Finite Group Databasebeta

← #11923405

Subgroups — #11923405

Order 1152.

704 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C3×#5945C211yesmaximal, characteristic
2#5819⋊C3C213yesmaximal, complemented
3C2×C3⋊(#131⋊C3)C213yesmaximal, complemented
4#87296C311yesmaximal, characteristic, complemented
5#85629—31nomaximal
6C3×2−1+4⋊C3C413yes—
7C3×#994C411yescharacteristic
8C2×C3×#41⋊C3C2213yes—
9C3×#948—41nomaximal
10C3⋊(#131⋊C3)C2211yescharacteristic, complemented
11C3⋊(#131⋊C3)C2213yescomplemented
12C3×#585C2×C311yescharacteristic
13#5945—21no—
14#5945D311yescharacteristic
15#5819—33no—
16#5819—31no—
17#5819C2×C313yescomplemented
18C2×#948—31no—
19C2×#948—33no—
20C2×#948C2×C313yescomplemented
21#568⋊C3—33no—
22C2×#131⋊C3—33no—
23C32×#41—41no—
24C3×#41⋊C3C2×C413yes—
25C3×#41⋊C3C2311yescharacteristic
26C2×C3×SL(2,𝔽3)C2×C413yes—
27C3×#208—43no—
28C2×C3×C3⋊C8—43no—
29#3573—31noSylow
30C3×2−1+4—31no—
31C3×2−1+4C3×C413yes—
32C3×H(5,𝔽2)—33no—
33C3×H(5,𝔽2)C3×C411yescharacteristic
34C2×C3×#41—33no—
35C2×C3×#41—31no—
36C2×C3×#41C22×C313yes—
372−1+4⋊C3—23no—
382−1+4⋊C3Dic313yes—
39#994—21no—
40#994Dic311yescharacteristic
41C2×#41⋊C3—23no—
42C2×#41⋊C3C2×D313yes—
43C3×#131—121no—
44#948—36no—
45#948—31no—
46#948—33no—
47#948—36no—
48#948—33no—
49#948—33no—
50#948—33no—
51#948A411yescharacteristic
52#948C22×C311yescharacteristic
53#948C22×C313yes—
54C2×#208—36no—
55C2×#208—33no—
56C2×#208—33no—
57C2×#208—33no—
58C22×C3⋊C8—32no—
59C22×C3⋊C8—33no—
60#131⋊C3—31no—
61#131⋊C3—33no—
62C32×Q8—41no—
63C32×D4—43no—
64C2×C32×C4—43no—
65C3×SL(2,𝔽3)C22×C411yescharacteristic
66C3×C3⋊C8—41no—
67C3×C3⋊C8—43no—
68#585C3×D311yescharacteristic
69#568—33no—
70#568—93no—
71#568—91no—
72C2×#131—33no—
73C2×#131—91no—
74C2×#131—93no—
75C3×#41—36no—
76C3×#41—31no—
77C3×#41—33no—
78C3×#41—36no—
79C3×#41—33no—
80C3×#41—33no—
81C3×#41—33no—
82C3×#41—41no—
83C3×#41—81no—
84C3×#41C2×C3×C413yes—
85C3×#41C2×A411yescharacteristic
86C3×#41C23×C311yescharacteristic
87C2×C3×Q8—33no—
88C2×C3×Q8—31no—
89C2×C3×Q8C2×C3×C413yes—
90C2×C3×D4—33no—
91C2×C3×D4—36no—
92C2×C3×D4—33no—
93C2×C3×D4—33no—
94C22×C3×C4—32no—
95C22×C3×C4—33no—
96#41⋊C3—23no—
97#41⋊C3—21no—
98#41⋊C3C2×Dic313yes—
99#41⋊C3C22×D311yescharacteristic
100C2×SL(2,𝔽3)—23no—
101C2×SL(2,𝔽3)C2×Dic313yes—
102C3×M16—123no—
103C2×C3×C8—123no—
104#208—36no—
105#208—33no—
106#208—36no—
107#208—33no—
108#208—31no—
109#208C2×A413yes—
110C2×C3⋊C8—36no—
111C2×C3⋊C8—33no—
112C2×C3⋊C8—33no—
113C2×C3⋊C8—31no—
114C2×C3⋊C8—33no—
115C2×C3⋊C8—33no—
116C2×C3⋊C8C2×A413yes—
117C22×C32—43no—
118C32×C4—43no—
119C32×C4—41no—
1202−1+4—31no—
1212−1+4C3×Dic313yes—
122H(5,𝔽2)—33no—
123H(5,𝔽2)C3×Dic311yescharacteristic
124C2×#41—33no—
125C2×#41—31no—
126C2×#41C2×C3×D313yes—
127#131—31no—
128#131—31no—
129#131—33no—
130#131—93no—
131#131—93no—
132#131—96no—
133#131—91no—
134#131—96no—
135#131—93no—
136#131—93no—
137C2×M16—93no—
138C2×M16—96no—
139C2×M16—93no—
140C2×M16—93no—
141C22×C8—93no—
142C22×C8—92no—
143C23×C3—32no—
144C23×C3—33no—
145C3×Q8—33no—
146C3×Q8—33no—
147C3×Q8—41no—
148C3×Q8—81no—
149C3×Q8C4×A411yescharacteristic
150C3×Q8C22×C3×C411yescharacteristic
151C3×D4—33no—
152C3×D4—31no—
153C3×D4—36no—
154C3×D4—36no—
155C3×D4—33no—
156C3×D4—43no—
157C3×D4—83no—
158C3×D4C4×A413yes—
159C2×C3×C4—36no—
160C2×C3×C4—33no—
161C2×C3×C4—33no—
162C2×C3×C4—33no—
163C2×C3×C4—33no—
164C2×C3×C4—31no—
165C2×C3×C4—43no—
166C2×C3×C4—83no—
167C2×C3×C4C22×A413yes—
168SL(2,𝔽3)—21no—
169SL(2,𝔽3)C22×Dic311yescharacteristic
170C3×C8—121no—
171C3×C8—123no—
172C3⋊C8—33no—
173C3⋊C8—31no—
174C3⋊C8C22×A411yescharacteristic
175C3⋊C8C22×A413yes—
176C2×C32—41no—
177C2×C32—83no—
178#41—36no—
179#41—31no—
180#41—33no—
181#41—36no—
182#41—33no—
183#41—33no—
184#41—33no—
185#41C2×C3×Dic313yes—
186#41D3×A411yescharacteristic
187#41C22×C3×D311yescharacteristic
188C2×Q8—33no—
189C2×Q8—31no—
190C2×Q8C2×C3×Dic313yes—
191C2×D4—33no—
192C2×D4—36no—
193C2×D4—33no—
194C2×D4—33no—
195C22×C4—32no—
196C22×C4—33no—
197M16—33no—
198M16—96no—
199M16—96no—
200M16—93no—
201M16—93no—
202M16—91no—
203C2×C8—33no—
204C2×C8—91no—
205C2×C8—93no—
206C2×C8—93no—
207C2×C8—93no—
208C2×C8—96no—
209C2×C8—93no—
210C22×C3—33no—
211C22×C3—31no—
212C22×C3—43no—
213C22×C3—83no—
214C22×C3—123no—
215C22×C3—122no—
216C22×C3C2×C4×A413yes—
217C3×C4—33no—
218C3×C4—31no—
219C3×C4—43no—
220C3×C4—41no—
221C3×C4—83no—
222C3×C4—81no—
223C3×C4C2×C4×A413yes—
224C3×C4C23×A411yescharacteristic
225C32—41noSylow
226C23—32no—
227C23—33no—
228Q8—33no—
229Q8—33no—
230Q8Dic3×A411yescharacteristic
231Q8C22×C3×Dic311yescharacteristic
232D4—33no—
233D4—31no—
234D4—36no—
235D4—36no—
236D4—33no—
237D4Dic3×A413yes—
238C2×C4—36no—
239C2×C4—33no—
240C2×C4—33no—
241C2×C4—33no—
242C2×C4—31no—
243C2×C4—33no—
244C2×C4C2×D3×A413yes—
245C8—31no—
246C8—33no—
247C8—91no—
248C8—93no—
249C2×C3—23no—
250C2×C3—41no—
251C2×C3—61no—
252C2×C3—63no—
253C2×C3—83no—
254C2×C3—81no—
255C2×C3—163no—
256C2×C3C22×C4×A411yescharacteristic
257C22—33no—
258C22—31no—
259C22—123no—
260C22—122no—
261C22C2×Dic3×A413yes—
262C4—31no—
263C4—33no—
264C4C2×Dic3×A413yes—
265C4C22×D3×A411yescharacteristic
266C3—41no—
267C3—81no—
268C3#8562911yescharacteristic, complemented
269C2—23no—
270C2—61no—
271C2—63no—
272C2C22×Dic3×A411yescharacteristic
subgroupG/H
center, FrattiniC4C22×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FittingC3×#585C2×C3
socleC2×C3C22×C4×A4
prime radicalC3×#5945C2
cosocleC3×#41C23×C3
hypercenter#41D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #5819⋊C3C2semidirect
3. C2×C3⋊(#131⋊C3)C2semidirect
4. #87296C3semidirect
10. C3⋊(#131⋊C3)C22semidirect
11. C3⋊(#131⋊C3)C22semidirect
17. #5819C2×C3semidirect
20. C2×#948C2×C3semidirect
268. C3#85629semidirect

Maximal subgroups

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

subgroupindex
C3×#59452
#5819⋊C32
C2×C3⋊(#131⋊C3)2
#872963
#856293
C3×#9484