Finite Group Databasebeta

← C22×#3755⋊C3 #13153654

Subgroups — C22×#3755⋊C3 #13153654

Order 1728.

1199 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1199 conjugacy classes of subgroups — too many to draw. Only its 292 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
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C2×#3755⋊C3C2112yesmaximal, direct product factor
4C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
5C22×#3755C311yesmaximal, characteristic, complemented
6C22×C3×#215—31nomaximal
7C23×(C3⋊D3)⋊C3—21no—
8C2×Q8×H(3,𝔽3)—23no—
9C2×Q8×H(3,𝔽3)C2216yescomplemented
10C22×C4×H(3,𝔽3)—21no—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)—23no—
13C2×C4⋊((C3⋊D3)⋊C3)C2216yes—
14#3755⋊C3C22116yesdirect product factor
15C2×(C3⋊(C3⋊C8))⋊C3C2216yescomplemented
16C22×C32×Q8—21no—
17C22×C32×Q8D311yescharacteristic, complemented
18C22×C32×Q8C2×C311yescharacteristic, complemented
19C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
20C22×C3×D12—31no—
21C2×#3755C2×C3112yescomplemented
22C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
23C2×C3×#215—312no—
24C22×C3×C3⋊C8—31no—
25C23×H(3,𝔽3)D411yescharacteristic
26C22×(C3⋊D3)⋊C3—26no—
27C22×(C3⋊D3)⋊C3—44no—
28Q8×H(3,𝔽3)—26no—
29Q8×H(3,𝔽3)C2314yescomplemented
30C2×C4×H(3,𝔽3)—26no—
31C2×C4×H(3,𝔽3)D413yes—
32C2×C4×H(3,𝔽3)C2313yes—
33C4⋊((C3⋊D3)⋊C3)—26no—
34C4⋊((C3⋊D3)⋊C3)C2314yes—
35(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
36C22×C3×SD16—91no—
37C22×#215—31no—
38C22×#215—91no—
39C23×C3⋊D3—21no—
40C23×C3×D3—61no—
41C2×C32×Q8—26no—
42C2×C32×Q8—26no—
43C2×C32×Q8—23no—
44C2×C32×Q8—23no—
45C2×C32×Q8C2×D316yescomplemented
46C2×C32×Q8C22×C316yescomplemented
47C22×C32×C4—22no—
48C22×C32×C4—21no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C2×C4⋊(C3⋊D3)—23no—
54C2×C4⋊(C3⋊D3)C22×C316yes—
55C2×C3×D12—36no—
56C2×C3×D12—63no—
57#3755C22×C3116yescomplemented
58C2×C3⋊(C3⋊C8)C22×C316yescomplemented
59C3×#215—316no—
60C2×C3×C3⋊C8—36no—
61C22×H(3,𝔽3)SD1614yescomplemented
62C22×H(3,𝔽3)C2×D413yes—
63C2×(C3⋊D3)⋊C3—24no—
64C2×(C3⋊D3)⋊C3—412no—
65C4×H(3,𝔽3)—24no—
66C4×H(3,𝔽3)C2×D413yes—
67C4×H(3,𝔽3)C2411yescharacteristic
68C22×C3×Q8—31no—
69C22×C3×Q8—31no—
70C22×C3×Q8—61no—
71C22×C3×Q8C3×D311yescharacteristic
72C22×C3×D4—91no—
73C22×D12—31no—
74C22×D12—91no—
75C2×C3×SD16—912no—
76C22×C3×C8—91no—
77C2×#215—312no—
78C2×#215—912no—
79C22×C3⋊C8—31no—
80C22×C3⋊C8—91no—
81C23×C32—21no—
82C23×C32#6711yescharacteristic
83C23×C32C3×D411yescharacteristic
84C22×C3⋊D3—26no—
85C22×C3⋊D3—44no—
86C22×C3×D3—66no—
87C22×C3×D3—124no—
88C32×Q8—212no—
89C32×Q8—26no—
90C32×Q8—26no—
91C32×Q8—24no—
92C32×Q8C22×D314yescomplemented
93C32×Q8C23×C314yescomplemented
94C2×C32×C4—212no—
95C2×C32×C4—26no—
96C2×C32×C4—26no—
97C2×C32×C4—23no—
98C2×C32×C4—23no—
99C2×C32×C4#6713yes—
100C2×C32×C4C3×D413yes—
101C2×C32×C4C22×D313yes—
102C2×C32×C4C23×C313yes—
103C4⋊(C3⋊D3)—26no—
104C4⋊(C3⋊D3)C23×C314yes—
105C3×D12—34no—
106C3×D12—66no—
107C3⋊(C3⋊C8)C23×C314yescomplemented
108C3×C3⋊C8—34no—
109C22×SD16—91noSylow
110C2×H(3,𝔽3)C2×SD1616yescomplemented
111C2×H(3,𝔽3)C22×D411yescharacteristic
112(C3⋊D3)⋊C3—44no—
113C24×C3—181no—
114C23×D3—61no—
115C23×D3—181no—
116C2×C3×Q8—23no—
117C2×C3×Q8—36no—
118C2×C3×Q8—36no—
119C2×C3×Q8—63no—
120C2×C3×Q8—63no—
121C2×C3×Q8—66no—
122C2×C3×Q8—66no—
123C2×C3×Q8C2×C3×D316yes—
124C2×C3×D4—96no—
125C2×C3×D4—183no—
126C22×C3×C4—21no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—61no—
130C22×C3×C4—61no—
131C22×C3×C4—62no—
132C22×C3×C4—61no—
133C22×C3×C4C2×C3×D311yescharacteristic
134C2×D12—36no—
135C2×D12—63no—
136C2×D12—96no—
137C2×D12—183no—
138C3×SD16—916no—
139C2×C3×C8—96no—
140#215—316no—
141#215—916no—
142C2×C3⋊C8—36no—
143C2×C3⋊C8—96no—
144C22×C32—24no—
145C22×C32—23no—
146C22×C32#21514yescomplemented
147C22×C32C3×SD1614yescomplemented
148C22×C32C2×#6713yes—
149C22×C32C2×C3×D413yes—
150C2×C3⋊D3—24no—
151C2×C3⋊D3—412no—
152C2×C3×D3—64no—
153C2×C3×D3—1212no—
154C32×C4—28no—
155C32×C4—24no—
156C32×C4—24no—
157C32×C4—21no—
158C32×C4—23no—
159C32×C4C2×#6713yes—
160C32×C4C2×C3×D413yes—
161C32×C4C23×D311yescharacteristic
162C32×C4C24×C311yescharacteristic
163C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
164C22×D4—91no—
165C2×SD16—912no—
166C22×C8—91no—
167H(3,𝔽3)C22×SD1611yescharacteristic, complemented, Sylow
168C23×C3—31no—
169C23×C3—31no—
170C23×C3—61no—
171C23×C3—186no—
172C23×C3—364no—
173C23×C3C3×#6711yescharacteristic
174C22×D3—66no—
175C22×D3—124no—
176C22×D3—186no—
177C22×D3—364no—
178C3×Q8—26no—
179C3×Q8—34no—
180C3×Q8—34no—
181C3×Q8—66no—
182C3×Q8—66no—
183C3×Q8—612no—
184C3×Q8—64no—
185C3×Q8C22×C3×D314yes—
186C3×D4—94no—
187C3×D4—186no—
188C2×C3×C4—26no—
189C2×C3×C4—33no—
190C2×C3×C4—33no—
191C2×C3×C4—33no—
192C2×C3×C4—33no—
193C2×C3×C4—66no—
194C2×C3×C4—66no—
195C2×C3×C4—612no—
196C2×C3×C4—63no—
197C2×C3×C4—63no—
198C2×C3×C4C3×#6713yes—
199C2×C3×C4C22×C3×D313yes—
200D12—34no—
201D12—66no—
202D12—94no—
203D12—186no—
204C3×C8—94no—
205C3⋊C8—34no—
206C3⋊C8—94no—
207C2×C32—26no—
208C2×C32—21no—
209C2×C32C2×#21516yescomplemented
210C2×C32C2×C3×SD1616yescomplemented
211C2×C32C22×#6711yescharacteristic
212C2×C32C22×C3×D411yescharacteristic
213C3⋊D3—44no—
214C3×D3—124no—
215C24—181no—
216C2×Q8—23no—
217C2×Q8C2×(C3⋊D3)⋊C316yescomplemented
218C2×D4—96no—
219C2×D4—183no—
220C22×C4—21no—
221C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
222SD16—916no—
223C2×C8—96no—
224C22×C3—33no—
225C22×C3—33no—
226C22×C3—34no—
227C22×C3—34no—
228C22×C3—63no—
229C22×C3—64no—
230C22×C3—184no—
231C22×C3—3612no—
232C22×C3C3×#21514yes—
233C22×C3C2×C3×#6713yes—
234C2×D3—64no—
235C2×D3—1212no—
236C2×D3—184no—
237C2×D3—3612no—
238C3×C4—24no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—33no—
242C3×C4—33no—
243C3×C4—64no—
244C3×C4—64no—
245C3×C4—68no—
246C3×C4—61no—
247C3×C4—63no—
248C3×C4C2×C3×#6713yes—
249C3×C4C23×C3×D311yescharacteristic
250C32—21no—
251C32C22×#21511yescharacteristic, complemented
252C32C22×C3×SD1611yescharacteristic, complemented
253C23—186no—
254C23—364no—
255C23#638⋊C311yescharacteristic
256Q8—26no—
257Q8C22×(C3⋊D3)⋊C314yescomplemented
258D4—94no—
259D4—186no—
260C2×C4—26no—
261C2×C4#638⋊C313yes—
262C2×C4C22×(C3⋊D3)⋊C313yes—
263C8—94no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—36no—
267C2×C3—36no—
268C2×C3—61no—
269C2×C3—66no—
270C2×C3—364no—
271C2×C3C2×C3×#21516yes—
272C2×C3C22×C3×#6711yescharacteristic
273D3—124no—
274D3—364no—
275C22—184no—
276C22—3612no—
277C22#3755⋊C314yesdirect product factor
278C22C2×#638⋊C313yes—
279C4—24no—
280C4C2×#638⋊C313yes—
281C4C23×(C3⋊D3)⋊C311yescharacteristic
282C3—31no—
283C3—31no—
284C3—61no—
285C3C22×C3×#21511yescharacteristic
286C2—364no—
287C2C2×#3755⋊C316yesdirect product factor
288C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC23#638⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
socleC23×C3C3×#67
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C22×C3×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
3. C2×#3755⋊C3C2direct (N × H)
4. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
5. C22×#3755C3semidirect
9. C2×Q8×H(3,𝔽3)C22semidirect
14. #3755⋊C3C22direct (N × H)
15. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
17. C22×C32×Q8D3semidirect
18. C22×C32×Q8C2×C3semidirect
21. C2×#3755C2×C3semidirect
22. C22×C3⋊(C3⋊C8)C2×C3semidirect
29. Q8×H(3,𝔽3)C23semidirect
35. (C3⋊(C3⋊C8))⋊C3C23semidirect
45. C2×C32×Q8C2×D3semidirect
46. C2×C32×Q8C22×C3semidirect
57. #3755C22×C3semidirect
58. C2×C3⋊(C3⋊C8)C22×C3semidirect
61. C22×H(3,𝔽3)SD16semidirect
92. C32×Q8C22×D3semidirect
93. C32×Q8C23×C3semidirect
107. C3⋊(C3⋊C8)C23×C3semidirect
110. C2×H(3,𝔽3)C2×SD16semidirect
146. C22×C32#215semidirect
147. C22×C32C3×SD16semidirect
163. C22×Q8(C3⋊D3)⋊C3semidirect
167. H(3,𝔽3)C22×SD16semidirect
209. C2×C32C2×#215semidirect
210. C2×C32C2×C3×SD16semidirect
217. C2×Q8C2×(C3⋊D3)⋊C3semidirect
251. C32C22×#215semidirect
252. C32C22×C3×SD16semidirect
257. Q8C22×(C3⋊D3)⋊C3semidirect
277. C22#3755⋊C3direct (N × H)
287. C2C2×#3755⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C22×C4⋊((C3⋊D3)⋊C3)2
C2×#3755⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
C22×#37553
C22×C3×#2153