Finite Group Databasebeta

← #49120219

Subgroups — #49120219

Order 36224.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#31827090C211yesmaximal, characteristic
2#31826623C211yesmaximal, characteristic, complemented
3#31826620C211yesmaximal, characteristic, complemented
4#31826378C211yesmaximal, characteristic
5#31826201C211yesmaximal, characteristic, complemented
6#31826188C211yesmaximal, characteristic
7#328×C283C211yesmaximal, characteristic, complemented
8C22×Dic566C2211yescharacteristic
9C2×Dic1132—21no—
10C2×#17985382—21no—
11#22583590—21no—
12#22583589C2211yescharacteristic
13#22583588—21no—
14#22583588—21no—
15#22583587—21no—
16#22583586—21no—
17C2×(C2×D283)⋊C4—21no—
18C2×(C2×D283)⋊C4C2211yescharacteristic
19C22⋊C4×D283—21no—
20#22583535—21no—
21#22583533—21no—
22C22⋊(C283⋊C8)C2211yescharacteristic
23C283⋊#95—21no—
24C22⋊Q8×C283C2211yescharacteristic
25C22⋊C8×C283C2211yescharacteristic
26#95×C283C2211yescharacteristic
27C23×D283D411yescharacteristic
28C2×Dic566—21no—
29C2×Dic566—21no—
30C2×Dic566—21no—
31C2×Dic566—42no—
32C2×Dic566D411yescharacteristic
33C22×Dic283—21no—
34C22×Dic283D411yescharacteristic
35C2×C4×D283—21no—
36C2×C4×D283—21no—
37Dic1132—42no—
38#17985382—42no—
39C2×C283⋊C8—21no—
40C4⋊Dic283—21no—
41C4⋊Dic283D411yescharacteristic
42Dic283⋊C4—21no—
43Dic283⋊C4—21no—
44Dic283⋊C4—21no—
45C22⋊Dic283—21no—
46(C2×D283)⋊C4—21no—
47(C2×D283)⋊C4—21no—
48(C2×D283)⋊C4—21no—
49(C2×D283)⋊C4—21no—
50(C2×D283)⋊C4—42no—
51C2×Q8×C283D411yescharacteristic
52C22×C4×C283—21no—
53C22×C4×C283C2311yescharacteristic
54C2×C8×C283—21no—
55C4⋊C4×C283—21no—
56C4⋊C4×C283D411yescharacteristic
57C22⋊C4×C283—21no—
58C22×D283—21no—
59C22×D283—21no—
60C22×D283—41no—
61C22×D283—41no—
62C22×D283Q1612yescomplemented
63Dic566—22no—
64Dic566—22no—
65Dic566—41no—
66Dic566—41no—
67Dic566—41no—
68Dic566—81no—
69C2×Dic283—22no—
70C2×Dic283—21no—
71C2×Dic283—21no—
72C2×Dic283—21no—
73C2×Dic283—41no—
74C2×Dic283—41no—
75C2×Dic283—41no—
76C4×D283—81no—
77C4×D283—81no—
78C283⋊C8—41no—
79C23×C283C2×D411yescharacteristic
80Q8×C283—41no—
81C2×C4×C283—21no—
82C2×C4×C283—21no—
83C2×C4×C283—41no—
84C2×C4×C283—41no—
85C2×C4×C283—41no—
86C2×C4×C283—41no—
87C2×C4×C283C2×D411yescharacteristic
88C2×C4×C283C2×D411yescharacteristic
89C8×C283—41no—
90C22×C283—21no—
91C22×C283—21no—
92C22×C283#12011yescharacteristic
93C22×C283C2×Q1611yescharacteristic
94C22×C283#13711yescharacteristic
95C2×D283—22no—
96C2×D283—41no—
97C2×D283—41no—
98C2×D283—42no—
99C2×D283—81no—
100C4×C283—21no—
101C4×C283—21no—
102C4×C283—41no—
103C4×C283—41no—
104C4×C283—81no—
105Dic283—41no—
106Dic283—41no—
107Dic283—41no—
108Dic283—42no—
109C2×C283—21no—
110C2×C283—21no—
111C2×C283#45111yescharacteristic
112C2×C283#45511yescharacteristic
113C2×C283B(4,𝔽2)11yescharacteristic
114D283—42no—
115C283#160211yescharacteristic, complemented, Sylow
116#1602—2831nomaximal, Sylow
117#543—2831no—
118#451—2831no—
119#451—2831no—
120#393—2831no—
121#328—2831no—
122#328D28311yescharacteristic, complemented
123#323—2831no—
124C22×Q8—2831no—
125C2×Q16—5661no—
126C2×Q16—5661no—
127C22⋊Q8—2831no—
128C22⋊Q8—5661no—
129C22⋊Q8—5661no—
130C22⋊Q8—5661no—
131C22⋊Q8—5661no—
132C22⋊Q8—5661no—
133C22⋊Q8C2×D28311yescharacteristic
134C2×C22⋊C4—2831no—
135C2×C22⋊C4—5661no—
136C2×C22⋊C4—5661no—
137#103—5661no—
138#103—5661no—
139C22⋊C8—2831no—
140C22⋊C8C2×D28311yescharacteristic
141#95—5661no—
142#95C2×D28311yescharacteristic
143C24—2831no—
144C2×Q8—2831no—
145C2×Q8—5661no—
146C2×Q8—5661no—
147C2×Q8—5661no—
148C2×Q8—11322no—
149C2×Q8D113211yescharacteristic
150C22×C4—21no—
151C22×C4—2831no—
152C22×C4—5661no—
153C22×C4—5661no—
154C22×C4—5661no—
155C22×C4C22×D28311yescharacteristic
156Q16—11322no—
157Q16—11322no—
158C2×C8—21no—
159C2×C8—5661no—
160C4⋊C4—21no—
161C4⋊C4—2831no—
162C4⋊C4—5661no—
163C4⋊C4—5661no—
164C4⋊C4—5661no—
165C4⋊C4—5661no—
166C4⋊C4D113211yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—5661no—
169C22⋊C4—5661no—
170C22⋊C4—5661no—
171C22⋊C4—5661no—
172C22⋊C4—5661no—
173C22⋊C4—11322no—
174C23—2832no—
175C23—5661no—
176C23—5661no—
177C23—11321no—
178C23—11321no—
179C23D4×D28311yescharacteristic
180Q8—41no—
181Q8—5662no—
182Q8—5662no—
183Q8—11321no—
184Q8—11321no—
185Q8—11321no—
186Q8—22641no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—5662no—
194C2×C4—5661no—
195C2×C4—5661no—
196C2×C4—5661no—
197C2×C4—11321no—
198C2×C4—11321no—
199C2×C4—11321no—
200C2×C4—22641no—
201C2×C4—22641no—
202C2×C4D4×D28311yescharacteristic
203C2×C4C2×D113211yescharacteristic
204C8—41no—
205C8—11321no—
206C22—21no—
207C22—21no—
208C22—5662no—
209C22—11322no—
210C22—11321no—
211C22—11321no—
212C22—22641no—
213C22#2258358011yescharacteristic
214C22Q16×D28311yescharacteristic
215C22#2258363911yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—11321no—
222C4—11321no—
223C4—11322no—
224C4—11321no—
225C2—21no—
226C2—21no—
227C2—11322no—
228C2#3182662211yescharacteristic
229C2#3182664411yescharacteristic
230C2#3182665311yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#328×C283C2
socleC22×C283#120
prime radicalC23×D283D4
hypercenter#328D283
hypocenterC283#1602
LCSG ▷ C22×C4×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #31826623C2semidirect
3. #31826620C2semidirect
5. #31826201C2semidirect
7. #328×C283C2semidirect
62. C22×D283Q16semidirect
115. C283#1602semidirect
122. #328D283semidirect

Maximal subgroups

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

subgroupindex
#318270902
#318266232
#318266202
#318263782
#318262012
#318261882
#328×C2832
#1602283