Finite Group Databasebeta

← #49122076

Subgroups — #49122076

Order 36224.

344 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#22583597C211yesmaximal, characteristic
2C2×#22583535C211yesmaximal, characteristic, complemented
3C2×#22583533C211yesmaximal, characteristic, complemented
4C2×#22583531C211yesmaximal, characteristic
5C2×#22583520C211yesmaximal, characteristic, complemented
6#31826334C211yesmaximal, characteristic, complemented
7C2×#102×C283C211yesmaximal, characteristic
8C22×Dic566C2211yescharacteristic
9C22×C283⋊C8C2211yescharacteristic
10#22583597—22no—
11#22583597C414yescomplemented
12C2×C4⋊Dic283C2211yescharacteristic
13C2×Dic283⋊C4—22no—
14C2×C22⋊Dic283—22no—
15C2×C4×Dic283C2211yescharacteristic
16#22583535—22no—
17#22583533—22no—
18#22583531—22no—
19#22583520—22no—
20C22×D4×C283C2211yescharacteristic
21C22×C8×C283C2211yescharacteristic
22C2×C4⋊C4×C283C2211yescharacteristic
23#102×C283—22no—
24C2×Dic566—21no—
25C2×Dic566—21no—
26C2×Dic566—21no—
27C2×Dic566—21no—
28C2×Dic566—21no—
29C2×Dic566C2×C412yes—
30C22×Dic283—21no—
31C22×Dic283—21no—
32C22×Dic283D412yes—
33C2×C283⋊C8—21no—
34C2×C283⋊C8—21no—
35C2×C283⋊C8D412yes—
36C4⋊Dic283—21no—
37C4⋊Dic283—21no—
38Dic283⋊C4—44no—
39C22⋊Dic283—28no—
40C4×Dic283C2×C412yes—
41C4×Dic283D412yes—
42C24×C283—21no—
43C2×D4×C283—21no—
44C2×D4×C283—21no—
45C2×D4×C283—21no—
46C2×D4×C283—21no—
47C2×D4×C283—21no—
48C2×D4×C283C2×C412yes—
49C22×C4×C283—21no—
50C22×C4×C283C2311yescharacteristic
51C2×C8×C283—21no—
52C2×C8×C283—21no—
53C2×C8×C283D412yes—
54C4⋊C4×C283—21no—
55C4⋊C4×C283—21no—
56Dic566—22no—
57Dic566—22no—
58Dic566—41no—
59Dic566—41no—
60C2×Dic283—22no—
61C2×Dic283—22no—
62C2×Dic283—22no—
63C2×Dic283—41no—
64C2×Dic283—41no—
65C2×Dic283—41no—
66C2×Dic283—42no—
67C2×Dic283C22⋊C414yes—
68C283⋊C8—22no—
69C23×C283—22no—
70C23×C283—41no—
71C23×C283—41no—
72C23×C283—42no—
73C23×C283—42no—
74C23×C283C2×D411yescharacteristic
75D4×C283—22no—
76D4×C283—22no—
77D4×C283—41no—
78D4×C283—41no—
79C2×C4×C283—41no—
80C2×C4×C283—42no—
81C2×C4×C283C22×C411yescharacteristic
82C2×C4×C283C2×D411yescharacteristic
83C2×C4×C283C2×D411yescharacteristic
84C2×C4×C283C2×D411yescharacteristic
85C2×C4×C283#4111yescharacteristic
86C2×C4×C283#4111yescharacteristic
87C8×C283—22no—
88C22×C283—42no—
89C22×C283—44no—
90C22×C283—82no—
91C22×C283—81no—
92C22×C283—81no—
93C22×C283C2×C22⋊C411yescharacteristic
94C22×C283#12411yescharacteristic
95C22×C283#12711yescharacteristic
96C22×C283AGL(1,ℤ/8)12yes—
97C22×C283#13712yes—
98C4×C283—81no—
99C4×C283C4×D411yescharacteristic
100C4×C283C4×D411yescharacteristic
101C4×C283#12111yescharacteristic
102C4×C283#12111yescharacteristic
103Dic283—24no—
104Dic283—42no—
105Dic283—81no—
106C2×C283—82no—
107C2×C283#39011yescharacteristic
108C2×C283#41712yes—
109C2×C283#48912yes—
110C2×C283#49611yescharacteristic
111C2×C283#49711yescharacteristic
112C283#195911yescharacteristic, complemented, Sylow
113#1959—2831nomaximal, Sylow
114C2×#124—2831no—
115C2×#103—2831no—
116C2×#103—2831no—
117C2×#102—2831no—
118C2×#102D28311yescharacteristic
119C2×#97—2831no—
120#384—2831no—
121C22×Q8—2831no—
122C22×D4C2×D28311yescharacteristic
123C22×C8—2831no—
124C22×C8C2×D28311yescharacteristic
125#124—2834no—
126#124—5662no—
127C2×C4⋊C4—2831no—
128C2×C4⋊C4—5662no—
129C2×C4⋊C4C2×D28311yescharacteristic
130C2×C22⋊C4—5662no—
131C2×C42—2831no—
132#103—5662no—
133#103—5662no—
134#102—22no—
135#102—5662no—
136#97—5662no—
137C24—21no—
138C2×Q8—2832no—
139C2×Q8—5661no—
140C2×Q8—5661no—
141C2×Q8—5661no—
142C2×Q8—5661no—
143C2×Q8—5661no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C4×D28312yes—
150C22×C4—21no—
151C22×C4—2832no—
152C22×C4—5661no—
153C22×C4—5661no—
154C22×C4C22×D28311yescharacteristic
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—2832no—
158C2×C8—5661no—
159C2×C8—5661no—
160C2×C8C22⋊D28312yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—5661no—
164C4⋊C4—5661no—
165C4⋊C4—11324no—
166C22⋊C4—5668no—
167C42—2832no—
168C42—2832no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—42no—
173C23—42no—
174C23D4×D28311yescharacteristic
175Q8—5662no—
176Q8—5662no—
177Q8—11321no—
178Q8—11321no—
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—2834no—
186C2×C4—5662no—
187C2×C4—5662no—
188C2×C4—5662no—
189C2×C4—11321no—
190C2×C4—11321no—
191C2×C4—11321no—
192C2×C4—11322no—
193C2×C4C2×C4×D28311yescharacteristic
194C2×C4D4×D28311yescharacteristic
195C2×C4D4×D28311yescharacteristic
196C2×C4C2×C22⋊D28311yescharacteristic
197C2×C4#1798539111yescharacteristic
198C2×C4#1798539211yescharacteristic
199C8—22no—
200C8—5662no—
201C22—44no—
202C22—42no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22C22⋊C4×D28311yescharacteristic
207C22#2258359611yescharacteristic
208C22#2258360611yescharacteristic
209C22#2258363412yes—
210C22#2258363912yes—
211C4—81no—
212C4—5664no—
213C4—11322no—
214C4—22641no—
215C4(C22⋊D283)⋊C411yescharacteristic
216C4C4×C22⋊D28311yescharacteristic
217C4#2258358211yescharacteristic
218C4#2258358311yescharacteristic
219C2—82no—
220C2#3182636511yescharacteristic
221C2#3182646312yes—
222C2#3182682312yes—
223C2#3182685311yescharacteristic
224C2#3182685611yescharacteristic
subgroupG/H
centerC23D4×D283
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
FittingC2×#102×C283C2
socleC23×C283C2×D4
prime radicalC22×D4×C283C22
cosocleC22×C4×C283C23
hypercenterC2×#102D283
hypocenterC283#1959
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#22583535C2semidirect
3. C2×#22583533C2semidirect
5. C2×#22583520C2semidirect
6. #31826334C2semidirect
11. #22583597C4semidirect
112. C283#1959semidirect

Maximal subgroups

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

subgroupindex
C2×#225835972
C2×#225835352
C2×#225835332
C2×#225835312
C2×#225835202
#318263342
C2×#102×C2832
#1959283