Finite Group Databasebeta

← #49131228

Subgroups — #49131228

Order 36224.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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#31827039C211yesmaximal, characteristic
2#31826945C211yesmaximal, characteristic
3#31826805C212yesmaximal, complemented
4#31826804C212yesmaximal, complemented
5#31826799C212yesmaximal, complemented
6#31826794C212yesmaximal, complemented
7#31826507C211yesmaximal, characteristic, complemented
8#31826501C211yesmaximal, characteristic, complemented
9C2×#22583534C211yesmaximal, characteristic, complemented
10#31826433C211yesmaximal, characteristic
11#419×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×C283⋊C8C2211yescharacteristic
14#22583608—21no—
15#22583599—21no—
16#22583593—22no—
17#22583586C2212yes—
18#22583586C2212yes—
19C4×Dic566—21no—
20C4×D1132—21no—
21(C22⋊D283)⋊C4—22no—
22#22583569C2211yescharacteristic
23#22583568C2212yes—
24C2×C4⋊Dic283C2211yescharacteristic
25C4⋊C4×D283C2212yes—
26#22583544C2212yes—
27#22583543C2211yescharacteristic
28#22583543C2211yescharacteristic
29#22583542C2212yes—
30#22583541C2211yescharacteristic
31#22583541C2211yescharacteristic
32#22583534C2212yes—
33#22583534C2212yes—
34(C2×D283)⋊C8C2212yes—
35(C2×D283)⋊C8C2212yes—
36C22×Q8×C283C2211yescharacteristic
37C2×M16×C283C2211yescharacteristic
38#117×C283C2211yescharacteristic
39#103×C283C2212yes—
40#103×C283C2212yes—
41#17985392—22no—
42#17985392—22no—
43C2×Dic566D411yescharacteristic
44C2×D1132D411yescharacteristic
45C2×C22⋊D283D412yes—
46C22×Dic283—21no—
47C2×C4×D283—22no—
48C2×C4×D283C2312yes—
49C2×C283⋊C8—22no—
50C2×C283⋊C8C2311yescharacteristic
51C2×C283⋊C8C2311yescharacteristic
52C4⋊Dic283—21no—
53C4⋊Dic283C2312yes—
54C4⋊Dic283C2311yescharacteristic
55C4⋊Dic283C2311yescharacteristic
56Dic283⋊C4—22no—
57Dic283⋊C4—22no—
58Dic283⋊C4—22no—
59C22⋊Dic283—21no—
60(C2×D283)⋊C4—22no—
61(C2×D283)⋊C4—22no—
62(C2×D283)⋊C4—22no—
63C4×Dic283—22no—
64C4×Dic283—21no—
65C2×Q8×C283—21no—
66C2×Q8×C283—22no—
67C2×Q8×C283—22no—
68C2×Q8×C283C2312yes—
69C22×C4×C283—21no—
70C22×C4×C283C2311yescharacteristic
71M16×C283—22no—
72C2×C8×C283C2312yes—
73C4⋊C4×C283C2311yescharacteristic
74C4⋊C4×C283C2311yescharacteristic
75C22⋊C4×C283—21no—
76C42×C283—21no—
77C22×D283C2×D412yes—
78Dic566—22no—
79D1132—22no—
80C22⋊D283—24no—
81C2×Dic283—21no—
82C2×Dic283—21no—
83C2×Dic283—22no—
84C2×Dic283—81no—
85C2×Dic283C2×D412yes—
86C4×D283—22no—
87C4×D283—22no—
88C4×D283—82no—
89C283⋊C8—21no—
90C283⋊C8—21no—
91C23×C283C2×D411yescharacteristic
92Q8×C283—22no—
93Q8×C283—22no—
94Q8×C283—42no—
95C2×C4×C283—21no—
96C2×C4×C283—21no—
97C2×C4×C283—22no—
98C2×C4×C283—42no—
99C2×C4×C283C2×D411yescharacteristic
100C2×C4×C283#4112yes—
101C2×C4×C283#4112yes—
102C2×C4×C283C2411yescharacteristic
103C8×C283—22no—
104C22×C283—21no—
105C22×C283—21no—
106C22×C283#12312yes—
107C22×C283C22×D411yescharacteristic
108C2×D283—22no—
109C2×D283—22no—
110C2×D283—22no—
111C4×C283—42no—
112C4×C283—41no—
113C4×C283—41no—
114C4×C283#12312yes—
115C4×C283C2×#4111yescharacteristic
116C4×C283C2×#4111yescharacteristic
117Dic283—22no—
118Dic283—42no—
119Dic283—41no—
120Dic283—41no—
121C2×C283—22no—
122C2×C283C2×#12311yescharacteristic
123C2×C283#57711yescharacteristic
124C2×C283#57811yescharacteristic
125D283—42no—
126C283#309711yescharacteristic, complemented, Sylow
127#3097—2831nomaximal, Sylow
128#536—2831no—
129#520—2831no—
130#485—2832no—
131#485—2832no—
132#484—2832no—
133#483—2832no—
134#428—2831no—
135#427—2831no—
136#419D28311yescharacteristic, complemented
137C2×#103—2831no—
138#408—2831no—
139C2×#41—2831no—
140C22×Q8C2×D28311yescharacteristic
141C2×M16C2×D28311yescharacteristic
142C22×C8—2831no—
143C4⋊Q8—5661no—
144#124—5661no—
145#123—5662no—
146C22⋊Q8—2832no—
147C22⋊Q8—2832no—
148C4×Q8—5661no—
149C4×D4—5661no—
150C4×D4—5662no—
151#117—2831no—
152#117—2832no—
153#117C2×D28311yescharacteristic
154C2×C4⋊C4—2831no—
155C2×C4⋊C4—2832no—
156#107—2832no—
157#107—2831no—
158#107—2831no—
159#106—2832no—
160#106—2831no—
161#106—2831no—
162#103—2832no—
163#103—2832no—
164#103C2×D28312yes—
165#103C2×D28312yes—
166C22⋊C8—2832no—
167C22⋊C8—2832no—
168#41—5662no—
169#41—5662no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—2831no—
174C2×Q8C22×D28312yes—
175C2×D4—2832no—
176C2×D4—2831no—
177C22×C4—21no—
178C22×C4—2832no—
179C22×C4—5662no—
180C22×C4—5661no—
181C22×C4C22×D28311yescharacteristic
182M16—22no—
183C2×C8—2831no—
184C2×C8—2831no—
185C2×C8—5662no—
186C2×C8C22×D28312yes—
187C4⋊C4—2832no—
188C4⋊C4—2831no—
189C4⋊C4—2831no—
190C4⋊C4—5662no—
191C4⋊C4—5662no—
192C4⋊C4—5662no—
193C4⋊C4—5661no—
194C4⋊C4C22×D28311yescharacteristic
195C4⋊C4C22×D28311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—5662no—
198C22⋊C4—5662no—
199C22⋊C4—5662no—
200C22⋊C4—5661no—
201C42—21no—
202C42—5662no—
203C42—5661no—
204C23—2832no—
205C23D4×D28311yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—42no—
209Q8—5662no—
210D4—5664no—
211D4—5662no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—2832no—
217C2×C4—5662no—
218C2×C4—5662no—
219C2×C4—5661no—
220C2×C4—5661no—
221C2×C4—5662no—
222C2×C4—22642no—
223C2×C4—22641no—
224C2×C4D4×D28311yescharacteristic
225C2×C4#1798539112yes—
226C2×C4#1798539212yes—
227C2×C4C23×D28311yescharacteristic
228C8—22no—
229C8—5661no—
230C8—5661no—
231C22—21no—
232C22—21no—
233C22—5662no—
234C22—5662no—
235C22—5662no—
236C22#2258359212yes—
237C22C2×D4×D28311yescharacteristic
238C4—42no—
239C4—41no—
240C4—41no—
241C4—5662no—
242C4—11322no—
243C4—11321no—
244C4—11321no—
245C4#2258359212yes—
246C4C2×#1798539111yescharacteristic
247C4C2×#1798539211yescharacteristic
248C2—22no—
249C2—11322no—
250C2C2×#2258359211yescharacteristic
251C2#3182727411yescharacteristic
252C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#419×C283C2
socleC22×C283C22×D4
prime radicalC2×#17985392C22
hypercenter#419D283
hypocenterC283#3097
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #31826805C2semidirect
4. #31826804C2semidirect
5. #31826799C2semidirect
6. #31826794C2semidirect
7. #31826507C2semidirect
8. #31826501C2semidirect
9. C2×#22583534C2semidirect
11. #419×C283C2semidirect
126. C283#3097semidirect
136. #419D283semidirect

Maximal subgroups

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

subgroupindex
#318270392
#318269452
#318268052
#318268042
#318267992
#318267942
#318265072
#318265012
C2×#225835342
#318264332
#419×C2832
#3097283