Finite Group Databasebeta

← #49122401

Subgroups — #49122401

Order 36224.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#22583609C211yesmaximal, characteristic
2C2×#22583535C211yesmaximal, characteristic
3C2×#22583533C211yesmaximal, characteristic
4#31826370C211yesmaximal, characteristic
5#31826227C211yesmaximal, characteristic
6#31826219C211yesmaximal, characteristic
7#340×C283C211yesmaximal, characteristic
8C22×Dic566—21no—
9C22×Dic566C2211yescharacteristic
10C22×C283⋊C8C2211yescharacteristic
11#22583609—22no—
12#22583609—22no—
13C2×C4⋊Dic283C2211yescharacteristic
14C2×Dic283⋊C4—21no—
15C2×C4×Dic283C2211yescharacteristic
16#22583535—22no—
17#22583533—22no—
18C283⋊#95—21no—
19C283⋊#95—21no—
20C22×C8×C283C2211yescharacteristic
21C2×C4⋊C4×C283C2211yescharacteristic
22C2×C4⋊C4×C283C2211yescharacteristic
23C2×Dic566—24no—
24C2×Dic566—21no—
25C2×Dic566—21no—
26C2×Dic566—21no—
27C2×Dic566—21no—
28C2×Dic566—21no—
29C2×Dic566—42no—
30C2×Dic566—42no—
31C2×Dic566Q812yescomplemented
32C22×Dic283—21no—
33C22×Dic283—21no—
34C22×Dic283D411yescharacteristic
35C22×Dic283D411yescharacteristic
36C2×C283⋊C8—21no—
37C2×C283⋊C8—21no—
38C2×C283⋊C8—21no—
39C4⋊Dic283—21no—
40C4⋊Dic283D412yes—
41Dic283⋊C4—24no—
42C4×Dic283—21no—
43C4×Dic283—21no—
44C22×C4×C283—21no—
45C22×C4×C283—21no—
46C22×C4×C283C2311yescharacteristic
47C2×C8×C283—21no—
48C2×C8×C283—21no—
49C2×C8×C283—21no—
50C4⋊C4×C283—21no—
51C4⋊C4×C283—21no—
52C4⋊C4×C283—21no—
53C4⋊C4×C283D412yes—
54Dic566—22no—
55Dic566—22no—
56Dic566—48no—
57Dic566—41no—
58Dic566—41no—
59C2×Dic283—21no—
60C2×Dic283—21no—
61C2×Dic283—21no—
62C2×Dic283—21no—
63C2×Dic283—22no—
64C2×Dic283—22no—
65C2×Dic283—41no—
66C2×Dic283—41no—
67C2×Dic283—41no—
68C2×Dic283—41no—
69C2×Dic283SD1612yes—
70C2×Dic283Q1612yes—
71C283⋊C8—41no—
72C23×C283C2×D411yescharacteristic
73C2×C4×C283—22no—
74C2×C4×C283—41no—
75C2×C4×C283—41no—
76C2×C4×C283—41no—
77C2×C4×C283—41no—
78C2×C4×C283—41no—
79C2×C4×C283C2×D411yescharacteristic
80C2×C4×C283C2×D411yescharacteristic
81C2×C4×C283C2×Q811yescharacteristic
82C2×C4×C283#4111yescharacteristic
83C2×C4×C283#4111yescharacteristic
84C2×C4×C283#4111yescharacteristic
85C8×C283—41no—
86C22×C283#12011yescharacteristic
87C22×C283C22⋊Q811yescharacteristic
88C22×C283#12411yescharacteristic
89C22×C283C2×SD1611yescharacteristic
90C22×C283C2×Q1611yescharacteristic
91C22×C283AGL(1,ℤ/8)11yescharacteristic
92C22×C283#13711yescharacteristic
93C4×C283—42no—
94C4×C283—81no—
95C4×C283C22⋊Q811yescharacteristic
96C4×C283C22⋊Q811yescharacteristic
97C4×C283#12411yescharacteristic
98C4×C283#12411yescharacteristic
99Dic283—22no—
100Dic283—22no—
101Dic283—42no—
102Dic283—42no—
103C2×C283#39311yescharacteristic
104C2×C283#45011yescharacteristic
105C2×C283#45111yescharacteristic
106C2×C283#47511yescharacteristic
107C2×C283#47711yescharacteristic
108C2×C283#48711yescharacteristic
109C2×C283#48911yescharacteristic
110C283#202411yescharacteristic, complemented, Sylow
111#2024—2831nomaximal, Sylow
112C2×C4⋊Q8—2831no—
113C2×#103—2831no—
114C2×#103—2831no—
115#391—2831no—
116#340—2831no—
117#340D28311yescharacteristic
118#336—2831no—
119C22×Q8—2831no—
120C22×Q8—5661no—
121C22×C8—2831no—
122C22×C8C2×D28311yescharacteristic
123C4⋊Q8—5662no—
124C4⋊Q8—5662no—
125C2×C4⋊C4—2831no—
126C2×C4⋊C4—5661no—
127C2×C4⋊C4C2×D28311yescharacteristic
128C2×C4⋊C4C2×D28311yescharacteristic
129C2×C42—2831no—
130#103—5662no—
131#103—5662no—
132#95—5661no—
133#95—5661no—
134C2×Q8—2832no—
135C2×Q8—5664no—
136C2×Q8—5661no—
137C2×Q8—5661no—
138C2×Q8—5661no—
139C2×Q8—5661no—
140C2×Q8—5661no—
141C2×Q8—11322no—
142C2×Q8—11322no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—2831no—
146C22×C4—2831no—
147C22×C4—5661no—
148C22×C4—5661no—
149C22×C4C22×D28311yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—5661no—
154C2×C8—5661no—
155C2×C8—5661no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—2832no—
160C4⋊C4—5664no—
161C4⋊C4—5661no—
162C4⋊C4D113212yes—
163C42—5661no—
164C42—5661no—
165C23D4×D28311yescharacteristic
166Q8—5662no—
167Q8—5662no—
168Q8—11328no—
169Q8—11321no—
170Q8—11321no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—2832no—
178C2×C4—2832no—
179C2×C4—5661no—
180C2×C4—5661no—
181C2×C4—5661no—
182C2×C4—5661no—
183C2×C4—5662no—
184C2×C4—5662no—
185C2×C4—11321no—
186C2×C4—11321no—
187C2×C4—11321no—
188C2×C4—11321no—
189C2×C4D4×D28311yescharacteristic
190C2×C4C2×D113211yescharacteristic
191C2×C4Q8×D28311yescharacteristic
192C2×C4#1798539111yescharacteristic
193C2×C4#1798539211yescharacteristic
194C2×C4#1798539211yescharacteristic
195C8—41no—
196C8—11321no—
197C22#2258358011yescharacteristic
198C22#2258358811yescharacteristic
199C22#2258359611yescharacteristic
200C22SD16×D28311yescharacteristic
201C22Q16×D28311yescharacteristic
202C22#2258363411yescharacteristic
203C22#2258363911yescharacteristic
204C4—42no—
205C4—81no—
206C4—5662no—
207C4—5662no—
208C4—11322no—
209C4—11322no—
210C4#2258358811yescharacteristic
211C4#2258359011yescharacteristic
212C4#2258359611yescharacteristic
213C4#2258359811yescharacteristic
214C2#3182637811yescharacteristic
215C2#3182661711yescharacteristic
216C2#3182662211yescharacteristic
217C2#3182674911yescharacteristic
218C2#3182675911yescharacteristic
219C2#3182681411yescharacteristic
220C2#3182682311yescharacteristic
subgroupG/H
centerC23D4×D283
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#340×C283C2
socle, prime radicalC23×C283C2×D4
hypercenter#340D283
hypocenterC283#2024
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic566Q8semidirect
110. C283#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#225836092
C2×#225835352
C2×#225835332
#318263702
#318262272
#318262192
#340×C2832
#2024283