Finite Group Databasebeta

← #49168149

Subgroups — #49168149

Order 37504.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#34036927C212yesmaximal, complemented
2#34036703C211yesmaximal, characteristic, complemented
3#34036312C214yesmaximal, complemented
4#34036302C214yesmaximal, complemented
5#34036146C212yesmaximal, complemented
6C2×#22627543C211yesmaximal, characteristic, complemented
7C2×2−1+4×C293C211yesmaximal, characteristic, complemented
8C2×#17989802C2211yescharacteristic
9#22627655—22no—
10#22627654C2214yescomplemented
11#22627650—22no—
12#22627649C2214yescomplemented
13#22627644—24no—
14C2×#17989786C2212yes—
15#22627622C2212yescomplemented
16#22627613C2212yescomplemented
17#22627607—22no—
18#22627600—22no—
19#22627583C2211yescharacteristic
20#22627551C2218yescomplemented
21#22627543C2214yescomplemented
222−1+4×C293—24no—
232−1+4×C293—22no—
242−1+4×C293C2214yes—
25C2×#41×C293—22no—
26C2×#41×C293—22no—
27C2×#41×C293C2212yes—
28C22×Q8×C293—22no—
29C22×Q8×C293C2211yescharacteristic
30#17989802—22no—
31#17989802—22no—
32C2×Dic586C2311yescharacteristic
33C2×D1172C2311yescharacteristic
34C2×C22⋊D293—21no—
35C2×C4×D293—21no—
36#17989792—24no—
37#17989790—24no—
38#17989789—24no—
39#17989787—24no—
40#17989786—22no—
41#17989786C2314yes—
42C2×#14556873—22no—
43C4⋊Dic293—21no—
44#17989781—24no—
45C22⋊Dic293—21no—
46#17989779—24no—
47C4×Dic293C2312yes—
48#41×C293—28no—
49#41×C293—28no—
50#41×C293—28no—
51#41×C293—28no—
52#41×C293—22no—
53#41×C293—22no—
54#41×C293D414yes—
55#41×C293C2314yes—
56C2×Q8×C293—22no—
57C2×Q8×C293—28no—
58C2×Q8×C293—28no—
59C2×Q8×C293—22no—
60C2×Q8×C293—22no—
61C2×Q8×C293D412yes—
62C2×Q8×C293D412yes—
63C2×Q8×C293C2312yes—
64C2×D4×C293—22no—
65C2×D4×C293—22no—
66C2×D4×C293D412yes—
67C22×C4×C293—24no—
68C22×C4×C293—22no—
69C22×C4×C293D412yes—
70C22×C4×C293C2311yescharacteristic
71C22×D293—21no—
72Dic586—22no—
73D1172—22no—
74C22⋊D293—41no—
75C22⋊D293—41no—
76C2×Dic293—22no—
77C2×Dic293—21no—
78C4×D293—42no—
79#14556873—24no—
80C23×C293—22no—
81C23×C293C2×D411yescharacteristic
82Q8×C293—28no—
83Q8×C293—22no—
84Q8×C293—24no—
85Q8×C293—24no—
86Q8×C293C2×D414yes—
87D4×C293—28no—
88D4×C293—24no—
89D4×C293—24no—
90D4×C293—22no—
91D4×C293C2×D414yes—
92C2×C4×C293—216no—
93C2×C4×C293—22no—
94C2×C4×C293—22no—
95C2×C4×C293—22no—
96C2×C4×C293—24no—
97C2×C4×C293—24no—
98C2×C4×C293C2×D414yes—
99C2×C4×C293C2×D411yescharacteristic
100C2×C4×C293C2×D412yes—
101C2×C4×C293C2×D412yes—
102C2×C4×C293C2411yescharacteristic
103C22×C293—24no—
104C22×C293—41no—
105C22×C293—44no—
106C22×C293#12011yescharacteristic
107C22×C293#12011yescharacteristic
108C22×C293C22×D411yescharacteristic
109C2×D293—42no—
110C2×D293—41no—
111C4×C293—24no—
112C4×C293—24no—
113C4×C293#12012yes—
114C4×C293C22×D412yes—
115Dic293—42no—
116Dic293—41no—
117C2×C293—21no—
118C2×C293—21no—
119C2×C293—21no—
120C2×C293—44no—
121C2×C293C2×#12011yescharacteristic
122D293—81no—
123C293#302011yescharacteristic, complemented, Sylow
124#3020—2931nomaximal, Sylow
125C2×2−1+4D29311yescharacteristic, complemented
126#575—2932no—
127#536—2931no—
128#456—2934no—
129#454—2934no—
130#421—2932no—
131C2×#101—2931no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D29314yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—2931no—
138C2×#41C2×D29312yes—
139C22×Q8—22no—
140C22×Q8C2×D29311yescharacteristic
141#137—2934no—
142#137—5862no—
143AGL(1,ℤ/8)—2934no—
144AGL(1,ℤ/8)—5862no—
145#135—5864no—
146C2×M16—2932no—
147C4⋊Q8—2932no—
148#124—2932no—
149#123—5862no—
150C22⋊Q8—5862no—
151#117—2931no—
152#104—2938no—
153#101—2934no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—5862no—
161#41—5862no—
162#41C22⋊D29314yes—
163#41C22×D29314yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—2931no—
170C2×Q8C22⋊D29312yes—
171C2×Q8C22⋊D29312yes—
172C2×Q8C22×D29312yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—2931no—
176C2×D4—5861no—
177C2×D4C22⋊D29312yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—5861no—
181C22×C4C22⋊D29312yes—
182C22×C4C22×D29311yescharacteristic
183Q16—5864no—
184SD16—5864no—
185SD16—5864no—
186D8—5864no—
187M16—2934no—
188M16—5862no—
189C2×C8—5862no—
190C4⋊C4—5861no—
191C4⋊C4—5864no—
192C22⋊C4—5861no—
193C22⋊C4—5864no—
194C42—2932no—
195C23—22no—
196C23—5861no—
197C23C2×C22⋊D29311yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—5862no—
203Q8C2×C22⋊D29314yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—5862no—
209D4—11721no—
210D4—11721no—
211D4C2×C22⋊D29314yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—5862no—
219C2×C4—5861no—
220C2×C4—11722no—
221C2×C4C2×C22⋊D29314yes—
222C2×C4C2×C22⋊D29311yescharacteristic
223C2×C4C2×C22⋊D29312yes—
224C2×C4C2×C22⋊D29312yes—
225C2×C4C23×D29311yescharacteristic
226C8—5864no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—11722no—
231C22—11721no—
232C22#2262759311yescharacteristic
233C22#2262759311yescharacteristic
234C22C22×C22⋊D29311yescharacteristic
235C4—24no—
236C4—24no—
237C4—11722no—
238C4—11721no—
239C4#2262759312yes—
240C4C22×C22⋊D29312yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—23441no—
246C2C2×#2262759311yescharacteristic
subgroupG/H
centerC2C2×#22627593
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×2−1+4×C293C2
socleC2×C293C2×#120
hypercenterC2×2−1+4D293
hypocenterC293#3020
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #34036927C2semidirect
2. #34036703C2semidirect
3. #34036312C2semidirect
4. #34036302C2semidirect
5. #34036146C2semidirect
6. C2×#22627543C2semidirect
7. C2×2−1+4×C293C2semidirect
10. #22627654C22semidirect
12. #22627649C22semidirect
15. #22627622C22semidirect
16. #22627613C22semidirect
20. #22627551C22semidirect
21. #22627543C22semidirect
123. C293#3020semidirect
125. C2×2−1+4D293semidirect

Maximal subgroups

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

subgroupindex
#340369272
#340367032
#340363122
#340363022
#340361462
C2×#226275432
C2×2−1+4×C2932
#3020293