Finite Group Databasebeta

← C22×#3752⋊C3 #13153554

Subgroups — C22×#3752⋊C3 #13153554

Order 1728.

2595 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C23×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C2112yesmaximal, direct product factor
5C22×#3752C311yesmaximal, characteristic, complemented
6C22×C3×(C2×D3)⋊C4—31nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×(C3⋊D3)⋊C3—24no—
9C23×(C3⋊D3)⋊C3—24no—
10C23×(C3⋊D3)⋊C3C418yescomplemented
11C23×(C3⋊D3)⋊C3C2216yes—
12C22×C4×H(3,𝔽3)—24no—
13C22×C4×H(3,𝔽3)C2216yescomplemented
14C22×(C3⋊Dic3)⋊C3—24no—
15C22×(C3⋊Dic3)⋊C3C2216yescomplemented
16#3752⋊C3C22116yesdirect product factor
17C24×C3⋊D3C2×C311yescharacteristic
18C24×C3×D3—31no—
19C23×C32×C4—21no—
20C23×C32×C4D311yescharacteristic, complemented
21C23×C32×C4C2×C311yescharacteristic, complemented
22C23×C3⋊Dic3C2×C311yescharacteristic, complemented
23C23×C3×Dic3—31no—
24C2×#3752C2×C3112yescomplemented
25C2×C3×(C2×D3)⋊C4—312no—
26C23×H(3,𝔽3)C2×C414yes—
27C23×H(3,𝔽3)D414yes—
28C23×H(3,𝔽3)D414yes—
29C23×H(3,𝔽3)C2313yes—
30C22×(C3⋊D3)⋊C3—232no—
31C22×(C3⋊D3)⋊C3—212no—
32C22×(C3⋊D3)⋊C3—212no—
33C22×(C3⋊D3)⋊C3C2×C4124yescomplemented
34C22×(C3⋊D3)⋊C3C2314yes—
35C2×C4×H(3,𝔽3)—212no—
36C2×C4×H(3,𝔽3)C2314yescomplemented
37C2×(C3⋊Dic3)⋊C3—212no—
38C2×(C3⋊Dic3)⋊C3C2314yescomplemented
39C22×C3×C22⋊C4—91no—
40C22×(C2×D3)⋊C4—31no—
41C22×(C2×D3)⋊C4—91no—
42C24×C32—21no—
43C24×C32C2×D311yescharacteristic
44C24×C32C22×C311yescharacteristic
45C23×C3⋊D3—24no—
46C23×C3⋊D3—24no—
47C23×C3⋊D3C3×C418yescomplemented
48C23×C3⋊D3C22×C316yes—
49C23×C3×D3—38no—
50C23×C3×D3—36no—
51C23×C3×D3—64no—
52C23×C3×D3—64no—
53C22×C32×C4—28no—
54C22×C32×C4—26no—
55C22×C32×C4—24no—
56C22×C32×C4—24no—
57C22×C32×C4C2×D316yescomplemented
58C22×C32×C4C22×C316yescomplemented
59C22×C3⋊Dic3—24no—
60C22×C3⋊Dic3C22×C316yescomplemented
61C22×C3×Dic3—36no—
62C22×C3×Dic3—64no—
63#3752C22×C3116yescomplemented
64C3×(C2×D3)⋊C4—316no—
65C22×H(3,𝔽3)C22⋊C4116yescomplemented
66C22×H(3,𝔽3)C22×C416yes—
67C22×H(3,𝔽3)C2×D416yes—
68C22×H(3,𝔽3)C2×D416yes—
69C22×H(3,𝔽3)C2411yescharacteristic
70C2×(C3⋊D3)⋊C3—248no—
71C2×(C3⋊D3)⋊C3—24no—
72C2×(C3⋊D3)⋊C3—24no—
73C2×(C3⋊D3)⋊C3C22×C418yescomplemented
74C4×H(3,𝔽3)—24no—
75(C3⋊Dic3)⋊C3—24no—
76C25×C3—91no—
77C24×D3—31no—
78C24×D3—91no—
79C23×C3×C4—31no—
80C23×C3×C4—31no—
81C23×C3×C4—61no—
82C23×C3×C4—91no—
83C23×C3×C4C3×D311yescharacteristic
84C23×Dic3—31no—
85C23×Dic3—91no—
86C2×C3×C22⋊C4—912no—
87C2×(C2×D3)⋊C4—312no—
88C2×(C2×D3)⋊C4—912no—
89C23×C32—24no—
90C23×C32—24no—
91C23×C32—24no—
92C23×C32—23no—
93C23×C32C4×D314yes—
94C23×C32D1214yes—
95C23×C32#6714yes—
96C23×C32C2×C3×C414yes—
97C23×C32C3×D414yes—
98C23×C32C3×D414yes—
99C23×C32C22×D313yes—
100C23×C32C23×C313yes—
101C22×C3⋊D3—232no—
102C22×C3⋊D3—212no—
103C22×C3⋊D3—212no—
104C22×C3⋊D3C2×C3×C4124yescomplemented
105C22×C3⋊D3C23×C314yes—
106C22×C3×D3—324no—
107C22×C3×D3—34no—
108C22×C3×D3—632no—
109C22×C3×D3—612no—
110C22×C3×D3—612no—
111C2×C32×C4—224no—
112C2×C32×C4—212no—
113C2×C32×C4—212no—
114C2×C32×C4—24no—
115C2×C32×C4C22×D314yescomplemented
116C2×C32×C4C23×C314yescomplemented
117C2×C3⋊Dic3—212no—
118C2×C3⋊Dic3C23×C314yescomplemented
119C2×C3×Dic3—34no—
120C2×C3×Dic3—612no—
121C22×C22⋊C4—91noSylow
122C2×H(3,𝔽3)C2×C22⋊C4112yescomplemented
123C2×H(3,𝔽3)C23×C411yescharacteristic
124C2×H(3,𝔽3)C22×D411yescharacteristic
125C2×H(3,𝔽3)C22×D411yescharacteristic
126(C3⋊D3)⋊C3—28no—
127C24×C3—31no—
128C24×C3—31no—
129C24×C3—61no—
130C24×C3—98no—
131C24×C3—96no—
132C24×C3—184no—
133C24×C3—184no—
134C24×C3C2×C3×D311yescharacteristic
135C23×D3—38no—
136C23×D3—36no—
137C23×D3—64no—
138C23×D3—64no—
139C23×D3—98no—
140C23×D3—96no—
141C23×D3—184no—
142C23×D3—184no—
143C22×C3×C4—24no—
144C22×C3×C4—36no—
145C22×C3×C4—36no—
146C22×C3×C4—64no—
147C22×C3×C4—64no—
148C22×C3×C4—68no—
149C22×C3×C4—66no—
150C22×C3×C4—96no—
151C22×C3×C4—184no—
152C22×C3×C4C2×C3×D316yes—
153C22×Dic3—36no—
154C22×Dic3—64no—
155C22×Dic3—96no—
156C22×Dic3—184no—
157C3×C22⋊C4—916no—
158(C2×D3)⋊C4—316no—
159(C2×D3)⋊C4—916no—
160C22×C32—21no—
161C22×C32—216no—
162C22×C32—26no—
163C22×C32—26no—
164C22×C32—26no—
165C22×C32(C2×D3)⋊C4116yescomplemented
166C22×C32C3×C22⋊C4116yescomplemented
167C22×C32C2×C4×D316yes—
168C22×C32C2×D1216yes—
169C22×C32C2×#6716yes—
170C22×C32C22×C3×C416yes—
171C22×C32C2×C3×D416yes—
172C22×C32C2×C3×D416yes—
173C22×C32C23×D311yescharacteristic
174C22×C32C24×C311yescharacteristic
175C2×C3⋊D3—248no—
176C2×C3⋊D3—24no—
177C2×C3⋊D3—24no—
178C2×C3⋊D3C22×C3×C418yescomplemented
179C2×C3×D3—38no—
180C2×C3×D3—648no—
181C2×C3×D3—64no—
182C2×C3×D3—64no—
183C32×C4—28no—
184C32×C4—24no—
185C32×C4—24no—
186C3⋊Dic3—24no—
187C3×Dic3—64no—
188C25—91no—
189C23×C4—91no—
190C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
191C2×C22⋊C4—912no—
192H(3,𝔽3)C22×C22⋊C411yescharacteristic, complemented, Sylow
193C23×C3—34no—
194C23×C3—34no—
195C23×C3—34no—
196C23×C3—34no—
197C23×C3—34no—
198C23×C3—34no—
199C23×C3—33no—
200C23×C3—33no—
201C23×C3—64no—
202C23×C3—64no—
203C23×C3—64no—
204C23×C3—63no—
205C23×C3—94no—
206C23×C3—924no—
207C23×C3—1812no—
208C23×C3—1812no—
209C23×C3—1832no—
210C23×C3C3×C4×D314yes—
211C23×C3C3×D1214yes—
212C23×C3C3×#6714yes—
213C23×C3C22×C3×D313yes—
214C22×D3—34no—
215C22×D3—324no—
216C22×D3—632no—
217C22×D3—612no—
218C22×D3—612no—
219C22×D3—924no—
220C22×D3—94no—
221C22×D3—1812no—
222C22×D3—1812no—
223C22×D3—1832no—
224C2×C3×C4—212no—
225C2×C3×C4—34no—
226C2×C3×C4—34no—
227C2×C3×C4—612no—
228C2×C3×C4—612no—
229C2×C3×C4—624no—
230C2×C3×C4—64no—
231C2×C3×C4—94no—
232C2×C3×C4—1812no—
233C2×C3×C4C22×C3×D314yes—
234C2×Dic3—34no—
235C2×Dic3—612no—
236C2×Dic3—94no—
237C2×Dic3—1812no—
238C2×C32—212no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32C2×(C2×D3)⋊C4112yescomplemented
243C2×C32C2×C3×C22⋊C4112yescomplemented
244C2×C32C22×C4×D311yescharacteristic
245C2×C32C22×D1211yescharacteristic
246C2×C32C22×#6711yescharacteristic
247C2×C32C23×C3×C411yescharacteristic
248C2×C32C22×C3×D411yescharacteristic
249C2×C32C22×C3×D411yescharacteristic
250C3⋊D3—28no—
251C3×D3—68no—
252C24—98no—
253C24—96no—
254C24—184no—
255C24—184no—
256C24C2×(C3⋊D3)⋊C311yescharacteristic
257C22×C4—24no—
258C22×C4—96no—
259C22×C4—184no—
260C22×C4C2×(C3⋊D3)⋊C316yescomplemented
261C22⋊C4—916no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—36no—
265C22×C3—36no—
266C22×C3—36no—
267C22×C3—36no—
268C22×C3—36no—
269C22×C3—36no—
270C22×C3—316no—
271C22×C3—316no—
272C22×C3—61no—
273C22×C3—66no—
274C22×C3—66no—
275C22×C3—66no—
276C22×C3—616no—
277C22×C3—98no—
278C22×C3—184no—
279C22×C3—1848no—
280C22×C3—184no—
281C22×C3C3×(C2×D3)⋊C4116yes—
282C22×C3C2×C3×C4×D316yes—
283C22×C3C2×C3×D1216yes—
284C22×C3C2×C3×#6716yes—
285C22×C3C23×C3×D311yescharacteristic
286C2×D3—38no—
287C2×D3—648no—
288C2×D3—64no—
289C2×D3—64no—
290C2×D3—98no—
291C2×D3—184no—
292C2×D3—1848no—
293C2×D3—184no—
294C3×C4—24no—
295C3×C4—64no—
296C3×C4—64no—
297C3×C4—68no—
298C3×C4—184no—
299Dic3—64no—
300Dic3—184no—
301C32—21no—
302C32C22×(C2×D3)⋊C411yescharacteristic, complemented
303C32C22×C3×C22⋊C411yescharacteristic, complemented
304C23—94no—
305C23—924no—
306C23—1812no—
307C23—1812no—
308C23—1832no—
309C23C4×(C3⋊D3)⋊C314yes—
310C23C4⋊((C3⋊D3)⋊C3)14yes—
311C23#638⋊C314yes—
312C23C22×(C3⋊D3)⋊C313yes—
313C2×C4—212no—
314C2×C4—94no—
315C2×C4—1812no—
316C2×C4C22×(C3⋊D3)⋊C314yescomplemented
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—312no—
322C2×C3—312no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—61no—
326C2×C3—61no—
327C2×C3—612no—
328C2×C3—61no—
329C2×C3—188no—
330C2×C3C2×C3×(C2×D3)⋊C4112yes—
331C2×C3C22×C3×C4×D311yescharacteristic
332C2×C3C22×C3×D1211yescharacteristic
333C2×C3C22×C3×#6711yescharacteristic
334D3—68no—
335D3—188no—
336C22—98no—
337C22—184no—
338C22—1848no—
339C22—184no—
340C22#3752⋊C3116yesdirect product factor
341C22C2×C4×(C3⋊D3)⋊C316yes—
342C22C2×C4⋊((C3⋊D3)⋊C3)16yes—
343C22C2×#638⋊C316yes—
344C22C23×(C3⋊D3)⋊C311yescharacteristic
345C4—24no—
346C4—184no—
347C3—31no—
348C3—31no—
349C3—61no—
350C3C22×C3×(C2×D3)⋊C411yescharacteristic
351C2—188no—
352C2C2×#3752⋊C3112yesdirect product factor
353C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
354C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
355C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC24C2×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC23×C4×H(3,𝔽3)C2
socleC24×C3C2×C3×D3
prime radicalC24×(C3⋊D3)⋊C3C2
cosocleC22×C32C24×C3
hypercenterC23×C4(C3⋊D3)⋊C3
hypocenterC32C22×C3×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C4×H(3,𝔽3)C2semidirect
3. C23×(C3⋊Dic3)⋊C3C2semidirect
4. C2×#3752⋊C3C2direct (N × H)
5. C22×#3752C3semidirect
10. C23×(C3⋊D3)⋊C3C4semidirect
13. C22×C4×H(3,𝔽3)C22semidirect
15. C22×(C3⋊Dic3)⋊C3C22semidirect
16. #3752⋊C3C22direct (N × H)
20. C23×C32×C4D3semidirect
21. C23×C32×C4C2×C3semidirect
22. C23×C3⋊Dic3C2×C3semidirect
24. C2×#3752C2×C3semidirect
33. C22×(C3⋊D3)⋊C3C2×C4semidirect
36. C2×C4×H(3,𝔽3)C23semidirect
38. C2×(C3⋊Dic3)⋊C3C23semidirect
47. C23×C3⋊D3C3×C4semidirect
57. C22×C32×C4C2×D3semidirect
58. C22×C32×C4C22×C3semidirect
60. C22×C3⋊Dic3C22×C3semidirect
63. #3752C22×C3semidirect
65. C22×H(3,𝔽3)C22⋊C4semidirect
73. C2×(C3⋊D3)⋊C3C22×C4semidirect
104. C22×C3⋊D3C2×C3×C4semidirect
115. C2×C32×C4C22×D3semidirect
116. C2×C32×C4C23×C3semidirect
118. C2×C3⋊Dic3C23×C3semidirect
122. C2×H(3,𝔽3)C2×C22⋊C4semidirect
165. C22×C32(C2×D3)⋊C4semidirect
166. C22×C32C3×C22⋊C4semidirect
178. C2×C3⋊D3C22×C3×C4semidirect
190. C23×C4(C3⋊D3)⋊C3semidirect
192. H(3,𝔽3)C22×C22⋊C4semidirect
242. C2×C32C2×(C2×D3)⋊C4semidirect
243. C2×C32C2×C3×C22⋊C4semidirect
260. C22×C4C2×(C3⋊D3)⋊C3semidirect
302. C32C22×(C2×D3)⋊C4semidirect
303. C32C22×C3×C22⋊C4semidirect
316. C2×C4C22×(C3⋊D3)⋊C3semidirect
340. C22#3752⋊C3direct (N × H)
352. C2C2×#3752⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C24×(C3⋊D3)⋊C32
C23×C4×H(3,𝔽3)2
C23×(C3⋊Dic3)⋊C32
C2×#3752⋊C32
C22×#37523
C22×C3×(C2×D3)⋊C43