Finite Group Databasebeta

← #49131791

Subgroups — #49131791

Order 36224.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#31827096C211yesmaximal, characteristic
2#31827093C211yesmaximal, characteristic
3#31826873C212yesmaximal, complemented
4#31826872C212yesmaximal, complemented
5#31826866C212yesmaximal, complemented
6#31826862C212yesmaximal, complemented
7#31826507C211yesmaximal, characteristic
8#31826501C211yesmaximal, characteristic
9C2×#22583541C211yesmaximal, characteristic
10#31826423C211yesmaximal, characteristic
11#428×C283C211yesmaximal, characteristic
12C22×C283⋊C8C2211yescharacteristic
13#22583610—21no—
14#22583609—21no—
15#22583609C2212yes—
16#22583609C2212yes—
17#22583602—21no—
18#22583601—21no—
19#22583601C2212yes—
20#22583601C2212yes—
21#22583589—22no—
22#22583587—22no—
23#22583569C2211yescharacteristic
24#22583569C2211yescharacteristic
25C2×C4⋊Dic283C2211yescharacteristic
26#22583544C2212yes—
27#22583543C2211yescharacteristic
28#22583543C2211yescharacteristic
29#22583542C2212yes—
30#22583541C2212yes—
31#22583541C2212yes—
32#22583541C2211yescharacteristic
33#22583541C2211yescharacteristic
34#22583520C2212yes—
35C8×Dic283C2212yes—
36C2×M16×C283C2211yescharacteristic
37#117×C283C2211yescharacteristic
38C2×C4⋊C4×C283C2211yescharacteristic
39#107×C283C2212yes—
40#106×C283C2212yes—
41C2×Dic566—22no—
42C2×Dic566—22no—
43C22×Dic283—21no—
44C2×C283⋊C8Q814yescomplemented
45C2×C283⋊C8C2311yescharacteristic
46C2×C283⋊C8C2311yescharacteristic
47C4⋊Dic283—22no—
48C4⋊Dic283—22no—
49C4⋊Dic283—21no—
50C4⋊Dic283D411yescharacteristic
51C4⋊Dic283D411yescharacteristic
52C4⋊Dic283C2312yes—
53C4⋊Dic283C2311yescharacteristic
54C4⋊Dic283C2311yescharacteristic
55Dic283⋊C4—22no—
56Dic283⋊C4—22no—
57Dic283⋊C4—22no—
58Dic283⋊C4—22no—
59C22⋊Dic283—21no—
60C22⋊Dic283D412yes—
61C4×Dic283—21no—
62C4×Dic283C2312yes—
63C22×C4×C283—21no—
64C22×C4×C283C2311yescharacteristic
65M16×C283Q814yescomplemented
66C2×C8×C283C2312yes—
67C4⋊C4×C283—21no—
68C4⋊C4×C283C2311yescharacteristic
69C4⋊C4×C283C2312yes—
70C4⋊C4×C283C2311yescharacteristic
71C22⋊C4×C283—21no—
72C42×C283—21no—
73Dic566—82no—
74Dic566—82no—
75C2×Dic283—21no—
76C2×Dic283—21no—
77C2×Dic283—22no—
78C2×Dic283—81no—
79C2×Dic283C2×D412yes—
80C2×Dic283C2×D412yes—
81C283⋊C8C2×Q812yes—
82C283⋊C8C2×Q812yes—
83C23×C283C2×D411yescharacteristic
84C2×C4×C283—21no—
85C2×C4×C283—21no—
86C2×C4×C283—22no—
87C2×C4×C283—81no—
88C2×C4×C283C2×D411yescharacteristic
89C2×C4×C283C2×Q812yes—
90C2×C4×C283C2×Q812yes—
91C2×C4×C283C2411yescharacteristic
92C8×C283C2×Q814yes—
93C22×C283—21no—
94C22×C283—21no—
95C22×C283C4⋊Q812yes—
96C22×C283C22×D411yescharacteristic
97C4×C283—41no—
98C4×C283—42no—
99C4×C283—41no—
100C4×C283C4⋊Q812yes—
101C4×C283C22×Q811yescharacteristic
102C4×C283C22×Q811yescharacteristic
103Dic283—22no—
104Dic283—22no—
105Dic283—41no—
106Dic283—42no—
107Dic283—41no—
108C2×C283—22no—
109C2×C283C2×C4⋊Q811yescharacteristic
110C2×C283#57711yescharacteristic
111C2×C283#57711yescharacteristic
112C283#316311yescharacteristic, complemented, Sylow
113#3163—2831nomaximal, Sylow
114#544—2831no—
115#544—2831no—
116#501—2832no—
117#501—2832no—
118#499—2832no—
119#498—2832no—
120#428—2831no—
121#428D28311yescharacteristic
122#427—2831no—
123C2×#106—2831no—
124#405—2831no—
125C2×M16C2×D28311yescharacteristic
126C22×C8—2831no—
127C4⋊Q8—2832no—
128C4⋊Q8—2832no—
129C4⋊Q8—5661no—
130C4⋊Q8—5661no—
131#125—2832no—
132#125—2832no—
133#125—5661no—
134#125—5661no—
135C22⋊Q8—5662no—
136C22⋊Q8—5662no—
137#117—2831no—
138#117—2831no—
139#117C2×D28311yescharacteristic
140C2×C4⋊C4—2831no—
141C2×C4⋊C4C2×D28311yescharacteristic
142#107—2832no—
143#107—2831no—
144#107—2831no—
145#107C2×D28312yes—
146#106—2832no—
147#106—2832no—
148#106—2832no—
149#106—2831no—
150#106—2831no—
151#106C2×D28312yes—
152#97—2832no—
153C4×C8—2832no—
154C2×Q8—5662no—
155C2×Q8—5662no—
156C22×C4—21no—
157C22×C4—5661no—
158C22×C4C22×D28311yescharacteristic
159M16Dic56614yescomplemented
160C2×C8—2834no—
161C2×C8—2831no—
162C2×C8—2831no—
163C2×C8C22×D28312yes—
164C4⋊C4—21no—
165C4⋊C4—2831no—
166C4⋊C4—2831no—
167C4⋊C4—2832no—
168C4⋊C4—2831no—
169C4⋊C4—2831no—
170C4⋊C4—5662no—
171C4⋊C4—5662no—
172C4⋊C4—5662no—
173C4⋊C4—5662no—
174C4⋊C4—5662no—
175C4⋊C4—5662no—
176C4⋊C4—5661no—
177C4⋊C4C22×D28312yes—
178C4⋊C4C22×D28311yescharacteristic
179C4⋊C4C22×D28311yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—2832no—
182C22⋊C4—5661no—
183C42—21no—
184C42—2832no—
185C42—5661no—
186C23D4×D28311yescharacteristic
187Q8—22642no—
188Q8—22642no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—2832no—
194C2×C4—2832no—
195C2×C4—5661no—
196C2×C4—5661no—
197C2×C4—5662no—
198C2×C4—22641no—
199C2×C4D4×D28311yescharacteristic
200C2×C4Q8×D28312yes—
201C2×C4C2×Dic56612yes—
202C2×C4C23×D28311yescharacteristic
203C8—2832no—
204C8—2832no—
205C8C2×Dic56614yes—
206C22—21no—
207C22—21no—
208C22#2258360912yes—
209C22C2×D4×D28311yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—5662no—
214C4—5662no—
215C4—11322no—
216C4—11321no—
217C4—11321no—
218C4#2258360912yes—
219C4C2×Q8×D28311yescharacteristic
220C4C22×Dic56611yescharacteristic
221C2—22no—
222C2C2×#2258360911yescharacteristic
223C2#3182727411yescharacteristic
224C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
prime radicalC23×C283C2×D4
hypercenter#428D283
hypocenterC283#3163
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. #31826873C2semidirect
4. #31826872C2semidirect
5. #31826866C2semidirect
6. #31826862C2semidirect
44. C2×C283⋊C8Q8semidirect
65. M16×C283Q8semidirect
112. C283#3163semidirect
159. M16Dic566semidirect

Maximal subgroups

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

subgroupindex
#318270962
#318270932
#318268732
#318268722
#318268662
#318268622
#318265072
#318265012
C2×#225835412
#318264232
#428×C2832
#3163283