Finite Group Databasebeta

← #49129742

Subgroups — #49129742

Order 36224.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#22583657C211yesmaximal, characteristic
2#31827231C212yesmaximal, complemented
3#31826449C211yesmaximal, characteristic, complemented
4#31826446C214yesmaximal, complemented
5C2×#22583528C212yesmaximal, complemented
6#101×D283C214yesmaximal, complemented
7#413×C283C211yesmaximal, characteristic, complemented
8#22583657—24no—
9#22583657—24no—
10C2×#17985392C412yes—
11C2×#17985392C2211yescharacteristic
12C2×#17985391C414yes—
13C2×#17985391C2212yes—
14C22×Dic566C412yes—
15C22×Dic566C2211yescharacteristic
16C2×Q8×D283C2212yes—
17#22583618—24no—
18#22583616—24no—
19C2×#17985376C2212yes—
20M16×D283C2218yes—
21#22583529C2212yes—
22#22583528C2218yes—
23#22583527C2212yes—
24C2×#41×C283C2211yescharacteristic
25C2×M16×C283C2212yes—
26#101×C283C2212yes—
27#100×C283C2212yes—
28#17985392—24no—
29#17985392—24no—
30#17985392D414yes—
31#17985392D414yes—
32#17985391—216no—
33#17985391—28no—
34C2×Dic566—24no—
35C2×Dic566—24no—
36C2×Dic566—22no—
37C2×Dic566—22no—
38C2×Dic566C2×C414yes—
39C2×Dic566C2×C412yes—
40C2×Dic566C2×C411yescharacteristic
41C2×Dic566C2312yes—
42Q8×D283—28no—
43C2×D1132C2×C411yescharacteristic
44C2×C22⋊D283C2×C414yes—
45C2×C22⋊D283C2×C412yes—
46C22×Dic283C2×C414yes—
47C22×Dic283C2×C412yes—
48C2×C4×D283C2×C414yes—
49C2×C4×D283C2312yes—
50#17985376—22no—
51#17985376C2314yes—
52#17985375—28no—
53C2×C283⋊C8—22no—
54C8×D283—28no—
55#41×C283—22no—
56#41×C283—22no—
57C2×Q8×C283C2311yescharacteristic
58C2×D4×C283C2×C412yes—
59C2×D4×C283C2311yescharacteristic
60C22×C4×C283C2×C412yes—
61C22×C4×C283C2311yescharacteristic
62M16×C283—22no—
63M16×C283C2314yes—
64C2×C8×C283—22no—
65C22×D283C22×C412yes—
66Dic566—22no—
67Dic566—22no—
68Dic566—24no—
69Dic566—28no—
70Dic566C22⋊C412yes—
71Dic566C22⋊C412yes—
72D1132C22⋊C412yes—
73D1132C22⋊C412yes—
74C22⋊D283—28no—
75C22⋊D283C22⋊C418yes—
76C2×Dic283—24no—
77C2×Dic283—28no—
78C2×Dic283C22×C414yes—
79C2×Dic283C22×C412yes—
80C4×D283—28no—
81C4×D283C2×D418yes—
82C283⋊C8—24no—
83C23×C283C22×C412yes—
84C23×C283C22×C411yescharacteristic
85Q8×C283—22no—
86D4×C283—24no—
87D4×C283—22no—
88C2×C4×C283—22no—
89C2×C4×C283—22no—
90C2×C4×C283C22×C412yes—
91C2×C4×C283C22×C411yescharacteristic
92C2×C4×C283C2×D412yes—
93C2×C4×C283C2×D412yes—
94C2×C4×C283C2411yescharacteristic
95C8×C283—24no—
96C22×C283—22no—
97C22×C283—41no—
98C22×C283—42no—
99C22×C283C2×C22⋊C412yes—
100C22×C283C23×C411yescharacteristic
101C2×D283—42no—
102C2×D283C2×C22⋊C414yes—
103C4×C283—22no—
104C4×C283C2×C22⋊C411yescharacteristic
105C4×C283C2×C22⋊C411yescharacteristic
106C4×C283C22×D412yes—
107Dic283—24no—
108Dic283C2×C22⋊C414yes—
109C2×C283—21no—
110C2×C283—22no—
111C2×C283—42no—
112C2×C283C22×C22⋊C411yescharacteristic
113D283—24no—
114C283#288911yescharacteristic, complemented, Sylow
115#2889—2831nomaximal, Sylow
116C2×2−1+4—2831no—
117#568—2832no—
118#413—2831no—
119#413—2834no—
120#413D28311yescharacteristic, complemented
121C2×#101—2832no—
122C2×#101—2834no—
1232−1+4—5664no—
1242−1+4—5664no—
125C2×#41—2834no—
126C2×#41—2832no—
127C2×#41—2832no—
128C2×#41—2831no—
129C2×#41C2×D28311yescharacteristic
130C22×Q8—2832no—
131C22×Q8—2832no—
132C22×Q8—2831no—
133#131—5664no—
134#131—5664no—
135C2×M16—2832no—
136C2×M16—2838no—
137C2×M16C2×D28312yes—
138#101—2832no—
139#101—2838no—
140#101C2×D28312yes—
141#100—2832no—
142#100C2×D28312yes—
143#41—22no—
144#41—22no—
145#41—2834no—
146#41—2834no—
147#41—5664no—
148#41—5664no—
149#41—5668no—
150#41—56616no—
151C2×Q8—2831no—
152C2×Q8—2834no—
153C2×Q8—2832no—
154C2×Q8—2832no—
155C2×Q8—5664no—
156C2×Q8—5664no—
157C2×Q8—5668no—
158C2×Q8—5662no—
159C2×Q8—5662no—
160C2×Q8C22×D28311yescharacteristic
161C2×D4—2832no—
162C2×D4—2834no—
163C2×D4—2831no—
164C2×D4C4×D28312yes—
165C2×D4C22×D28311yescharacteristic
166C22×C4—2832no—
167C22×C4—2834no—
168C22×C4—2832no—
169C22×C4—2834no—
170C22×C4C4×D28312yes—
171C22×C4C22×D28311yescharacteristic
172M16—22no—
173M16—2834no—
174M16—5668no—
175M16—5662no—
176M16C22×D28314yes—
177C2×C8—22no—
178C2×C8—5668no—
179C2×C8—5662no—
180C23—2832no—
181C23C2×C4×D28312yes—
182C23C2×C4×D28311yescharacteristic
183Q8—22no—
184Q8—2832no—
185Q8—2832no—
186Q8—5668no—
187Q8—5662no—
188Q8—5662no—
189Q8—5664no—
190D4—24no—
191D4—22no—
192D4—2838no—
193D4—2832no—
194D4—2832no—
195D4—5668no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—2832no—
199C2×C4—2838no—
200C2×C4—2834no—
201C2×C4—5668no—
202C2×C4—5668no—
203C2×C4—5664no—
204C2×C4C2×C4×D28312yes—
205C2×C4C2×C4×D28311yescharacteristic
206C2×C4D4×D28312yes—
207C2×C4D4×D28312yes—
208C2×C4C23×D28311yescharacteristic
209C8—24no—
210C8—5664no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—2834no—
215C22—11322no—
216C22C22⋊C4×D28312yes—
217C22C22×C4×D28311yescharacteristic
218C4—22no—
219C4—2834no—
220C4—5664no—
221C4C22⋊C4×D28311yescharacteristic
222C4C22⋊C4×D28311yescharacteristic
223C4C2×D4×D28312yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—5664no—
228C2C2×C22⋊C4×D28311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D283
commutatorC22×C283C23×C4
FrattiniC2×C4C23×D283
Fitting#413×C283C2
socleC2×C283C22×C22⋊C4
prime radicalC2×#22583657C2
cosocleC2×C4×C283C24
hypercenter#413D283
hypocenterC283#2889
LCSG ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #31827231C2semidirect
3. #31826449C2semidirect
4. #31826446C2semidirect
5. C2×#22583528C2semidirect
6. #101×D283C2semidirect
7. #413×C283C2semidirect
114. C283#2889semidirect
120. #413D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836572
#318272312
#318264492
#318264462
C2×#225835282
#101×D2832
#413×C2832
#2889283