Finite Group Databasebeta

← #49120583

Subgroups — #49120583

Order 36224.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#31827209C211yesmaximal, characteristic
2#31826833C211yesmaximal, characteristic, complemented
3#31826827C211yesmaximal, characteristic, complemented
4#31826206C211yesmaximal, characteristic, complemented
5#31826205C211yesmaximal, characteristic, complemented
6#31826204C211yesmaximal, characteristic, complemented
7#489×C283C211yesmaximal, characteristic, complemented
8#22583602C2211yescharacteristic
9#22583598C2211yescharacteristic
10#22583594—21no—
11#22583592—21no—
12#22583587—21no—
13#22583586—21no—
14#22583583—21no—
15#22583582—21no—
16#22583534—21no—
17#22583534—21no—
18#22583533—21no—
19#22583530—21no—
20#22583521C2211yescharacteristic
21#22583521C2211yescharacteristic
22C4⋊Q8×C283C2211yescharacteristic
23#124×C283C2211yescharacteristic
24#103×C283—21no—
25#102×C283—21no—
26#97×C283C2211yescharacteristic
27C2×Dic566D411yescharacteristic
28C2×D1132D411yescharacteristic
29C2×C22⋊D283—21no—
30C2×C22⋊D283—21no—
31C22×Dic283—21no—
32C2×C4×D283—21no—
33C2×C283⋊C8—21no—
34C2×C283⋊C8—21no—
35C4⋊Dic283—21no—
36C4⋊Dic283D411yescharacteristic
37C4⋊Dic283D411yescharacteristic
38Dic283⋊C4—21no—
39Dic283⋊C4—21no—
40Dic283⋊C4—21no—
41C22⋊Dic283—21no—
42(C2×D283)⋊C4—21no—
43(C2×D283)⋊C4—21no—
44(C2×D283)⋊C4—21no—
45C2×Q8×C283—21no—
46C2×Q8×C283D411yescharacteristic
47C2×D4×C283D411yescharacteristic
48C2×C8×C283—21no—
49C4⋊C4×C283—21no—
50C4⋊C4×C283—21no—
51C22⋊C4×C283—21no—
52C22⋊C4×C283—21no—
53C42×C283C2311yescharacteristic
54C22×D283—21no—
55Dic566—41no—
56D1132—41no—
57C22⋊D283—81no—
58C22⋊D283—81no—
59C2×Dic283—21no—
60C2×Dic283—21no—
61C2×Dic283—21no—
62C2×Dic283—81no—
63C4×D283—81no—
64C283⋊C8—41no—
65C283⋊C8—41no—
66C23×C283—21no—
67Q8×C283—41no—
68Q8×C283—42no—
69D4×C283—41no—
70C2×C4×C283—21no—
71C2×C4×C283—41no—
72C2×C4×C283C2×D411yescharacteristic
73C2×C4×C283C2×D411yescharacteristic
74C2×C4×C283C2×D411yescharacteristic
75C8×C283—41no—
76C22×C283—41no—
77C22×C283—41no—
78C22×C283—41no—
79C22×C283#12011yescharacteristic
80C2×D283—41no—
81C2×D283—41no—
82C2×D283—41no—
83C4×C283—21no—
84C4×C283—21no—
85C4×C283—21no—
86C4×C283—41no—
87C4×C283—81no—
88Dic283—41no—
89Dic283—41no—
90Dic283—41no—
91C2×C283—81no—
92C2×C283#45411yescharacteristic
93C2×C283#45511yescharacteristic
94C2×C283#45511yescharacteristic
95D283—81no—
96C283#166211yescharacteristic, complemented, Sylow
97#1662—2831nomaximal, Sylow
98#562—2831no—
99#491—2831no—
100#489—2831no—
101#489D28311yescharacteristic, complemented
102#330—2831no—
103#329—2831no—
104#329—2831no—
105C4⋊Q8C2×D28311yescharacteristic
106#125—2831no—
107#124—2831no—
108#124C2×D28311yescharacteristic
109#123—5661no—
110#123—5661no—
111C22⋊Q8—5661no—
112C22⋊Q8—5661no—
113#121—5661no—
114#121—5661no—
115#103—21no—
116#103—5661no—
117#103—5661no—
118#103—5661no—
119#102—21no—
120#102—5661no—
121#97—2831no—
122#97—2831no—
123#97C2×D28311yescharacteristic
124C2×Q8—21no—
125C2×Q8—2831no—
126C2×Q8C22⋊D28311yescharacteristic
127C2×D4—2831no—
128C2×D4—5661no—
129C2×D4—5661no—
130C2×D4C22⋊D28311yescharacteristic
131C22×C4—5661no—
132C22×C4—5661no—
133C2×C8—21no—
134C2×C8—5661no—
135C2×C8—5661no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—2831no—
139C4⋊C4—2831no—
140C4⋊C4—5661no—
141C4⋊C4—5661no—
142C4⋊C4—5661no—
143C4⋊C4—5661no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—5661no—
147C22⋊C4—5661no—
148C22⋊C4—5661no—
149C22⋊C4—5661no—
150C42C22×D28311yescharacteristic
151C23—21no—
152C23—5661no—
153Q8—41no—
154Q8—42no—
155Q8—11321no—
156D4—41no—
157D4—11321no—
158D4—22641no—
159D4—22641no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—5661no—
163C2×C4—5661no—
164C2×C4—5661no—
165C2×C4—22641no—
166C2×C4—22641no—
167C2×C4D4×D28311yescharacteristic
168C2×C4D4×D28311yescharacteristic
169C2×C4C2×C22⋊D28311yescharacteristic
170C8—41no—
171C8—11321no—
172C8—11321no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—11321no—
177C22—11321no—
178C22—11321no—
179C22#2258357811yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—81no—
185C4—11321no—
186C4—11321no—
187C4—11321no—
188C2—81no—
189C2—22641no—
190C2#3182663611yescharacteristic
191C2#3182664111yescharacteristic
192C2#3182664211yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#489×C283C2
socleC22×C283#120
prime radical#31827209C2
hypercenter#489D283
hypocenterC283#1662
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #31826833C2semidirect
3. #31826827C2semidirect
4. #31826206C2semidirect
5. #31826205C2semidirect
6. #31826204C2semidirect
7. #489×C283C2semidirect
96. C283#1662semidirect
101. #489D283semidirect

Maximal subgroups

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

subgroupindex
#318272092
#318268332
#318268272
#318262062
#318262052
#318262042
#489×C2832
#1662283