Finite Group Databasebeta

← #49133188

Subgroups — #49133188

Order 36224.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#31827181C211yesmaximal, characteristic, complemented
2#31827118C211yesmaximal, characteristic, complemented
3#31826805C212yesmaximal, complemented
4#31826752C211yesmaximal, characteristic, complemented
5#31826697C212yesmaximal, complemented
6#31826667C211yesmaximal, characteristic, complemented
7#31826613C212yesmaximal, complemented
8C4×D2264C211yesmaximal, characteristic, complemented
9C8×D1132C211yesmaximal, characteristic, complemented
10#31826460C212yesmaximal, complemented
11#487×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2212yes—
13C2×#17985379C2212yes—
14C2×D2264C2211yescharacteristic
15C2×C8×D283C2212yes—
16#22583599—22no—
17#22583595—22no—
18#22583586C2212yes—
19#22583585—21no—
20#22583585—21no—
21#22583585C2212yes—
22C4×D1132C2211yescharacteristic
23C4×D1132C2211yescharacteristic
24C4×D1132C2211yescharacteristic
25#22583568—21no—
26#22583568C2212yes—
27C4⋊C4×D283—21no—
28#22583544C2211yescharacteristic
29#22583541C2212yes—
30C4⋊(C283⋊C8)C2211yescharacteristic
31#22583534C2212yes—
32#22583532C2211yescharacteristic
33#22583532C2211yescharacteristic
34#22583530C2212yes—
35(C2×D283)⋊C8C2212yes—
36C4⋊Q8×C283C2211yescharacteristic
37C4⋊Q8×C283C2211yescharacteristic
38#103×C283C2212yes—
39#103×C283C2212yes—
40C4×C8×C283C2211yescharacteristic
41#17985393—24no—
42#17985393—42no—
43C2×D1132—22no—
44C2×D1132—22no—
45C2×D1132D411yescharacteristic
46C2×D1132C2311yescharacteristic
47C2×D1132C2311yescharacteristic
48C2×C4×D283—22no—
49C2×C4×D283—21no—
50C2×C4×D283—21no—
51C2×C4×D283C2312yes—
52#17985379—42no—
53D2264—22no—
54C2×C283⋊C8C2312yes—
55C8×D283—42no—
56C4⋊Dic283D411yescharacteristic
57C4⋊Dic283C2311yescharacteristic
58C4⋊Dic283C2311yescharacteristic
59Dic283⋊C4—22no—
60(C2×D283)⋊C4—22no—
61(C2×D283)⋊C4—22no—
62(C2×D283)⋊C4—22no—
63(C2×D283)⋊C4—21no—
64(C2×D283)⋊C4—21no—
65(C2×D283)⋊C4D412yes—
66C4×Dic283—22no—
67C2×Q8×C283—21no—
68C2×Q8×C283C2312yes—
69C2×C8×C283C2311yescharacteristic
70C2×C8×C283C2311yescharacteristic
71C4⋊C4×C283—21no—
72C4⋊C4×C283—21no—
73C4⋊C4×C283—21no—
74C4⋊C4×C283C2312yes—
75C42×C283C2311yescharacteristic
76C22×D283—21no—
77C22×D283—21no—
78C22×D283C2×D412yes—
79D1132—22no—
80D1132—21no—
81D1132—21no—
82D1132—21no—
83D1132—44no—
84D1132—82no—
85D1132#4112yes—
86C2×Dic283—21no—
87C2×Dic283—21no—
88C2×Dic283C2×D412yes—
89C4×D283—22no—
90C4×D283—22no—
91C4×D283—44no—
92C4×D283—42no—
93C4×D283—42no—
94C283⋊C8—22no—
95Q8×C283—22no—
96Q8×C283—22no—
97Q8×C283—81no—
98C2×C4×C283—22no—
99C2×C4×C283—22no—
100C2×C4×C283C2×D411yescharacteristic
101C2×C4×C283C2×D411yescharacteristic
102C2×C4×C283C2411yescharacteristic
103C8×C283—21no—
104C8×C283—21no—
105C22×C283C22×D411yescharacteristic
106C2×D283—22no—
107C2×D283—22no—
108C2×D283—22no—
109C2×D283—22no—
110C2×D283—41no—
111C2×D283—41no—
112C2×D283—41no—
113C2×D283—41no—
114C2×D283—41no—
115C4×C283—21no—
116C4×C283—42no—
117C4×C283—42no—
118C4×C283#13512yes—
119C4×C283C2×#4111yescharacteristic
120C4×C283H(5,𝔽2)11yescharacteristic
121Dic283—22no—
122Dic283—41no—
123Dic283—41no—
124C2×C283#54611yescharacteristic
125C2×C283C2×#13511yescharacteristic
126C2×C283#57711yescharacteristic
127D283—42no—
128D283—42no—
129D283—81no—
130C283#330311yescharacteristic, complemented, Sylow
131#3303—2831nomaximal, Sylow
132#555—2831no—
133#547—2831no—
134#487D28311yescharacteristic, complemented
135#485—2832no—
136#476—2831no—
137#465—2832no—
138#461—2831no—
139#449—2832no—
140C4×D8—2831no—
141C8×D4—2831no—
142#416—2832no—
143C2×#41—2832no—
144C2×SD16—2832no—
145C2×D8—2831no—
146C22×C8—2832no—
147C4⋊Q8C2×D28311yescharacteristic
148C4⋊Q8C2×D28311yescharacteristic
149#124—5662no—
150#123—5662no—
151C22⋊Q8—2832no—
152#121—2832no—
153#121—5661no—
154#121—5661no—
155C4×D4—2831no—
156C4×D4—2831no—
157C4×D4—2831no—
158#117—2832no—
159#117—5661no—
160C2×C4⋊C4—5661no—
161#107—2831no—
162#106—2832no—
163C4⋊C8—2831no—
164#103—2832no—
165#103C2×D28312yes—
166#103C2×D28312yes—
167#102—2831no—
168#102—2831no—
169#102—2832no—
170C22⋊C8—2832no—
171C4×C8C2×D28311yescharacteristic
172#41—5664no—
173#41—11322no—
174C2×Q8—21no—
175C2×Q8C22×D28312yes—
176C2×D4—2831no—
177C2×D4—2831no—
178C2×D4—2831no—
179C2×D4—5662no—
180C2×D4—5662no—
181C22×C4—2832no—
182C22×C4—5662no—
183C22×C4—5661no—
184C22×C4—5661no—
185SD16—11322no—
186D8—5662no—
187C2×C8—2832no—
188C2×C8—11322no—
189C2×C8C22×D28311yescharacteristic
190C2×C8C22×D28311yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—2831no—
195C4⋊C4—2831no—
196C4⋊C4—2831no—
197C4⋊C4—5662no—
198C4⋊C4C22×D28312yes—
199C22⋊C4—2832no—
200C22⋊C4—5662no—
201C22⋊C4—5662no—
202C22⋊C4—5662no—
203C22⋊C4—5661no—
204C22⋊C4—5661no—
205C42—5662no—
206C42C22×D28311yescharacteristic
207C23—2832no—
208C23—5661no—
209C23—5661no—
210Q8—22no—
211Q8—22no—
212Q8—81no—
213D4—2832no—
214D4—5662no—
215D4—5661no—
216D4—5661no—
217D4—5661no—
218D4—11324no—
219D4—22642no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—2832no—
223C2×C4—5662no—
224C2×C4—5662no—
225C2×C4—5661no—
226C2×C4—5661no—
227C2×C4—11324no—
228C2×C4—11322no—
229C2×C4—11322no—
230C2×C4D4×D28311yescharacteristic
231C2×C4D4×D28311yescharacteristic
232C2×C4C23×D28311yescharacteristic
233C8—21no—
234C8—21no—
235C8—5662no—
236C22—5662no—
237C22—5662no—
238C22—5662no—
239C22—5662no—
240C22—11321no—
241C22—11321no—
242C22—11321no—
243C22—11321no—
244C22—11321no—
245C22C2×D4×D28311yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—5662no—
250C4—11321no—
251C4—11321no—
252C4#2258363312yes—
253C4#41×D28311yescharacteristic
254C4#2258365611yescharacteristic
255C2—11322no—
256C2—11322no—
257C2—22641no—
258C2#3182710811yescharacteristic
259C2C2×#2258363311yescharacteristic
260C2#3182727111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#487×C283C2
socleC22×C283C22×D4
hypercenter#487D283
hypocenterC283#3303
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #31827181C2semidirect
2. #31827118C2semidirect
3. #31826805C2semidirect
4. #31826752C2semidirect
5. #31826697C2semidirect
6. #31826667C2semidirect
7. #31826613C2semidirect
8. C4×D2264C2semidirect
9. C8×D1132C2semidirect
10. #31826460C2semidirect
11. #487×C283C2semidirect
130. C283#3303semidirect
134. #487D283semidirect

Maximal subgroups

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

subgroupindex
#318271812
#318271182
#318268052
#318267522
#318266972
#318266672
#318266132
C4×D22642
C8×D11322
#318264602
#487×C2832
#3303283