Finite Group Databasebeta

← C22×#3828⋊C3 #13164902

Subgroups — C22×#3828⋊C3 #13164902

Order 1728.

3351 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
1C23×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C212yesmaximal, complemented
3C2×#3828⋊C3C2124yesmaximal, direct product factor
4C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
5C22×C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
6C22×#634⋊C3C211yesmaximal, characteristic, complemented
7C22×#3828C311yesmaximal, characteristic, complemented
8C22×C3×#235—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C22112yescomplemented
14C2×#638⋊C3C22124yescomplemented
15C22×(C3⋊Dic3)⋊C3C2212yes—
16#3828⋊C3C22164yesdirect product factor
17C2×C4⋊((C3⋊D3)⋊C3)C22112yescomplemented
18C2×C4×(C3⋊D3)⋊C3C22124yescomplemented
19C2×#634⋊C3C22112yescomplemented
20C23×C32×C4—21no—
21C23×C32×C4D311yescharacteristic, complemented
22C23×C32×C4C2×C311yescharacteristic, complemented
23C22×#638C2×C312yescomplemented
24C2×#3828C2×C3124yescomplemented
25C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
26C22×C4×C3⋊D3C2×C312yescomplemented
27C22×#634C2×C311yescharacteristic, complemented
28C22×C3×#67—32no—
29C2×C3×#235—324no—
30C22×C3×D12—31no—
31C22×C3×C4×D3—32no—
32C22×C3×Dic6—31no—
33C23×H(3,𝔽3)—24no—
34C23×H(3,𝔽3)C2316yes—
35C23×H(3,𝔽3)C2311yescharacteristic
36C22×(C3⋊D3)⋊C3—28no—
37C22×(C3⋊D3)⋊C3C23112yes—
38C2×C4×H(3,𝔽3)C2316yes—
39C2×C4×H(3,𝔽3)C2316yes—
40C2×C4×H(3,𝔽3)C23116yescomplemented
41#638⋊C3C23132yescomplemented
42C2×(C3⋊Dic3)⋊C3C23112yes—
43C4⋊((C3⋊D3)⋊C3)C23116yescomplemented
44C4×(C3⋊D3)⋊C3C23132yescomplemented
45#634⋊C3C23116yescomplemented
46C22×C3×#41—91no—
47C22×#235—31no—
48C22×#235—91no—
49C24×C32—21no—
50C24×C32C2×D311yescharacteristic
51C24×C32C22×C311yescharacteristic
52C23×C3⋊D3C22×C312yes—
53C23×C3×D3—32no—
54C22×C32×C4—21no—
55C22×C32×C4—21no—
56C22×C32×C4—212no—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D3112yescomplemented
60C22×C32×C4C22×C311yescharacteristic
61C22×C32×C4C22×C311yescharacteristic
62C22×C32×C4C22×C3112yescomplemented
63C2×#638C22×C3124yescomplemented
64C22×C3⋊Dic3C22×C312yes—
65#3828C22×C3164yescomplemented
66C2×C4⋊(C3⋊D3)C22×C3112yescomplemented
67C2×C4×C3⋊D3C22×C3124yescomplemented
68C2×#634C22×C3112yescomplemented
69C2×C3×#67—324no—
70C22×C3×Dic3—32no—
71C3×#235—364no—
72C2×C3×D12—312no—
73C2×C3×C4×D3—324no—
74C2×C3×Dic6—312no—
75C22×H(3,𝔽3)—212no—
76C22×H(3,𝔽3)#4114yescomplemented
77C22×H(3,𝔽3)C2414yes—
78C22×H(3,𝔽3)C2413yes—
79C2×(C3⋊D3)⋊C3—224no—
80C2×(C3⋊D3)⋊C3C2418yes—
81C4×H(3,𝔽3)C2414yes—
82C4×H(3,𝔽3)C2414yes—
83(C3⋊Dic3)⋊C3C2418yes—
84C2×C3×#41—924no—
85C22×C3×Q8—91no—
86C22×C3×D4—91no—
87C22×C3×D4—92no—
88C23×C3×C4—31no—
89C23×C3×C4—31no—
90C23×C3×C4—61no—
91C23×C3×C4—92no—
92C23×C3×C4C3×D311yescharacteristic
93C22×#67—32no—
94C22×#67—92no—
95C2×#235—324no—
96C2×#235—924no—
97C22×D12—31no—
98C22×D12—91no—
99C22×C4×D3—32no—
100C22×C4×D3—92no—
101C22×Dic6—31no—
102C22×Dic6—91no—
103C23×C32—28no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32—26no—
107C23×C32—21no—
108C23×C32C22×D316yes—
109C23×C32C22×D311yescharacteristic
110C23×C32C23×C316yes—
111C23×C32C23×C311yescharacteristic
112C22×C3⋊D3—28no—
113C22×C3⋊D3C23×C3112yes—
114C22×C3×D3—312no—
115C22×C3×D3—68no—
116C2×C32×C4—26no—
117C2×C32×C4—26no—
118C2×C32×C4—216no—
119C2×C32×C4C22×D316yes—
120C2×C32×C4C22×D316yes—
121C2×C32×C4C22×D3116yescomplemented
122C2×C32×C4C23×C316yes—
123C2×C32×C4C23×C316yes—
124C2×C32×C4C23×C3116yescomplemented
125#638C23×C3132yescomplemented
126C2×C3⋊Dic3C23×C3112yes—
127C4⋊(C3⋊D3)C23×C3116yescomplemented
128C4×C3⋊D3C23×C3132yescomplemented
129#634C23×C3116yescomplemented
130C3×#67—332no—
131C2×C3×Dic3—312no—
132C3×D12—316no—
133C3×C4×D3—332no—
134C3×Dic6—316no—
135C22×#41—91noSylow
136C2×H(3,𝔽3)—24no—
137C2×H(3,𝔽3)C2×#4116yescomplemented
138C2×H(3,𝔽3)C2511yescharacteristic
139(C3⋊D3)⋊C3—28no—
140C24×C3—31no—
141C24×C3—31no—
142C24×C3—61no—
143C24×C3—92no—
144C24×C3C2×C3×D311yescharacteristic
145C23×D3—32no—
146C23×D3—92no—
147C3×#41—964no—
148C2×C3×Q8—912no—
149C2×C3×D4—912no—
150C2×C3×D4—924no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—312no—
156C22×C3×C4—312no—
157C22×C3×C4—61no—
158C22×C3×C4—61no—
159C22×C3×C4—612no—
160C22×C3×C4—924no—
161C22×C3×C4—92no—
162C22×C3×C4C2×C3×D311yescharacteristic
163C22×C3×C4C2×C3×D311yescharacteristic
164C22×C3×C4C2×C3×D3112yes—
165C2×#67—324no—
166C2×#67—924no—
167C22×Dic3—32no—
168C22×Dic3—92no—
169#235—364no—
170#235—964no—
171C2×D12—312no—
172C2×D12—912no—
173C2×C4×D3—324no—
174C2×C4×D3—924no—
175C2×Dic6—312no—
176C2×Dic6—912no—
177C22×C32—224no—
178C22×C32—212no—
179C22×C32—212no—
180C22×C32—24no—
181C22×C32—23no—
182C22×C32—24no—
183C22×C32#23514yescomplemented
184C22×C32C3×#4114yescomplemented
185C22×C32C23×D314yes—
186C22×C32C23×D313yes—
187C22×C32C24×C314yes—
188C22×C32C24×C313yes—
189C2×C3⋊D3—224no—
190C2×C3⋊D3C24×C318yes—
191C2×C3×D3—38no—
192C2×C3×D3—624no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4C23×D314yes—
196C32×C4C23×D314yes—
197C32×C4C24×C314yes—
198C32×C4C24×C314yes—
199C3⋊Dic3C24×C318yes—
200C3×Dic3—38no—
201C2×#41—924no—
202C22×Q8—91no—
203C22×D4—92no—
204C22×D4—91no—
205C23×C4—92no—
206C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
207H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
208C23×C3—24no—
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—36no—
212C23×C3—36no—
213C23×C3—64no—
214C23×C3—64no—
215C23×C3—68no—
216C23×C3—66no—
217C23×C3—61no—
218C23×C3—912no—
219C23×C3—188no—
220C23×C3C22×C3×D316yes—
221C23×C3C22×C3×D311yescharacteristic
222C22×D3—312no—
223C22×D3—68no—
224C22×D3—912no—
225C22×D3—188no—
226C3×Q8—916no—
227C3×D4—916no—
228C3×D4—932no—
229C2×C3×C4—36no—
230C2×C3×C4—36no—
231C2×C3×C4—36no—
232C2×C3×C4—36no—
233C2×C3×C4—316no—
234C2×C3×C4—316no—
235C2×C3×C4—66no—
236C2×C3×C4—66no—
237C2×C3×C4—616no—
238C2×C3×C4—932no—
239C2×C3×C4—912no—
240C2×C3×C4C22×C3×D316yes—
241C2×C3×C4C22×C3×D316yes—
242C2×C3×C4C22×C3×D3116yes—
243#67—332no—
244#67—932no—
245C2×Dic3—312no—
246C2×Dic3—912no—
247D12—316no—
248D12—916no—
249C4×D3—332no—
250C4×D3—932no—
251Dic6—316no—
252Dic6—916no—
253C2×C32—28no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32—26no—
257C2×C32—21no—
258C2×C32C2×#23516yescomplemented
259C2×C32C2×C3×#4116yescomplemented
260C2×C32C24×D311yescharacteristic
261C2×C32C25×C311yescharacteristic
262C3⋊D3—28no—
263C3×D3—68no—
264C24—92no—
265C24C2×(C3⋊D3)⋊C311yescharacteristic
266#41—964no—
267C2×Q8—912no—
268C2×D4—924no—
269C2×D4—912no—
270C22×C4—92no—
271C22×C4—924no—
272C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
273C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
274C22×C4C2×(C3⋊D3)⋊C3112yescomplemented
275C22×C3—212no—
276C22×C3—33no—
277C22×C3—33no—
278C22×C3—34no—
279C22×C3—34no—
280C22×C3—34no—
281C22×C3—34no—
282C22×C3—612no—
283C22×C3—612no—
284C22×C3—624no—
285C22×C3—64no—
286C22×C3—63no—
287C22×C3—64no—
288C22×C3—98no—
289C22×C3—1824no—
290C22×C3C3×#23514yes—
291C22×C3C23×C3×D314yes—
292C22×C3C23×C3×D313yes—
293C2×D3—38no—
294C2×D3—624no—
295C2×D3—98no—
296C2×D3—1824no—
297C3×C4—34no—
298C3×C4—34no—
299C3×C4—34no—
300C3×C4—34no—
301C3×C4—64no—
302C3×C4—64no—
303C3×C4—98no—
304C3×C4C23×C3×D314yes—
305C3×C4C23×C3×D314yes—
306Dic3—38no—
307Dic3—98no—
308C32—21no—
309C32C22×#23511yescharacteristic, complemented
310C32C22×C3×#4111yescharacteristic, complemented
311C23—24no—
312C23—912no—
313C23—188no—
314C23C22×(C3⋊D3)⋊C316yes—
315C23C22×(C3⋊D3)⋊C311yescharacteristic
316Q8—916no—
317D4—932no—
318D4—916no—
319C2×C4—912no—
320C2×C4—932no—
321C2×C4C22×(C3⋊D3)⋊C316yes—
322C2×C4C22×(C3⋊D3)⋊C316yes—
323C2×C4C22×(C3⋊D3)⋊C3116yescomplemented
324C2×C3—24no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—36no—
328C2×C3—36no—
329C2×C3—64no—
330C2×C3—64no—
331C2×C3—68no—
332C2×C3—61no—
333C2×C3—66no—
334C2×C3—188no—
335C2×C3C2×C3×#23516yes—
336C2×C3C24×C3×D311yescharacteristic
337D3—68no—
338D3—188no—
339C22—212no—
340C22—98no—
341C22—1824no—
342C22#3828⋊C314yesdirect product factor
343C22C23×(C3⋊D3)⋊C314yes—
344C22C23×(C3⋊D3)⋊C313yes—
345C4—98no—
346C4C23×(C3⋊D3)⋊C314yes—
347C4C23×(C3⋊D3)⋊C314yes—
348C3—31no—
349C3—31no—
350C3—61no—
351C3C22×C3×#23511yescharacteristic
352C2—24no—
353C2—188no—
354C2C2×#3828⋊C316yesdirect product factor
355C2C24×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22×C4C2×(C3⋊D3)⋊C3
commutator, cosocleC2×C32C25×C3
FrattiniC2×C3C24×C3×D3
FittingC23×C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
hypercenterC23×C4(C3⋊D3)⋊C3
hypocenterC32C22×C3×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C4×H(3,𝔽3)C2semidirect
2. C22×#638⋊C3C2semidirect
3. C2×#3828⋊C3C2direct (N × H)
4. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
5. C22×C4×(C3⋊D3)⋊C3C2semidirect
6. C22×#634⋊C3C2semidirect
7. C22×#3828C3semidirect
13. C22×C4×H(3,𝔽3)C22semidirect
14. C2×#638⋊C3C22semidirect
16. #3828⋊C3C22direct (N × H)
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
18. C2×C4×(C3⋊D3)⋊C3C22semidirect
19. C2×#634⋊C3C22semidirect
21. C23×C32×C4D3semidirect
22. C23×C32×C4C2×C3semidirect
23. C22×#638C2×C3semidirect
24. C2×#3828C2×C3semidirect
25. C22×C4⋊(C3⋊D3)C2×C3semidirect
26. C22×C4×C3⋊D3C2×C3semidirect
27. C22×#634C2×C3semidirect
40. C2×C4×H(3,𝔽3)C23semidirect
41. #638⋊C3C23semidirect
43. C4⋊((C3⋊D3)⋊C3)C23semidirect
44. C4×(C3⋊D3)⋊C3C23semidirect
45. #634⋊C3C23semidirect
59. C22×C32×C4C2×D3semidirect
62. C22×C32×C4C22×C3semidirect
63. C2×#638C22×C3semidirect
65. #3828C22×C3semidirect
66. C2×C4⋊(C3⋊D3)C22×C3semidirect
67. C2×C4×C3⋊D3C22×C3semidirect
68. C2×#634C22×C3semidirect
76. C22×H(3,𝔽3)#41semidirect
121. C2×C32×C4C22×D3semidirect
124. C2×C32×C4C23×C3semidirect
125. #638C23×C3semidirect
127. C4⋊(C3⋊D3)C23×C3semidirect
128. C4×C3⋊D3C23×C3semidirect
129. #634C23×C3semidirect
137. C2×H(3,𝔽3)C2×#41semidirect
183. C22×C32#235semidirect
184. C22×C32C3×#41semidirect
206. C23×C4(C3⋊D3)⋊C3semidirect
207. H(3,𝔽3)C22×#41semidirect
258. C2×C32C2×#235semidirect
259. C2×C32C2×C3×#41semidirect
274. C22×C4C2×(C3⋊D3)⋊C3semidirect
309. C32C22×#235semidirect
310. C32C22×C3×#41semidirect
323. C2×C4C22×(C3⋊D3)⋊C3semidirect
342. C22#3828⋊C3direct (N × H)
354. C2C2×#3828⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×H(3,𝔽3)2
C22×#638⋊C32
C2×#3828⋊C32
C22×C4⋊((C3⋊D3)⋊C3)2
C22×C4×(C3⋊D3)⋊C32
C22×#634⋊C32
C22×#38283
C22×C3×#2353