Finite Group Databasebeta

← #49122131

Subgroups — #49122131

Order 36224.

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

normal subgroups only

This group has 304 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
1C22×#17985382C211yesmaximal, characteristic
2C2×#22583541C211yesmaximal, characteristic
3C2×#22583534C211yesmaximal, characteristic
4C2×#22583533C211yesmaximal, characteristic
5C2×#22583521C211yesmaximal, characteristic
6#31826371C211yesmaximal, characteristic
7#391×C283C211yesmaximal, characteristic
8C22×Dic566C2211yescharacteristic
9C2×#17985382—22no—
10C2×#17985382—22no—
11C2×#17985382C414yescomplemented
12C22×C283⋊C8C2211yescharacteristic
13C22×C283⋊C8C2211yescharacteristic
14C2×C4⋊Dic283C2211yescharacteristic
15#22583541—22no—
16#22583534—22no—
17#22583533—22no—
18#22583521—22no—
19C283⋊#95—22no—
20C22×Q8×C283C2211yescharacteristic
21C2×C4⋊C4×C283C2211yescharacteristic
22C2×C42×C283C2211yescharacteristic
23#95×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—
32#17985382—28no—
33C2×C283⋊C8—21no—
34C2×C283⋊C8—21no—
35C2×C283⋊C8C2×C412yes—
36C2×C283⋊C8D412yes—
37C2×C283⋊C8D412yes—
38C2×C283⋊C8D412yes—
39C4⋊Dic283—21no—
40C4⋊Dic283—21no—
41C2×Q8×C283—21no—
42C2×Q8×C283—21no—
43C2×Q8×C283—21no—
44C2×Q8×C283—21no—
45C2×Q8×C283—21no—
46C2×Q8×C283C2×C412yes—
47C22×C4×C283—21no—
48C22×C4×C283—21no—
49C22×C4×C283D412yes—
50C22×C4×C283C2311yescharacteristic
51C4⋊C4×C283—21no—
52C4⋊C4×C283—21no—
53C42×C283—21no—
54C42×C283—21no—
55Dic566—22no—
56Dic566—22no—
57Dic566—41no—
58Dic566—41no—
59C2×Dic283—22no—
60C2×Dic283—41no—
61C2×Dic283—41no—
62C2×Dic283—41no—
63C2×Dic283—42no—
64C283⋊C8—22no—
65C283⋊C8C22⋊C414yes—
66C23×C283C2×D411yescharacteristic
67Q8×C283—22no—
68Q8×C283—22no—
69Q8×C283—41no—
70Q8×C283—41no—
71C2×C4×C283—22no—
72C2×C4×C283—22no—
73C2×C4×C283—22no—
74C2×C4×C283—22no—
75C2×C4×C283—41no—
76C2×C4×C283—41no—
77C2×C4×C283—41no—
78C2×C4×C283—42no—
79C2×C4×C283C22×C411yescharacteristic
80C2×C4×C283C2×D411yescharacteristic
81C2×C4×C283C2×D411yescharacteristic
82C2×C4×C283C2×D411yescharacteristic
83C2×C4×C283#4111yescharacteristic
84C2×C4×C283#4111yescharacteristic
85C22×C283C4×D411yescharacteristic
86C22×C283#12111yescharacteristic
87C22×C283#12711yescharacteristic
88C22×C283#13712yes—
89C22×C283#13712yes—
90C4×C283—42no—
91C4×C283—42no—
92C4×C283—81no—
93C4×C283C2×C22⋊C411yescharacteristic
94C4×C283C4×D411yescharacteristic
95C4×C283#12111yescharacteristic
96C4×C283#12411yescharacteristic
97Dic283—42no—
98Dic283—81no—
99C2×C283#39011yescharacteristic
100C2×C283#44112yes—
101C2×C283#47012yes—
102C2×C283#49711yescharacteristic
103C2×C283#49711yescharacteristic
104C283#197111yescharacteristic, complemented, Sylow
105#1971—2831nomaximal, Sylow
106C22×Q16—2831no—
107C2×#106—2831no—
108C2×#103—2831no—
109C2×#103—2831no—
110C2×#97—2831no—
111#391—2831no—
112#391D28311yescharacteristic
113C22×Q8—2831no—
114C22×Q8C2×D28311yescharacteristic
115C2×Q16—2834no—
116C2×Q16—5662no—
117C2×Q16—5662no—
118C22×C8—2831no—
119C22×C8—2831no—
120C2×C4⋊C4—2831no—
121C2×C4⋊C4C2×D28311yescharacteristic
122C2×C42C2×D28311yescharacteristic
123#106—5662no—
124#103—5662no—
125#103—5662no—
126#97—5662no—
127#95—22no—
128#95—5662no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—2832no—
135C2×Q8—5661no—
136C2×Q8—5661no—
137C2×Q8—5661no—
138C2×Q8—5661no—
139C2×Q8—5661no—
140C2×Q8C4×D28312yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—5661no—
144C22×C4—5661no—
145C22×C4C22⋊D28312yes—
146C22×C4C22×D28311yescharacteristic
147Q16—5668no—
148C2×C8—2832no—
149C2×C8—2832no—
150C2×C8—2832no—
151C2×C8—2832no—
152C2×C8—5661no—
153C2×C8—5661no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—5661no—
157C4⋊C4—5661no—
158C42—21no—
159C42—21no—
160C23C2×C22⋊D28311yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—5662no—
166Q8—5662no—
167Q8—11321no—
168Q8—11321no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—5662no—
178C2×C4—11321no—
179C2×C4—11321no—
180C2×C4—11321no—
181C2×C4—11322no—
182C2×C4C2×C4×D28311yescharacteristic
183C2×C4D4×D28311yescharacteristic
184C2×C4D4×D28311yescharacteristic
185C2×C4D4×D28311yescharacteristic
186C2×C4#1798539111yescharacteristic
187C2×C4#1798539211yescharacteristic
188C8—2834no—
189C8—5662no—
190C22C4×C22⋊D28311yescharacteristic
191C22#2258358211yescharacteristic
192C22#2258360611yescharacteristic
193C22#2258364012yes—
194C22#2258364112yes—
195C4—42no—
196C4—42no—
197C4—81no—
198C4—11322no—
199C4—22641no—
200C4C22⋊C4×D28311yescharacteristic
201C4(C22⋊D283)⋊C411yescharacteristic
202C4#2258358311yescharacteristic
203C4#2258359611yescharacteristic
204C2#3182636511yescharacteristic
205C2#3182657012yes—
206C2#3182672112yes—
207C2#3182685711yescharacteristic
208C2#3182686011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D283
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
Fitting#391×C283C2
socle, prime radicalC23×C283C2×D4
cosocleC22×C4×C283C23
hypercenter#391D283
hypocenterC283#1971
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#17985382C4semidirect
104. C283#1971semidirect

Maximal subgroups

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

subgroupindex
C22×#179853822
C2×#225835412
C2×#225835342
C2×#225835332
C2×#225835212
#318263712
#391×C2832
#1971283