Finite Group Databasebeta

← #49131232

Subgroups — #49131232

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#31827041C211yesmaximal, characteristic
2#31826946C211yesmaximal, characteristic
3#31826809C212yesmaximal, complemented
4#31826808C212yesmaximal, complemented
5#31826802C212yesmaximal, complemented
6#31826797C212yesmaximal, complemented
7#31826501C211yesmaximal, characteristic, complemented
8#31826475C211yesmaximal, characteristic, complemented
9C2×#22583533C211yesmaximal, characteristic, complemented
10#31826433C211yesmaximal, characteristic
11#428×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×Dic566C2211yescharacteristic
14C22×C283⋊C8C2211yescharacteristic
15#22583610—21no—
16#22583598—21no—
17#22583594—22no—
18#22583590C2212yes—
19#22583590C2212yes—
20#22583575—21no—
21#22583572—21no—
22C4×C22⋊D283—22no—
23#22583569C2211yescharacteristic
24#22583568C2212yes—
25C4⋊C4×D283C2212yes—
26#22583543C2211yescharacteristic
27#22583543C2211yescharacteristic
28#22583541C2211yescharacteristic
29#22583541C2211yescharacteristic
30#22583535C2212yes—
31#22583535C2212yes—
32#22583533C2212yes—
33#22583533C2212yes—
34(C2×D283)⋊C8C2212yes—
35(C2×D283)⋊C8C2212yes—
36C2×M16×C283C2211yescharacteristic
37#117×C283C2211yescharacteristic
38C2×C4⋊C4×C283C2211yescharacteristic
39#107×C283C2212yes—
40#106×C283C2212yes—
41#17985392—22no—
42#17985392—22no—
43C2×Dic566—21no—
44C2×Dic566—22no—
45C2×Dic566—22no—
46C2×Dic566D411yescharacteristic
47C2×Dic566C2312yes—
48C2×D1132D411yescharacteristic
49C2×C22⋊D283D412yes—
50C22×Dic283—21no—
51C2×C4×D283—22no—
52C2×C4×D283C2312yes—
53C2×C283⋊C8—22no—
54C2×C283⋊C8C2311yescharacteristic
55C2×C283⋊C8C2311yescharacteristic
56C4⋊Dic283—22no—
57C4⋊Dic283—22no—
58C4⋊Dic283C2311yescharacteristic
59C4⋊Dic283C2311yescharacteristic
60Dic283⋊C4—22no—
61C22⋊Dic283—21no—
62(C2×D283)⋊C4—22no—
63(C2×D283)⋊C4—22no—
64(C2×D283)⋊C4—22no—
65C4×Dic283—22no—
66C4×Dic283—21no—
67C22×C4×C283—21no—
68C22×C4×C283C2311yescharacteristic
69M16×C283—22no—
70C2×C8×C283C2312yes—
71C4⋊C4×C283—21no—
72C4⋊C4×C283C2312yes—
73C4⋊C4×C283C2311yescharacteristic
74C4⋊C4×C283C2311yescharacteristic
75C22⋊C4×C283—21no—
76C42×C283—21no—
77C22×D283C2×D412yes—
78Dic566—22no—
79Dic566—22no—
80Dic566—22no—
81Dic566—42no—
82D1132—22no—
83C22⋊D283—24no—
84C2×Dic283—21no—
85C2×Dic283—21no—
86C2×Dic283—22no—
87C2×Dic283—42no—
88C2×Dic283C2×D412yes—
89C4×D283—22no—
90C4×D283—22no—
91C4×D283—82no—
92C283⋊C8—21no—
93C283⋊C8—21no—
94C23×C283C2×D411yescharacteristic
95C2×C4×C283—21no—
96C2×C4×C283—21no—
97C2×C4×C283—22no—
98C2×C4×C283—81no—
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—41no—
112C4×C283—42no—
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#428D28311yescharacteristic, complemented
135#427—2831no—
136#419—2831no—
137C2×#103—2831no—
138#408—2831no—
139C2×#41—2831no—
140C22×Q8—2831no—
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—2832no—
155C2×C4⋊C4C2×D28311yescharacteristic
156#107—2831no—
157#107—2831no—
158#107C2×D28312yes—
159#106—2831no—
160#106—2831no—
161#106C2×D28312yes—
162#103—2832no—
163#103—2832no—
164#103—2832no—
165#103—2832no—
166C22⋊C8—2832no—
167C22⋊C8—2832no—
168#41—5662no—
169#41—5662no—
170C2×Q8—2831no—
171C2×Q8—2832no—
172C2×Q8—5661no—
173C2×Q8—5662no—
174C2×Q8—5662no—
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—21no—
188C4⋊C4—2831no—
189C4⋊C4—2831no—
190C4⋊C4—5662no—
191C4⋊C4—5662no—
192C4⋊C4—5662no—
193C4⋊C4C22×D28312yes—
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—5662no—
207Q8—5662no—
208Q8—5662no—
209Q8—11322no—
210D4—5664no—
211D4—5662no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—81no—
216C2×C4—2832no—
217C2×C4—5662no—
218C2×C4—5662no—
219C2×C4—5661no—
220C2×C4—5661no—
221C2×C4—5662no—
222C2×C4—11322no—
223C2×C4—22642no—
224C2×C4D4×D28311yescharacteristic
225C2×C4#1798539212yes—
226C2×C4#1798539312yes—
227C2×C4C23×D28311yescharacteristic
228C8—22no—
229C8—5661no—
230C8—5661no—
231C22—21no—
232C22—21no—
233C22—5662no—
234C22—5662no—
235C22—5662no—
236C22#2258359512yes—
237C22C2×D4×D28311yescharacteristic
238C4—42no—
239C4—41no—
240C4—41no—
241C4—5662no—
242C4—11322no—
243C4—11321no—
244C4—11321no—
245C4#2258359512yes—
246C4C2×#1798539211yescharacteristic
247C4C2×#1798539311yescharacteristic
248C2—22no—
249C2—11322no—
250C2C2×#2258359511yescharacteristic
251C2#3182727511yescharacteristic
252C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
prime radicalC2×#17985392C22
hypercenter#428D283
hypocenterC283#3097
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #31826809C2semidirect
4. #31826808C2semidirect
5. #31826802C2semidirect
6. #31826797C2semidirect
7. #31826501C2semidirect
8. #31826475C2semidirect
9. C2×#22583533C2semidirect
11. #428×C283C2semidirect
126. C283#3097semidirect
134. #428D283semidirect

Maximal subgroups

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

subgroupindex
#318270412
#318269462
#318268092
#318268082
#318268022
#318267972
#318265012
#318264752
C2×#225835332
#318264332
#428×C2832
#3097283