Finite Group Databasebeta

← #49122293

Subgroups — #49122293

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×#22583643C211yesmaximal, characteristic, complemented
2#31827034C211yesmaximal, characteristic
3C2×#22583598C211yesmaximal, characteristic, complemented
4C2×#22583538C211yesmaximal, characteristic, complemented
5#31826473C211yesmaximal, characteristic, complemented
6C2×#22583526C211yesmaximal, characteristic, complemented
7#339×C283C211yesmaximal, characteristic, complemented
8C2×#17985392C2211yescharacteristic
9C22×Dic566C2211yescharacteristic
10C22×D1132C2211yescharacteristic
11#22583643—22no—
12#22583643—22no—
13C2×Dic1132—21no—
14C2×#17985381—21no—
15C2×#17985376C2211yescharacteristic
16#22583607—21no—
17#22583598—21no—
18#22583598—22no—
19#22583598—22no—
20#22583585—22no—
21#22583580—22no—
22C2×(C2×D283)⋊C4—22no—
23#22583538—22no—
24#22583533—22no—
25#22583526—22no—
26C2×M16×C283C2211yescharacteristic
27#117×C283C2211yescharacteristic
28C2×C42×C283C2211yescharacteristic
29C23×D283D412yes—
30#17985392—21no—
31#17985392—21no—
32#17985392—21no—
33#17985392—21no—
34C2×Dic566—21no—
35C2×Dic566—21no—
36C2×Dic566—21no—
37C2×Dic566—21no—
38C2×Dic566—21no—
39C2×Dic566D411yescharacteristic
40C2×Dic566D412yescomplemented
41C2×D1132—24no—
42C2×D1132—21no—
43C2×D1132—21no—
44C2×D1132—21no—
45C2×D1132—21no—
46C2×D1132—21no—
47C2×D1132—21no—
48C2×D1132D411yescharacteristic
49C2×C22⋊D283—21no—
50C22×Dic283—21no—
51C2×C4×D283—21no—
52Dic1132—24no—
53#17985381—24no—
54#17985376—21no—
55#17985376—21no—
56C2×C283⋊C8—21no—
57(C2×D283)⋊C4—28no—
58(C2×D283)⋊C4—24no—
59C22×C4×C283—21no—
60C22×C4×C283C2311yescharacteristic
61M16×C283—21no—
62M16×C283D412yescomplemented
63C2×C8×C283—21no—
64C4⋊C4×C283D412yes—
65C22⋊C4×C283—21no—
66C42×C283—21no—
67C42×C283—21no—
68C42×C283D412yescomplemented
69C22×D283—22no—
70C22×D283—22no—
71C22×D283—21no—
72C22×D283—22no—
73C22×D283—44no—
74Dic566—22no—
75Dic566—21no—
76Dic566—22no—
77Dic566—21no—
78Dic566—41no—
79Dic566—41no—
80D1132—21no—
81D1132—21no—
82D1132—41no—
83D1132—41no—
84D1132—41no—
85D1132—41no—
86D1132—84no—
87C22⋊D283—41no—
88C22⋊D283—41no—
89C2×Dic283—21no—
90C2×Dic283—22no—
91C2×Dic283—41no—
92C2×Dic283—41no—
93C4×D283—41no—
94C4×D283—41no—
95C283⋊C8—41no—
96C23×C283C2×D411yescharacteristic
97C2×C4×C283—22no—
98C2×C4×C283—22no—
99C2×C4×C283—42no—
100C2×C4×C283C2×D411yescharacteristic
101C2×C4×C283C2×D411yescharacteristic
102C2×C4×C283C2×D411yescharacteristic
103C2×C4×C283C2×D411yescharacteristic
104C2×C4×C283C2×D411yescharacteristic
105C2×C4×C283#4111yescharacteristic
106C8×C283—22no—
107C22×C283—22no—
108C22×C283#12011yescharacteristic
109C22×C283#12011yescharacteristic
110C22×C283#12111yescharacteristic
111C2×D283—44no—
112C2×D283—41no—
113C2×D283—42no—
114C2×D283—42no—
115C2×D283—44no—
116C2×D283—82no—
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#39211yescharacteristic
128C2×C283#45512yes—
129D283—81no—
130D283—82no—
131C283#200411yescharacteristic, complemented, Sylow
132#2004—2831nomaximal, Sylow
133C2×#137—2831no—
134#535—2831no—
135C2×#124—2831no—
136C2×#104—2831no—
137#419—2831no—
138C2×#100—2831no—
139#339D28311yescharacteristic, complemented
140C2×#41—2831no—
141C22×Q8—2831no—
142C22×D4—2831no—
143#137—5662no—
144#137—5662no—
145C2×Q16—5661no—
146C2×SD16—5661no—
147C2×M16—2831no—
148C2×M16C2×D28311yescharacteristic
149#127—5661no—
150#124—5662no—
151#124—5661no—
152#124—5662no—
153#121—5662no—
154#120—5662no—
155#117C2×D28311yescharacteristic
156C2×C22⋊C4—5662no—
157C2×C42C2×D28311yescharacteristic
158#104—5662no—
159#103—5662no—
160#100—5662no—
161C24—2832no—
162#41—5661no—
163#41—5661no—
164#41—5661no—
165#41—5661no—
166C2×Q8—2831no—
167C2×Q8—2832no—
168C2×Q8—5661no—
169C2×Q8—5661no—
170C2×Q8—5661no—
171C2×Q8—5661no—
172C2×Q8—5661no—
173C2×D4—2831no—
174C2×D4—5664no—
175C2×D4—5661no—
176C2×D4—5661no—
177C2×D4—5661no—
178C2×D4—5661no—
179C2×D4—5661no—
180C2×D4—5661no—
181C2×D4—5661no—
182C22×C4—21no—
183C22×C4—5661no—
184C22×C4—5661no—
185C22×C4C22×D28311yescharacteristic
186Q16—5664no—
187SD16—5664no—
188M16—21no—
189M16—5661no—
190M16—5661no—
191M16D113212yescomplemented
192C2×C8—21no—
193C2×C8—5661no—
194C4⋊C4D113212yes—
195C22⋊C4—21no—
196C22⋊C4—5668no—
197C22⋊C4—5664no—
198C42—21no—
199C42—21no—
200C42D113212yescomplemented
201C23—5662no—
202C23—5662no—
203C23—5661no—
204C23—5662no—
205C23—11324no—
206C23D4×D28311yescharacteristic
207Q8—5662no—
208Q8—5661no—
209Q8—5662no—
210Q8—5661no—
211Q8—11321no—
212Q8—11321no—
213D4—5661no—
214D4—5661no—
215D4—11321no—
216D4—11321no—
217D4—11321no—
218D4—11321no—
219D4—11321no—
220D4—11321no—
221D4—22644no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—5662no—
226C2×C4—5661no—
227C2×C4—11321no—
228C2×C4—11321no—
229C2×C4—11321no—
230C2×C4—11321no—
231C2×C4D4×D28311yescharacteristic
232C2×C4D4×D28311yescharacteristic
233C2×C4C2×D113211yescharacteristic
234C2×C4C2×D113211yescharacteristic
235C2×C4C2×D113211yescharacteristic
236C2×C4#1798539311yescharacteristic
237C8—22no—
238C8—11321no—
239C22—22no—
240C22—11324no—
241C22—11321no—
242C22—11322no—
243C22—11322no—
244C22—11324no—
245C22—22642no—
246C22#2258358011yescharacteristic
247C22#2258358011yescharacteristic
248C22#2258358511yescharacteristic
249C4—42no—
250C4—42no—
251C4—11321no—
252C4—11322no—
253C4#2258358011yescharacteristic
254C4#2258358511yescharacteristic
255C4#2258358511yescharacteristic
256C4#2258360711yescharacteristic
257C2—21no—
258C2—21no—
259C2—22642no—
260C2—22641no—
261C2#3182637511yescharacteristic
262C2#3182664412yes—
subgroupG/H
centerC22#22583580
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#339×C283C2
socleC22×C283#120
prime radical#31827034C2
hypercenter#339D283
hypocenterC283#2004
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#22583643C2semidirect
3. C2×#22583598C2semidirect
4. C2×#22583538C2semidirect
5. #31826473C2semidirect
6. C2×#22583526C2semidirect
7. #339×C283C2semidirect
40. C2×Dic566D4semidirect
62. M16×C283D4semidirect
68. C42×C283D4semidirect
131. C283#2004semidirect
139. #339D283semidirect
191. M16D1132semidirect
200. C42D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836432
#318270342
C2×#225835982
C2×#225835382
#318264732
C2×#225835262
#339×C2832
#2004283