Finite Group Databasebeta

← #49133923

Subgroups — #49133923

Order 36224.

362 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 123 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#31827200C211yesmaximal, characteristic, complemented
2#31827175C211yesmaximal, characteristic, complemented
3#31826820C212yesmaximal, complemented
4#31826812C211yesmaximal, characteristic, complemented
5#31826766C212yesmaximal, complemented
6#31826761C211yesmaximal, characteristic, complemented
7#31826681C212yesmaximal, complemented
8#31826661C211yesmaximal, characteristic
9C4×#17985381C211yesmaximal, characteristic, complemented
10C4×D2264C212yesmaximal, complemented
11C8×Q8×C283C211yesmaximal, characteristic, complemented
12C2×#17985381C2211yescharacteristic
13C2×D2264C2212yes—
14#22583610C2211yescharacteristic
15#22583607C2211yescharacteristic
16#22583602C2212yes—
17#22583599—21no—
18#22583598C2212yes—
19#22583595—22no—
20#22583588—22no—
21#22583586—21no—
22C4×Dic566C2211yescharacteristic
23C4×D1132C2212yes—
24C4×D1132C2212yes—
25C4×D1132C2211yescharacteristic
26#22583572—21no—
27C4⋊C4×D283—21no—
28#22583544C2212yes—
29#22583542C2211yescharacteristic
30#22583535C2212yes—
31#22583535C2211yescharacteristic
32#22583532C2212yes—
33#22583532C2212yes—
34#22583532C2211yescharacteristic
35#22583532C2212yes—
36C4×Q8×C283C2211yescharacteristic
37C4⋊C8×C283C2211yescharacteristic
38C4⋊C8×C283C2212yes—
39C4×C8×C283C2212yes—
40C4×C8×C283C2211yescharacteristic
41C2×Dic566—21no—
42C2×Dic566C2311yescharacteristic
43C2×D1132—21no—
44C2×D1132—21no—
45C2×D1132C2312yes—
46C2×D1132C2311yescharacteristic
47C2×C4×D283—21no—
48C2×C4×D283—22no—
49C2×C4×D283—22no—
50#17985381—22no—
51D2264—24no—
52C4⋊Dic283—21no—
53C4⋊Dic283C2311yescharacteristic
54C4⋊Dic283C2311yescharacteristic
55C4⋊Dic283C2312yes—
56Dic283⋊C4—22no—
57Dic283⋊C4—21no—
58Dic283⋊C4—22no—
59(C2×D283)⋊C4—22no—
60(C2×D283)⋊C4—22no—
61(C2×D283)⋊C4—21no—
62(C2×D283)⋊C4—22no—
63(C2×D283)⋊C4—22no—
64C4×Dic283—21no—
65C2×Q8×C283D411yescharacteristic
66C2×C8×C283C2312yes—
67C2×C8×C283C2311yescharacteristic
68C2×C8×C283C2311yescharacteristic
69C4⋊C4×C283D411yescharacteristic
70C4⋊C4×C283D412yes—
71C42×C283C2312yes—
72C42×C283C2311yescharacteristic
73C22×D283—21no—
74C22×D283—22no—
75Dic566—21no—
76Dic566—21no—
77Dic566—81no—
78D1132—22no—
79D1132—21no—
80D1132—22no—
81D1132—42no—
82D1132—81no—
83D1132#4112yes—
84C2×Dic283—21no—
85C2×Dic283—21no—
86C2×Dic283—21no—
87C2×Dic283—22no—
88C4×D283—42no—
89C4×D283—82no—
90C4×D283—82no—
91Q8×C283—22no—
92C2×C4×C283C2×D412yes—
93C2×C4×C283C2×D411yescharacteristic
94C2×C4×C283C2×D412yes—
95C2×C4×C283C2×D411yescharacteristic
96C2×C4×C283C2411yescharacteristic
97C8×C283—22no—
98C8×C283—21no—
99C8×C283—21no—
100C22×C283C22×D411yescharacteristic
101C2×D283—22no—
102C2×D283—41no—
103C2×D283—41no—
104C2×D283—42no—
105C2×D283—42no—
106C2×D283—42no—
107C4×C283—22no—
108C4×C283—22no—
109C4×C283—21no—
110C4×C283#13512yes—
111C4×C283C2×#4111yescharacteristic
112C4×C2832−1+411yescharacteristic
113Dic283—41no—
114Dic283—41no—
115Dic283—41no—
116Dic283—42no—
117C2×C283#54811yescharacteristic
118C2×C283C2×#13511yescharacteristic
119C2×C283#57611yescharacteristic
120D283—42no—
121D283—82no—
122C283#336111yescharacteristic, complemented, Sylow
123#3361—2831nomaximal, Sylow
124#559—2831no—
125#554—2831no—
126#488—2832no—
127#486—2831no—
128#478—2832no—
129#477—2831no—
130#463—2832no—
131#459—2831no—
132C8×Q8D28311yescharacteristic, complemented
133C4×SD16—2831no—
134C4×D8—2832no—
135C2×SD16—2831no—
136C2×D8—2832no—
137C4⋊Q8—2831no—
138#127—2831no—
139#125—2832no—
140#124—2832no—
141#124—5661no—
142#123—5662no—
143C22⋊Q8—5661no—
144C22⋊Q8—5662no—
145C4×Q8—2831no—
146C4×Q8C2×D28311yescharacteristic
147C4×D4—2832no—
148C4×D4—2831no—
149C4×D4—2832no—
150C4×D4—5661no—
151C2×C4⋊C4—5661no—
152#107—2832no—
153#106—2831no—
154C4⋊C8C2×D28311yescharacteristic
155C4⋊C8C2×D28312yes—
156#103—2831no—
157#103—2832no—
158#102—2831no—
159#102—2832no—
160#102—2832no—
161#102—2832no—
162C4×C8C2×D28312yes—
163C4×C8C2×D28311yescharacteristic
164C2×Q8—2831no—
165C2×Q8—5661no—
166C2×Q8D113211yescharacteristic
167C2×D4—2831no—
168C2×D4—2832no—
169C2×D4—5661no—
170C2×D4—5661no—
171C22×C4—5661no—
172C22×C4—5662no—
173C22×C4—5662no—
174SD16—5662no—
175D8—5664no—
176C2×C8C22×D28311yescharacteristic
177C2×C8C22×D28311yescharacteristic
178C2×C8C22×D28312yes—
179C4⋊C4—2831no—
180C4⋊C4—2832no—
181C4⋊C4—2831no—
182C4⋊C4—5662no—
183C4⋊C4—5661no—
184C4⋊C4—5662no—
185C4⋊C4—5661no—
186C4⋊C4D113211yescharacteristic
187C4⋊C4D113212yes—
188C22⋊C4—5661no—
189C22⋊C4—5662no—
190C22⋊C4—5662no—
191C22⋊C4—5662no—
192C22⋊C4—5662no—
193C42—5661no—
194C42C22×D28312yes—
195C42C22×D28311yescharacteristic
196C23—5661no—
197C23—5662no—
198Q8—22no—
199Q8—5661no—
200Q8—5661no—
201Q8—22641no—
202D4—2832no—
203D4—5662no—
204D4—5661no—
205D4—5662no—
206D4—11322no—
207D4—22641no—
208C2×C4—5661no—
209C2×C4—5661no—
210C2×C4—5661no—
211C2×C4—5662no—
212C2×C4—11322no—
213C2×C4—22642no—
214C2×C4—22642no—
215C2×C4C2×D113212yes—
216C2×C4C2×D113212yes—
217C2×C4C2×D113211yescharacteristic
218C2×C4C2×D113211yescharacteristic
219C2×C4C23×D28311yescharacteristic
220C8—22no—
221C8—21no—
222C8—21no—
223C22—5662no—
224C22—11321no—
225C22—11321no—
226C22—11322no—
227C22—11322no—
228C22—11322no—
229C22C22×D113211yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—11321no—
234C4—11321no—
235C4—11322no—
236C4—11321no—
237C4#2258363212yes—
238C4#41×D28311yescharacteristic
239C4#2258365811yescharacteristic
240C2—11322no—
241C2—22642no—
242C2#3182712711yescharacteristic
243C2C2×#2258363211yescharacteristic
244C2#3182727011yescharacteristic
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC8×Q8×C283C2
socleC22×C283C22×D4
prime radical#31826661C2
hypercenterC8×Q8D283
hypocenterC283#3361
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
1. #31827200C2semidirect
2. #31827175C2semidirect
3. #31826820C2semidirect
4. #31826812C2semidirect
5. #31826766C2semidirect
6. #31826761C2semidirect
7. #31826681C2semidirect
9. C4×#17985381C2semidirect
10. C4×D2264C2semidirect
11. C8×Q8×C283C2semidirect
122. C283#3361semidirect
132. C8×Q8D283semidirect

Maximal subgroups

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

subgroupindex
#318272002
#318271752
#318268202
#318268122
#318267662
#318267612
#318266812
#318266612
C4×#179853812
C4×D22642
C8×Q8×C2832
#3361283