Finite Group Databasebeta

← #26136614

Subgroups — #26136614

Order 12032.

398 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 79 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×#15688922C212yesmaximal, complemented
2#20241557C212yesmaximal, complemented
3#20241555C212yesmaximal, complemented
4C47×#3152C211yesmaximal, characteristic
5C2×#13093943—22no—
6#15688922—22no—
7#15688922C2214yes—
8C2×#137×C47—22no—
9C47×#532C411yescharacteristic
10C2×C4⋊Q8×C47C2211yescharacteristic
11C47×#497—22no—
12C47×#494C411yescharacteristic
13C47×#494C2211yescharacteristic
14C47×#492C411yescharacteristic
15C47×#492C2211yescharacteristic
16C4×M16×C47C411yescharacteristic
17C2×#10625329—42no—
18#13093943—24no—
19#13093943—42no—
20C2×#41×C47—21no—
21C22×Q8×C47D412yes—
22#137×C47—44no—
23#137×C47—44no—
24C2×Q16×C47—22no—
25C2×Q16×C47—22no—
26C2×SD16×C47—22no—
27C2×SD16×C47—22no—
28C2×M16×C47—21no—
29C4⋊Q8×C47—41no—
30C4⋊Q8×C47C2×C411yescharacteristic
31C4⋊Q8×C47C2×C411yescharacteristic
32C4⋊Q8×C47D412yes—
33C4⋊Q8×C47C2311yescharacteristic
34#127×C47C2×C411yescharacteristic
35#124×C47—21no—
36#121×C47—41no—
37C4×D4×C47—41no—
38C2×C4⋊C4×C47—41no—
39C2×C42×C47C2×C411yescharacteristic
40#97×C47—21no—
41C4×C8×C47C2×C411yescharacteristic
42C4×C8×C47C2×C411yescharacteristic
43#41×C47—42no—
44#41×C47—42no—
45C2×Q8×C47—22no—
46C2×Q8×C47—21no—
47C2×Q8×C47—42no—
48C2×Q8×C47—42no—
49C2×Q8×C47C22⋊C412yes—
50C2×Q8×C47C2×D412yes—
51C2×D4×C47—21no—
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C22×C4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47—41no—
57C22×C4×C47C22⋊C411yescharacteristic
58Q16×C47—44no—
59Q16×C47—44no—
60SD16×C47—44no—
61SD16×C47—44no—
62M16×C47—24no—
63C2×C8×C47—21no—
64C2×C8×C47—21no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—42no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—41no—
69C22⋊C4×C47—42no—
70C42×C47—21no—
71C42×C47C22⋊C411yescharacteristic
72C42×C47C22×C411yescharacteristic
73#10625329—44no—
74#10625329—82no—
75C2×C47⋊C8—82no—
76C23×C47—41no—
77C23×C47#9911yescharacteristic
78Q8×C47—42no—
79Q8×C47—42no—
80Q8×C47—42no—
81Q8×C47—82no—
82D4×C47—42no—
83D4×C47—82no—
84D4×C47—82no—
85C2×C4×C47—22no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—21no—
89C2×C4×C47—22no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—41no—
94C2×C4×C47—81no—
95C2×C4×C47—81no—
96C2×C4×C47#9911yescharacteristic
97C2×C4×C47C2×C22⋊C411yescharacteristic
98C8×C47—22no—
99C8×C47—22no—
100C47⋊C8—84no—
101#13590—471nomaximal, Sylow
102C22×C47—41no—
103C22×C47—82no—
104C22×C47—81no—
105C22×C47#35312yes—
106C22×C47C2×#9911yescharacteristic
107C4×C47—22no—
108C4×C47—21no—
109C4×C47—21no—
110C4×C47—41no—
111C4×C47—42no—
112C4×C47—42no—
113C4×C47—81no—
114#3152D4711yescharacteristic
115C2×#356—472no—
116#1411—472no—
117#1410—472no—
118C2×C47—21no—
119C2×C47—22no—
120C2×C47—161no—
121C2×C47C2×#35311yescharacteristic
122C2×#137—22no—
123#532Dic4711yescharacteristic
124C2×C4⋊Q8C2×D4711yescharacteristic
125#497—22no—
126#494Dic4711yescharacteristic
127#494C2×D4711yescharacteristic
128#492Dic4711yescharacteristic
129#492C2×D4711yescharacteristic
130C2×#101—942no—
131C4×M16Dic4711yescharacteristic
132#356—474no—
133#356—942no—
134C47#1359011yescharacteristic, complemented, Sylow
135C2×#41—21no—
136C22×Q8C22⋊D4712yes—
137#137—44no—
138#137—44no—
139C2×Q16—22no—
140C2×Q16—22no—
141C2×SD16—22no—
142C2×SD16—22no—
143C2×M16—21no—
144C2×M16—1882no—
145C4⋊Q8—41no—
146C4⋊Q8C2×Dic4711yescharacteristic
147C4⋊Q8C2×Dic4711yescharacteristic
148C4⋊Q8C22⋊D4712yes—
149C4⋊Q8C22×D4711yescharacteristic
150#127C2×Dic4711yescharacteristic
151#124—21no—
152#121—41no—
153C4×D4—41no—
154C2×C4⋊C4—41no—
155C2×C42C2×Dic4711yescharacteristic
156#101—944no—
157#101—1882no—
158#97—21no—
159C4×C8C2×Dic4711yescharacteristic
160C4×C8C2×Dic4711yescharacteristic
161#41—42no—
162#41—42no—
163C2×Q8—22no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C22⋊Dic4712yes—
168C2×Q8C2×C22⋊D4712yes—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4C22⋊Dic4711yescharacteristic
176Q16—44no—
177Q16—44no—
178SD16—44no—
179SD16—44no—
180M16—24no—
181M16—3762no—
182M16—1884no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—3762no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C22⋊C4—42no—
191C42—21no—
192C42C22⋊Dic4711yescharacteristic
193C42C22×Dic4711yescharacteristic
194C23—41no—
195C23#1309393911yescharacteristic
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—82no—
200D4—42no—
201D4—82no—
202D4—82no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4#1309393911yescharacteristic
215C2×C4C2×C22⋊Dic4711yescharacteristic
216C8—22no—
217C8—22no—
218C8—3764no—
219C22—41no—
220C22—82no—
221C22—81no—
222C22#1568891612yes—
223C22C2×#1309393911yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—42no—
230C4—81no—
231C2—21no—
232C2—22no—
233C2—161no—
234C2C2×#1568891611yescharacteristic
subgroupG/H
centerC2C2×#15688916
commutatorC42×C47C22×C4
FrattiniC4⋊Q8C22×D47
FittingC47×#3152C2
socleC2×C47C2×#353
prime radicalC47×#532C4
cosocleC4⋊Q8×C47C23
hypercenter#3152D47
hypocenterC47#13590
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#15688922C2semidirect
2. #20241557C2semidirect
3. #20241555C2semidirect
134. C47#13590semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
#202415572
#202415552
C47×#31522
#1359047