Finite Group Databasebeta

← #49121512

Subgroups — #49121512

Order 36224.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22583575C211yesmaximal, characteristic
2C2×#22583535C211yesmaximal, characteristic
3C2×#22583533C211yesmaximal, characteristic
4#31826361C211yesmaximal, characteristic
5#31826227C211yesmaximal, characteristic
6#31826219C211yesmaximal, characteristic
7C2×#107×C283C211yesmaximal, characteristic
8C22×Dic566C2211yescharacteristic
9C22×C283⋊C8C2211yescharacteristic
10#22583575—22no—
11#22583575—22no—
12C2×C4⋊Dic283C2211yescharacteristic
13C2×Dic283⋊C4—21no—
14C2×Dic283⋊C4—21no—
15C2×C4×Dic283—21no—
16C2×C4×Dic283C2211yescharacteristic
17#22583535C414yescomplemented
18#22583533—22no—
19C283⋊#95—21no—
20C22×C8×C283C2211yescharacteristic
21C2×C4⋊C4×C283C2211yescharacteristic
22C2×C4⋊C4×C283C2211yescharacteristic
23#107×C283—22no—
24C2×Dic566—21no—
25C2×Dic566—21no—
26C2×Dic566—21no—
27C2×Dic566C2×C412yes—
28C2×Dic566D412yes—
29C2×Dic566Q812yes—
30C22×Dic283—21no—
31C22×Dic283—21no—
32C22×Dic283D411yescharacteristic
33C22×Dic283D411yescharacteristic
34C2×C283⋊C8—21no—
35C2×C283⋊C8—21no—
36C2×C283⋊C8—21no—
37C4⋊Dic283—21no—
38C4⋊Dic283C2×C412yes—
39Dic283⋊C4—24no—
40Dic283⋊C4—42no—
41C4×Dic283—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×C283C2×C412yes—
50C4⋊C4×C283—21no—
51C4⋊C4×C283—21no—
52C4⋊C4×C283—21no—
53C4⋊C4×C283D412yes—
54Dic566—22no—
55Dic566—22no—
56Dic566—22no—
57Dic566C4⋊C414yescomplemented
58C2×Dic283—22no—
59C2×Dic283—21no—
60C2×Dic283—22no—
61C2×Dic283—22no—
62C2×Dic283—21no—
63C2×Dic283—22no—
64C2×Dic283—21no—
65C2×Dic283—21no—
66C2×Dic283—21no—
67C2×Dic283—41no—
68C2×Dic283—41no—
69C2×Dic283Q1612yes—
70C283⋊C8—41no—
71C23×C283C2×D411yescharacteristic
72C2×C4×C283—22no—
73C2×C4×C283—41no—
74C2×C4×C283—41no—
75C2×C4×C283—41no—
76C2×C4×C283—41no—
77C2×C4×C283—41no—
78C2×C4×C283C22×C411yescharacteristic
79C2×C4×C283C2×D411yescharacteristic
80C2×C4×C283C2×D411yescharacteristic
81C2×C4×C283C2×Q811yescharacteristic
82C2×C4×C283#4111yescharacteristic
83C2×C4×C283#4111yescharacteristic
84C8×C283—22no—
85C22×C283C4×D411yescharacteristic
86C22×C283#12011yescharacteristic
87C22×C283C22⋊Q811yescharacteristic
88C22×C283C2×Q1611yescharacteristic
89C22×C283#13511yescharacteristic
90C22×C283AGL(1,ℤ/8)11yescharacteristic
91C22×C283#13711yescharacteristic
92C4×C283—42no—
93C4×C283—81no—
94C4×C283C2×C4⋊C411yescharacteristic
95C4×C283C4×D411yescharacteristic
96C4×C283C22⋊Q811yescharacteristic
97C4×C283#12311yescharacteristic
98Dic283—24no—
99Dic283—22no—
100Dic283—42no—
101Dic283—41no—
102C2×C283#38511yescharacteristic
103C2×C283C4×Q1611yescharacteristic
104C2×C283#44011yescharacteristic
105C2×C283#44911yescharacteristic
106C2×C283#45111yescharacteristic
107C2×C283#47711yescharacteristic
108C2×C283#47911yescharacteristic
109C283#186311yescharacteristic, complemented, Sylow
110#1863—2831nomaximal, Sylow
111C2×C4×Q8—2831no—
112C2×#107D28311yescharacteristic
113C2×#103—2831no—
114C2×#103—2831no—
115#389—2831no—
116#340—2831no—
117#336—2831no—
118C22×Q8—2831no—
119C22×C8—2831no—
120C22×C8C2×D28311yescharacteristic
121C4×Q8—5662no—
122C4×Q8—5662no—
123C2×C4⋊C4—2831no—
124C2×C4⋊C4—5661no—
125C2×C4⋊C4—5661no—
126C2×C4⋊C4C2×D28311yescharacteristic
127C2×C4⋊C4C2×D28311yescharacteristic
128C2×C42—2831no—
129C2×C42—5661no—
130#107—22no—
131#103—2834no—
132#103—5662no—
133#95—5661no—
134C2×Q8—2832no—
135C2×Q8—2832no—
136C2×Q8—2832no—
137C2×Q8—5661no—
138C2×Q8—5661no—
139C2×Q8—5661no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—2831no—
143C22×C4—2831no—
144C22×C4—5661no—
145C22×C4—5661no—
146C22×C4C22×D28311yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—5661no—
150C2×C8—5661no—
151C2×C8—5661no—
152C2×C8C4×D28312yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2832no—
157C4⋊C4—5664no—
158C4⋊C4—5661no—
159C4⋊C4—11322no—
160C4⋊C4C22⋊D28312yes—
161C42—5664no—
162C42—5661no—
163C42—5661no—
164C23D4×D28311yescharacteristic
165Q8—2834no—
166Q8—5662no—
167Q8—5662no—
168Q8—5662no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—2832no—
176C2×C4—5662no—
177C2×C4—5661no—
178C2×C4—5662no—
179C2×C4—5662no—
180C2×C4—5661no—
181C2×C4—5662no—
182C2×C4—5661no—
183C2×C4—5661no—
184C2×C4—5661no—
185C2×C4—11321no—
186C2×C4—11321no—
187C2×C4C2×C4×D28311yescharacteristic
188C2×C4D4×D28311yescharacteristic
189C2×C4C2×C22⋊D28311yescharacteristic
190C2×C4Q8×D28311yescharacteristic
191C2×C4#1798539211yescharacteristic
192C2×C4#1798539311yescharacteristic
193C8—22no—
194C8—11321no—
195C22#2258357211yescharacteristic
196C22#2258357811yescharacteristic
197C22#2258358811yescharacteristic
198C22Q16×D28311yescharacteristic
199C22#2258362911yescharacteristic
200C22#2258363411yescharacteristic
201C22#2258364211yescharacteristic
202C4—42no—
203C4—81no—
204C4—5664no—
205C4—5662no—
206C4—11322no—
207C4—11321no—
208C4C4⋊C4×D28311yescharacteristic
209C4C4×C22⋊D28311yescharacteristic
210C4#2258358611yescharacteristic
211C4#2258359511yescharacteristic
212C2#3182633611yescharacteristic
213C2#3182655911yescharacteristic
214C2#3182656411yescharacteristic
215C2#3182660711yescharacteristic
216C2#3182661911yescharacteristic
217C2#3182675911yescharacteristic
218C2#3182677111yescharacteristic
subgroupG/H
centerC23D4×D283
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
FittingC2×#107×C283C2
socle, prime radicalC23×C283C2×D4
cosocleC22×C4×C283C23
hypercenterC2×#107D283
hypocenterC283#1863
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
17. #22583535C4semidirect
57. Dic566C4⋊C4semidirect
109. C283#1863semidirect

Maximal subgroups

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

subgroupindex
C2×#225835752
C2×#225835352
C2×#225835332
#318263612
#318262272
#318262192
C2×#107×C2832
#1863283