yesmaximal, characteristic, complemented
4#31826772C212yesmaximal, complemented
5#31826755C211yesmaximal, characteristic, complemented
6#31826677C212yesmaximal
7#31826667C211yesmaximal, characteristic, complemented
8Q8×C283⋊C8C211yesmaximal, characteristic, complemented
9C4×#17985379C212yesmaximal, complemented
10C4×#17985377C211yesmaximal, characteristic
11#556×C283C211yesmaximal, characteristic, complemented
12C2×#17985379C2212yes—
13C2×#17985377C2211yescharacteristic
14#22583610C2211yescharacteristic
15#22583602C2212yes—
16#22583588—22no—
17#22583586—21no—
18C4×D1132C2211yescharacteristic
19C4×D1132C2212yes—
20C4⋊C4×D283—21no—
21#22583543C2211yescharacteristic
22#22583541C2212yes—
23C4⋊(C283⋊C8)C2211yescharacteristic
24C4⋊(C283⋊C8)C2212yes—
25#22583534C2212yes—
26#22583534C2212yes—
27#22583534C2212yes—
28#22583531C2212yes—
29#22583531C2211yescharacteristic
30#22583530C2211yescharacteristic
31C4×C283⋊C8C2211yescharacteristic
32C4×C283⋊C8C2212yes—
33C4⋊Q8×C283C2211yescharacteristic
34#126×C283—22no—
35#124×C283C2212yes—
36C22⋊Q8×C283—21no—
37C4×Q8×C283C2212yes—
38C4×Q8×C283C2211yescharacteristic
39C4×D4×C283C2211yescharacteristic
40#117×C283—21no—
41C2×Dic566—21no—
42C2×D1132C2311yescharacteristic
43C2×C4×D283—21no—
44C2×C4×D283—22no—
45#17985379—24no—
46#17985377—22no—
47C2×C283⋊C8C2311yescharacteristic
48C2×C283⋊C8C2312yes—
49C2×C283⋊C8C2311yescharacteristic
50C4⋊Dic283—21no—
51C4⋊Dic283C2311yescharacteristic
52C4⋊Dic283C2312yes—
53Dic283⋊C4—22no—
54Dic283⋊C4—22no—
55(C2×D283)⋊C4—22no—
56(C2×D283)⋊C4—21no—
57C2×Q8×C283D411yescharacteristic
58C2×Q8×C283C2312yes—
59C2×D4×C283C2311yescharacteristic
60C22×C4×C283—21no—
61C4⋊C4×C283—21no—
62C4⋊C4×C283—22no—
63C4⋊C4×C283—21no—
64C4⋊C4×C283D411yescharacteristic
65C4⋊C4×C283D412yes—
66C4⋊C4×C283C2311yescharacteristic
67C22⋊C4×C283—22no—
68C22⋊C4×C283—21no—
69C22⋊C4×C283—22no—
70C42×C283—22no—
71C42×C283C2312yes—
72C42×C283C2311yescharacteristic
73C22×D283—21no—
74Dic566—81no—
75D1132—21no—
76D1132#4112yes—
77C2×Dic283—21no—
78C2×Dic283—22no—
79C4×D283—42no—
80C4×D283—82no—
81C283⋊C8—21no—
82C283⋊C8—22no—
83C283⋊C8—21no—
84C23×C283—21no—
85Q8×C283—22no—
86Q8×C283—22no—
87Q8×C283—22no—
88D4×C283—21no—
89D4×C283—21no—
90C2×C4×C283—21no—
91C2×C4×C283—22no—
92C2×C4×C283—42no—
93C2×C4×C283C2×D412yes—
94C2×C4×C283C2×D411yescharacteristic
95C2×C4×C283C2×D412yes—
96C2×C4×C283C2×D411yescharacteristic
97C2×C4×C283C2411yescharacteristic
98C22×C283—41no—
99C22×C283—41no—
100C22×C283—41no—
101C22×C283C22×D411yescharacteristic
102C2×D283—22no—
103C2×D283—41no—
104C2×D283—41no—
105C4×C283—22no—
106C4×C283—22no—
107C4×C283—21no—
108C4×C283—42no—
109C4×C283—41no—
110C4×C283#13512yes—
111C4×C283C2×#4111yescharacteristic
112C4×C2832−1+411yescharacteristic
113Dic283—41no—
114Dic283—42no—
115C2×C283—81no—
116C2×C283#54811yescharacteristic
117C2×C283C2×#13511yescharacteristic
118C2×C283#57711yescharacteristic
119D283—42no—
120C283#336011yescharacteristic, complemented, Sylow
121#3360—2831nomaximal, Sylow
122#556D28311yescharacteristic, complemented
123#554—2831no—
124#488—2832no—
125#487—2831no—
126#479—2832no—
127#476—2831no—
128#463—2832no—
129#461—2831no—
130C8×Q8—2831no—
131C4×SD16—2832no—
132C4×D8—2831no—
133C2×SD16—2832no—
134C2×D8—2831no—
135C4⋊Q8—2831no—
136C4⋊Q8C2×D28311yescharacteristic
137#126—22no—
138#125—2832no—
139#124C2×D28312yes—
140C22⋊Q8—21no—
141C22⋊Q8—5661no—
142C22⋊Q8—5662no—
143C4×Q8C2×D28312yes—
144C4×Q8C2×D28311yescharacteristic
145C4×D4—2831no—
146C4×D4—2832no—
147C4×D4C2×D28311yescharacteristic
148#117—21no—
149C2×C4⋊C4—5661no—
150#107—2831no—
151#106—2832no—
152C4⋊C8—2831no—
153C4⋊C8—2832no—
154#103—2832no—
155#103—2832no—
156#103—2832no—
157#102—2831no—
158#102—2832no—
159#102—2831no—
160C4×C8—2831no—
161C4×C8—2832no—
162C2×Q8—5661no—
163C2×Q8C22⋊D28311yescharacteristic
164C2×Q8C22×D28312yes—
165C2×D4—2831no—
166C2×D4C22×D28311yescharacteristic
167C22×C4—21no—
168C22×C4—5661no—
169C22×C4—5662no—
170SD16—5664no—
171D8—5662no—
172C2×C8—2831no—
173C2×C8—2832no—
174C2×C8—2831no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—2832no—
179C4⋊C4—2831no—
180C4⋊C4—5662no—
181C4⋊C4—5662no—
182C4⋊C4—5661no—
183C4⋊C4C22⋊D28311yescharacteristic
184C4⋊C4C22⋊D28312yes—
185C4⋊C4C22×D28311yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—5661no—
190C22⋊C4—5662no—
191C42—22no—
192C42C22×D28311yescharacteristic
193C42C22×D28312yes—
194C23—21no—
195C23—5661no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—22641no—
200D4—21no—
201D4—21no—
202D4—2832no—
203D4—5661no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—5661no—
208C2×C4—5662no—
209C2×C4—11322no—
210C2×C4—22642no—
211C2×C4C2×C22⋊D28312yes—
212C2×C4C2×C22⋊D28312yes—
213C2×C4C2×C22⋊D28311yescharacteristic
214C2×C4C2×C22⋊D28311yescharacteristic
215C2×C4C23×D28311yescharacteristic
216C8—5661no—
217C8—5662no—
218C8—5661no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—5662no—
223C22—11321no—
224C22—11321no—
225C22C22×C22⋊D28311yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—11321no—
232C4—11322no—
233C4#2258363012yes—
234C4#41×D28311yescharacteristic
235C4#2258365711yescharacteristic
236C2—81no—
237C2—11322no—
238C2#3182712211yescharacteristic
239C2C2×#2258363011yescharacteristic
240C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#556×C283C2
socleC22×C283C22×D4
prime radicalC2×#17985377C22
hypercenter#556D283
hypocenterC283#3360
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #31827175C2semidirect
2. #31826819C2semidirect
3. #31826815C2semidirect
4. #31826772C2semidirect
5. #31826755C2semidirect
7. #31826667C2semidirect
8. Q8×C283⋊C8C2semidirect
9. C4×#17985379C2semidirect
11. #556×C283C2semidirect
120. C283#3360semidirect
122. #556D283semidirect

Maximal subgroups

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

subgroupindex
#318271752
#318268192
#318268152
#318267722
#318267552
#318266772
#318266672
Q8×C283⋊C82
C4×#179853792
C4×#179853772
#556×C2832
#3360283