Finite Group Databasebeta

← #11796367

Subgroups — #11796367

Order 1152.

568 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 52 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#10596139C211yesmaximal, characteristic
2#10593832C212yesmaximal, complemented
3#10590211C212yesmaximal, complemented
4#10590143C212yesmaximal, complemented
5C2×#3832C2211yescharacteristic
6C2×#3830C411yescharacteristic, complemented
7#64286—22no—
8C32×#124C411yescharacteristic, complemented
9#64153C411yescharacteristic, complemented
10#64149C2211yescharacteristic
11#64093—22no—
12#64090—22no—
13#64084C2211yescharacteristic
14C42×C3⋊D3C411yescharacteristic, complemented
15#63774—22no—
16#63774C2214yes—
17#5668—62no—
18C2×C32×Q8C2×C411yescharacteristic
19C2×C32×D4C2×C411yescharacteristic
20C2×#638—21no—
21C2×#638—22no—
22C22×C3⋊Dic3—21no—
23#3832—41no—
24#3832—41no—
25#3830—41no—
26#3830—41no—
27C2×C4⋊(C3⋊D3)D412yes—
28C2×C4⋊(C3⋊D3)C2311yescharacteristic
29C2×C4×C3⋊D3—21no—
30C2×C4×C3⋊D3C2×C411yescharacteristic
31C2×C4×C3⋊D3D412yes—
32C2×#634C2×C411yescharacteristic
33C22⋊#157—24no—
34C4⋊#157—42no—
35C4×#157—42no—
36C32×C22⋊C4—22no—
37C32×C42C2×C411yescharacteristic
38#3752—22no—
39#3750—22no—
40C4×C3⋊Dic3—21no—
41C4×C3⋊Dic3C2×C411yescharacteristic
42#2126—91nomaximal, Sylow
43C2×#239—62no—
44C2×#237—62no—
45C3×#124—22no—
46#944—62no—
47#940—62no—
48#884—68no—
49#67⋊C4—68no—
50#875—62no—
51C42×D3—62no—
52C23×C32—21no—
53C22×C3⋊D3C22⋊C411yescharacteristic
54C22×C3⋊D3C2×D412yes—
55C2×#157—44no—
56C32×Q8—41no—
57C32×D4—41no—
58C2×C32×C4—21no—
59C2×C32×C4C22⋊C412yes—
60C2×C32×C4C22×C411yescharacteristic
61#638—44no—
62#638—42no—
63C2×C3⋊Dic3—21no—
64C2×C3⋊Dic3—21no—
65C2×C3⋊Dic3—41no—
66C2×C3⋊Dic3—41no—
67C2×C3⋊Dic3C22⋊C411yescharacteristic
68C4⋊(C3⋊D3)—42no—
69C4⋊(C3⋊D3)—41no—
70C4×C3⋊D3—21no—
71C4×C3⋊D3—21no—
72C4×C3⋊D3—22no—
73C4×C3⋊D3—22no—
74C4×C3⋊D3—42no—
75#634—41no—
76#532—91no—
77#410—92no—
78#354—92no—
79#353—92no—
80C2×C3×Q8—22no—
81C2×C3×D4—22no—
82C2×#67—68no—
83C2×#67—64no—
84C22×Dic3—64no—
85#239—124no—
86#239—124no—
87#237—124no—
88#237—124no—
89C2×D12—62no—
90C2×D12—64no—
91C2×C4×D3—62no—
92C2×C4×D3—64no—
93C2×C4×D3—64no—
94C2×Dic6—62no—
95C3×C22⋊C4—28no—
96C3×C42—22no—
97(C2×D3)⋊C4—68no—
98Dic3⋊C4—68no—
99C4×Dic3—62no—
100C4×Dic3—64no—
101C22×C32—42no—
102C22×C32—41no—
103C22×C32C2×C22⋊C411yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3⋊D3—22no—
106C2×C3⋊D3—41no—
107C2×C3⋊D3—42no—
108#157—84no—
109C32×C4—21no—
110C32×C4—22no—
111C32×C4—41no—
112C3⋊Dic3—22no—
113C3⋊Dic3—41no—
114C3⋊Dic3#9911yescharacteristic
115C3⋊Dic3#9911yescharacteristic
116C2×#41—91no—
117C2×#41—91no—
118C4⋊Q8—91no—
119#127—91no—
120#124—91no—
121#124#15711yescharacteristic, complemented
122#121—182no—
123C4×D4—182no—
124#117—182no—
125C2×C42—91no—
126#99—94no—
127#99—182no—
128C23×C3—24no—
129C22×D3—62no—
130C22×D3—64no—
131C3×Q8—44no—
132C3×D4—44no—
133C2×C3×C4—24no—
134C2×C3×C4—22no—
135C2×C3×C4—24no—
136#67—1216no—
137#67—128no—
138C2×Dic3—62no—
139C2×Dic3—64no—
140C2×Dic3—64no—
141C2×Dic3—124no—
142C2×Dic3—124no—
143D12—124no—
144D12—128no—
145C4×D3—64no—
146C4×D3—64no—
147C4×D3—124no—
148C4×D3—124no—
149C4×D3—128no—
150Dic6—124no—
151C2×C32—21no—
152C2×C32—81no—
153C2×C32C2×#9911yescharacteristic
154C3⋊D3—41no—
155C3⋊D3—42no—
156#41—361no—
157#41—361no—
158#41—361no—
159#41—361no—
160C2×Q8—91no—
161C2×Q8C2×#15711yescharacteristic
162C2×D4—91no—
163C2×D4—92no—
164C2×D4—182no—
165C2×D4—181no—
166C2×D4C2×#15711yescharacteristic
167C22×C4—91no—
168C22×C4—92no—
169C22×C4—181no—
170C22×C4—181no—
171C4⋊C4—182no—
172C4⋊C4—362no—
173C22⋊C4—22no—
174C22⋊C4—182no—
175C22⋊C4—184no—
176C42—91no—
177C42—181no—
178C42—362no—
179C42C2×#15711yescharacteristic
180C22×C3—22no—
181C22×C3—44no—
182C22×C3—48no—
183C2×D3—64no—
184C2×D3—124no—
185C2×D3—124no—
186C2×D3—128no—
187C3×C4—24no—
188C3×C4—44no—
189C3×C4—44no—
190Dic3—62no—
191Dic3—62no—
192Dic3—68no—
193Dic3—124no—
194C32#212611yescharacteristic, complemented, Sylow
195C23—21no—
196C23—92no—
197C23—91no—
198Q8—41no—
199Q8—361no—
200D4—41no—
201D4—361no—
202D4—362no—
203D4—364no—
204D4—362no—
205C2×C4—21no—
206C2×C4—91no—
207C2×C4—182no—
208C2×C4—182no—
209C2×C4—181no—
210C2×C4—181no—
211C2×C4—181no—
212C2×C4—181no—
213C2×C4—362no—
214C2×C4—361no—
215C2×C4—361no—
216C2×C4—364no—
217C2×C4C22⋊#15712yes—
218C2×C4C22×#15711yescharacteristic
219C2×C3—24no—
220C2×C3—22no—
221C2×C3—84no—
222D3—124no—
223D3—244no—
224C22—42no—
225C22—41no—
226C22—182no—
227C22—181no—
228C22—361no—
229C22—362no—
230C22C2×C22⋊#15711yescharacteristic
231C4—21no—
232C4—22no—
233C4—41no—
234C4—91no—
235C4—91no—
236C4—182no—
237C4—361no—
238C4—724no—
239C3—22no—
240C2—21no—
241C2—81no—
242C2—362no—
243C2—361no—
244C2#1059381711yescharacteristic
subgroupG/H
centerC2#10593817
commutatorC2×C32×C4C22×C4
FrattiniC2×C4C22×#157
FittingC32×#124C4
socleC2×C32C2×#99
prime radical#10596139C2
cosocleC2×C4⋊(C3⋊D3)C23
hypercenter#124#157
hypocenterC32#2126
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #10593832C2semidirect
3. #10590211C2semidirect
4. #10590143C2semidirect
6. C2×#3830C4semidirect
8. C32×#124C4semidirect
9. #64153C4semidirect
14. C42×C3⋊D3C4semidirect
121. #124#157semidirect
194. C32#2126semidirect

Maximal subgroups

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

subgroupindex
#105961392
#105938322
#105902112
#105901432
#21269