Finite Group Databasebeta

← #49122290

Subgroups — #49122290

Order 36224.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#22583641C211yesmaximal, characteristic, complemented
2#31827032C211yesmaximal, characteristic
3C2×#22583538C211yesmaximal, characteristic, complemented
4#31826474C211yesmaximal, characteristic, complemented
5C2×#22583527C211yesmaximal, characteristic, complemented
6#31826225C211yesmaximal, characteristic, complemented
7C2×#124×C283C211yesmaximal, characteristic, complemented
8C2×#17985392C2211yescharacteristic
9#22583641—22no—
10#22583641—22no—
11C2×#17985382—21no—
12C2×#17985379—21no—
13C2×#17985376C2211yescharacteristic
14C2×#17985376C2211yescharacteristic
15#22583606—21no—
16#22583597—21no—
17#22583581—22no—
18#22583579—22no—
19#22583569C2211yescharacteristic
20#22583538—22no—
21#22583534—22no—
22#22583527—22no—
23C22×Q8×C283C2211yescharacteristic
24C22×D4×C283C2211yescharacteristic
25#124×C283—22no—
26#124×C283—22no—
27C2×C22⋊C4×C283—22no—
28C2×C42×C283C2211yescharacteristic
29#17985392—21no—
30#17985392—21no—
31#17985392—21no—
32#17985392—21no—
33C2×Dic566D411yescharacteristic
34C2×D1132D411yescharacteristic
35C2×C22⋊D283—24no—
36C2×C22⋊D283—21no—
37C2×C4×D283—21no—
38#17985382—24no—
39#17985379—24no—
40#17985376—21no—
41#17985376—21no—
42#17985376—21no—
43#17985376D412yescomplemented
44C2×C283⋊C8—21no—
45C2×C283⋊C8—21no—
46C4⋊Dic283D412yes—
47C22⋊Dic283—24no—
48C22⋊Dic283—21no—
49C4×Dic283—21no—
50C24×C283D412yes—
51C2×Q8×C283—21no—
52C2×Q8×C283—21no—
53C2×Q8×C283—21no—
54C2×Q8×C283—21no—
55C2×Q8×C283—21no—
56C2×Q8×C283D412yescomplemented
57C2×D4×C283—21no—
58C2×D4×C283—21no—
59C2×D4×C283—21no—
60C2×D4×C283—21no—
61C2×D4×C283—21no—
62C2×D4×C283—21no—
63C22×C4×C283—21no—
64C22×C4×C283—21no—
65C22×C4×C283C2311yescharacteristic
66C22⋊C4×C283—28no—
67C42×C283—21no—
68C42×C283D412yescomplemented
69C22×D283—21no—
70Dic566—21no—
71Dic566—21no—
72D1132—21no—
73D1132—21no—
74C22⋊D283—41no—
75C22⋊D283—41no—
76C22⋊D283—84no—
77C2×Dic283—22no—
78C2×Dic283—21no—
79C4×D283—41no—
80C4×D283—41no—
81C283⋊C8—22no—
82C283⋊C8—41no—
83C23×C283—22no—
84C23×C283—22no—
85C23×C283—22no—
86C23×C283—44no—
87C23×C283C2×D411yescharacteristic
88Q8×C283—22no—
89Q8×C283—22no—
90Q8×C283—41no—
91Q8×C283—41no—
92D4×C283—41no—
93D4×C283—41no—
94D4×C283—41no—
95D4×C283—41no—
96C2×C4×C283—22no—
97C2×C4×C283—22no—
98C2×C4×C283—42no—
99C2×C4×C283—41no—
100C2×C4×C283—41no—
101C2×C4×C283C2×D411yescharacteristic
102C2×C4×C283C2×D411yescharacteristic
103C2×C4×C283C2×D411yescharacteristic
104C2×C4×C283C2×D411yescharacteristic
105C2×C4×C283C2×D411yescharacteristic
106C2×C4×C283#4111yescharacteristic
107C22×C283—22no—
108C22×C283—44no—
109C22×C283—42no—
110C22×C283—44no—
111C22×C283—82no—
112C22×C283#12011yescharacteristic
113C22×C283#12011yescharacteristic
114C22×C283#12111yescharacteristic
115C2×D283—41no—
116C2×D283—42no—
117C4×C283—42no—
118C4×C283—42no—
119C4×C283#12011yescharacteristic
120C4×C283#12111yescharacteristic
121C4×C283#12111yescharacteristic
122C4×C283#12711yescharacteristic
123Dic283—41no—
124Dic283—42no—
125C2×C283—21no—
126C2×C283—21no—
127C2×C283—82no—
128C2×C283#39211yescharacteristic
129C2×C283#45512yes—
130D283—81no—
131C283#200411yescharacteristic, complemented, Sylow
132#2004—2831nomaximal, Sylow
133C2×#137—2831no—
134#535—2831no—
135C2×#124D28311yescharacteristic, complemented
136C2×#104—2831no—
137#419—2831no—
138C2×#100—2831no—
139#339—2831no—
140C2×#41—2831no—
141C22×Q8C2×D28311yescharacteristic
142C22×D4C2×D28311yescharacteristic
143#137—5662no—
144#137—5662no—
145C2×Q16—5661no—
146C2×SD16—5661no—
147C2×M16—2831no—
148C2×M16—2831no—
149#127—5661no—
150#124—22no—
151#124—22no—
152#124—5661no—
153#121—5662no—
154#120—5662no—
155#117—2831no—
156C2×C22⋊C4—22no—
157C2×C42C2×D28311yescharacteristic
158#104—5662no—
159#103—5662no—
160#100—5662no—
161C24C22⋊D28312yes—
162#41—5661no—
163#41—5661no—
164#41—5661no—
165#41—5661no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—2831no—
172C2×Q8C22⋊D28312yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—2831no—
180C2×D4—5661no—
181C2×D4—5664no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—5661no—
185C22×C4C22×D28311yescharacteristic
186Q16—5664no—
187SD16—5664no—
188M16—2832no—
189M16—5661no—
190M16—5661no—
191M16—5661no—
192C2×C8—5661no—
193C2×C8—5661no—
194C4⋊C4—2832no—
195C22⋊C4—28no—
196C22⋊C4—5664no—
197C22⋊C4—5661no—
198C42—21no—
199C42—5661no—
200C42C22⋊D28312yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—5661no—
206C23C2×C22⋊D28311yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—5661no—
212Q8—5661no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—5661no—
218D4—5661no—
219D4—11321no—
220D4—11321no—
221D4—22644no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—5662no—
228C2×C4—5661no—
229C2×C4—11321no—
230C2×C4—11321no—
231C2×C4D4×D28311yescharacteristic
232C2×C4D4×D28311yescharacteristic
233C2×C4D4×D28311yescharacteristic
234C2×C4C2×C22⋊D28311yescharacteristic
235C2×C4C2×C22⋊D28311yescharacteristic
236C2×C4#1798539111yescharacteristic
237C8—5662no—
238C8—11321no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—11321no—
245C22—11322no—
246C22#2258357811yescharacteristic
247C22#2258357911yescharacteristic
248C22#2258358211yescharacteristic
249C4—42no—
250C4—42no—
251C4—11321no—
252C4—11322no—
253C4#2258357811yescharacteristic
254C4#2258358111yescharacteristic
255C4#2258358211yescharacteristic
256C4#2258360611yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—22641no—
261C2#3182637311yescharacteristic
262C2#3182664212yes—
subgroupG/H
centerC22#22583578
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
FittingC2×#124×C283C2
socleC22×C283#120
prime radical#31827032C2
hypercenterC2×#124D283
hypocenterC283#2004
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#22583641C2semidirect
3. C2×#22583538C2semidirect
4. #31826474C2semidirect
5. C2×#22583527C2semidirect
6. #31826225C2semidirect
7. C2×#124×C283C2semidirect
43. #17985376D4semidirect
56. C2×Q8×C283D4semidirect
68. C42×C283D4semidirect
131. C283#2004semidirect
135. C2×#124D283semidirect
172. C2×Q8C22⋊D283semidirect
200. C42C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836412
#318270322
C2×#225835382
#318264742
C2×#225835272
#318262252
C2×#124×C2832
#2004283