Finite Group Databasebeta

← C3×#73322 #11800898

Subgroups — C3×#73322 #11800898

Order 1152.

1725 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#67C213yesmaximal
2C2×C32×#95C211yesmaximal, characteristic, complemented
3C3×C4×#217C213yesmaximal, complemented
4C3×#4666C213yesmaximal, complemented
5C3×D3×#95C211yesmaximal, characteristic, complemented
6C3×#4649C213yesmaximal, complemented
7C3×#4648C211yesmaximal, characteristic, complemented
8C3×#2275—31nomaximal
9#73322C311yesmaximal, characteristic, direct product factor
10C23×C32×C4C2213yes—
11C22×C3×#67C2211yescharacteristic
12C22×C3×C4×D3C2213yes—
13C2×C3×#217C2213yes—
14C3×C4×#67C4124yescomplemented
15C2×C3×(C2×D3)⋊C4C2213yes—
16C2×C3×Dic3⋊C4C2213yes—
17C2×C3×C4×Dic3C2216yes—
18C2×C3×C4×Dic3C2213yes—
19C32×#95C2211yescharacteristic
20C32×#95C2213yes—
21C3×C3⋊#95C2213yes—
22C3×C3⋊#95C2213yes—
23C2×C3×C4×D4—33no—
24C2×C4×#67C2×C313yes—
25C3×C4×C4⋊C4—33no—
26C3×C4×C22⋊C4—33no—
27C3×C4×C22⋊C4—33no—
28C3×#376—31no—
29C2×C3×#95—21no—
30C2×C3×#95—31no—
31C2×C3×#95D311yescharacteristic, complemented
32C2×C3×#95C2×C311yescharacteristic, complemented
33C4×#217C2×C313yescomplemented
34#4666C2×C313yescomplemented
35D3×#95C2×C311yescharacteristic, complemented
36#4649C2×C313yescomplemented
37#4648C2×C311yescharacteristic, complemented
38C24×C32C2311yescharacteristic
39C23×C3×D3C2311yescharacteristic
40C22×C32×C4—46no—
41C22×C32×C4C2×C4112yescomplemented
42C22×C32×C4C2313yes—
43C22×C32×C4C2313yes—
44C2×C3×#67C2×C4112yes—
45C22×C3×Dic3C2313yes—
46C22×C3×Dic3C2313yes—
47C22×C3×Dic3C2311yescharacteristic
48C2×C3×C4×D3—46no—
49C2×C3×C4×D3C2×C4112yescomplemented
50C3×#217C2×C4112yescomplemented
51C3×(C2×D3)⋊C4C2×C4112yescomplemented
52C3×Dic3⋊C4C2×C4112yescomplemented
53C3×C4×Dic3—46no—
54C3×C4×Dic3C2×C4112yescomplemented
55#2275—31noSylow
56C22×C3×D4—31no—
57C23×C3×C4—23no—
58C23×C3×C4—33no—
59C23×C3×C4C2×D313yes—
60C23×C3×C4C22×C313yes—
61C22×#67C22×C311yescharacteristic
62C22×C4×D3C22×C313yes—
63C3×C4×D4—324no—
64C2×C3×C4⋊C4—33no—
65C2×C3×C22⋊C4—33no—
66C2×C3×C22⋊C4—33no—
67C2×C3×C42—33no—
68C2×C3×C42—36no—
69C2×#217C22×C313yes—
70C4×#67C3×C4124yescomplemented
71C2×(C2×D3)⋊C4C22×C313yes—
72C2×Dic3⋊C4C22×C313yes—
73C2×C4×Dic3C22×C316yes—
74C2×C4×Dic3C22×C313yes—
75C3×#95—21no—
76C3×#95—23no—
77C3×#95—33no—
78C3×#95—33no—
79C3×#95C2×D311yescharacteristic
80C3×#95C2×D313yes—
81C3×#95C22×C311yescharacteristic
82C3×#95C22×C313yes—
83C3⋊#95C22×C313yes—
84C3⋊#95C22×C313yes—
85C23×C32—23no—
86C23×C32—21no—
87C23×C32C22×C416yes—
88C23×C32C2411yescharacteristic
89C22×C3×D3—23no—
90C22×C3×D3—21no—
91C22×C3×D3C22×C416yes—
92C2×C32×C4—26no—
93C2×C32×C4—26no—
94C2×C32×C4—412no—
95C2×C32×C4C22×C416yes—
96C2×C32×C4C22×C416yes—
97C3×#67C42116yescomplemented
98C2×C3×Dic3—23no—
99C2×C3×Dic3—26no—
100C2×C3×Dic3—23no—
101C2×C3×Dic3C22×C416yes—
102C2×C3×Dic3C22×C416yes—
103C2×C3×Dic3C22×C416yes—
104C3×C4×D3—412no—
105C2×C4×D4—33no—
106C4×C4⋊C4—33no—
107C4×C22⋊C4—33no—
108C4×C22⋊C4—33no—
109#376—31no—
110C2×#95—31no—
111C2×#95C3×D311yescharacteristic, complemented
112C24×C3—21no—
113C24×C3—31no—
114C24×C3C22×D311yescharacteristic
115C24×C3C23×C311yescharacteristic
116C23×D3C23×C311yescharacteristic
117C2×C3×D4—312no—
118C22×C3×C4—212no—
119C22×C3×C4—23no—
120C22×C3×C4—23no—
121C22×C3×C4—31no—
122C22×C3×C4—33no—
123C22×C3×C4—33no—
124C22×C3×C4—312no—
125C22×C3×C4—46no—
126C22×C3×C4—46no—
127C22×C3×C4—412no—
128C22×C3×C4—126no—
129C22×C3×C4C4×D3112yescomplemented
130C22×C3×C4C2×C3×C4112yescomplemented
131C22×C3×C4C22×D313yes—
132C22×C3×C4C22×D313yes—
133C22×C3×C4C23×C313yes—
134C22×C3×C4C23×C313yes—
135C2×#67C2×C3×C4112yes—
136C22×Dic3C23×C313yes—
137C22×Dic3C23×C313yes—
138C22×Dic3C23×C311yescharacteristic
139C2×C4×D3—46no—
140C2×C4×D3C2×C3×C4112yescomplemented
141C3×C4⋊C4—312no—
142C3×C22⋊C4—312no—
143C3×C22⋊C4—312no—
144C3×C42—312no—
145C3×C42—126no—
146#217C2×C3×C4112yescomplemented
147(C2×D3)⋊C4C2×C3×C4112yescomplemented
148Dic3⋊C4C2×C3×C4112yescomplemented
149C4×Dic3—46no—
150C4×Dic3C2×C3×C4112yescomplemented
151C22×C32—26no—
152C22×C32—26no—
153C22×C32C2×C4214yes—
154C22×C32C23×C413yes—
155C22×C32H(5,𝔽2)13yes—
156C22×C322−1+411yescharacteristic
157C2×C3×D3—26no—
158C2×C3×D3—26no—
159C2×C3×D3C2×C4214yes—
160C32×C4—26no—
161C32×C4—26no—
162C3×Dic3—26no—
163C3×Dic3—26no—
164C3×Dic3C2×C4214yes—
165C22×D4—31no—
166C23×C4—33no—
167C23×C4C2×C3×D313yes—
168C4×D4—324no—
169C2×C4⋊C4—33no—
170C2×C22⋊C4—33no—
171C2×C22⋊C4—33no—
172C2×C42—33no—
173C2×C42—36no—
174#95—33no—
175#95—33no—
176#95C2×C3×D311yescharacteristic
177#95C2×C3×D313yes—
178C23×C3—26no—
179C23×C3—22no—
180C23×C3—23no—
181C23×C3—21no—
182C23×C3—23no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—26no—
186C23×C3—36no—
187C23×C3—63no—
188C23×C3—61no—
189C23×C3C2×C4×D316yes—
190C23×C3C22×C3×C416yes—
191C23×C3C23×D311yescharacteristic
192C23×C3C24×C311yescharacteristic
193C22×D3—23no—
194C22×D3—21no—
195C22×D3C22×C3×C416yes—
196C3×D4—316no—
197C2×C3×C4—26no—
198C2×C3×C4—26no—
199C2×C3×C4—26no—
200C2×C3×C4—26no—
201C2×C3×C4—26no—
202C2×C3×C4—26no—
203C2×C3×C4—36no—
204C2×C3×C4—36no—
205C2×C3×C4—36no—
206C2×C3×C4—46no—
207C2×C3×C4—46no—
208C2×C3×C4—412no—
209C2×C3×C4—412no—
210C2×C3×C4—424no—
211C2×C3×C4—63no—
212C2×C3×C4—66no—
213C2×C3×C4—63no—
214C2×C3×C4—1212no—
215C2×C3×C4C2×C4×D316yes—
216C2×C3×C4C2×C4×D316yes—
217C2×C3×C4C22×C3×C416yes—
218C2×C3×C4C22×C3×C416yes—
219#67C3×C42116yescomplemented
220C2×Dic3—23no—
221C2×Dic3—26no—
222C2×Dic3—23no—
223C2×Dic3C22×C3×C416yes—
224C2×Dic3C22×C3×C416yes—
225C2×Dic3C22×C3×C416yes—
226C4×D3—412no—
227C2×C32—24no—
228C2×C32C22×C4211yescharacteristic
229C2×C32#51813yes—
230C2×C32#52013yes—
231C3×D3—24no—
232C24—31no—
233C24C22×C3×D311yescharacteristic
234C2×D4—312no—
235C22×C4—31no—
236C22×C4—33no—
237C22×C4—33no—
238C22×C4—312no—
239C22×C4—46no—
240C22×C4—126no—
241C22×C4C3×C4×D3112yescomplemented
242C22×C4C22×C3×D313yes—
243C22×C4C22×C3×D313yes—
244C4⋊C4—312no—
245C22⋊C4—312no—
246C22⋊C4—312no—
247C42—312no—
248C42—126no—
249C22×C3—212no—
250C22×C3—212no—
251C22×C3—26no—
252C22×C3—26no—
253C22×C3—26no—
254C22×C3—26no—
255C22×C3—23no—
256C22×C3—23no—
257C22×C3—21no—
258C22×C3—24no—
259C22×C3—34no—
260C22×C3—66no—
261C22×C3—66no—
262C22×C3C42×D314yes—
263C22×C3C2×C3×C4214yes—
264C22×C3C22×C4×D313yes—
265C22×C3#100413yes—
266C22×C3#100711yescharacteristic
267C22×C3C23×C3×C413yes—
268C22×C3C3×H(5,𝔽2)13yes—
269C22×C3C3×2−1+411yescharacteristic
270C2×D3—26no—
271C2×D3—26no—
272C2×D3C2×C3×C4214yes—
273C3×C4—26no—
274C3×C4—26no—
275C3×C4—26no—
276C3×C4—26no—
277C3×C4—34no—
278C3×C4—46no—
279C3×C4—46no—
280C3×C4—66no—
281C3×C4—66no—
282Dic3—26no—
283Dic3—26no—
284Dic3C2×C3×C4214yes—
285C32#227511yescharacteristic, complemented, Sylow
286C23—23no—
287C23—21no—
288C23—36no—
289C23—63no—
290C23—61no—
291C23C2×C3×C4×D316yes—
292C23C23×C3×D311yescharacteristic
293D4—316no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—36no—
297C2×C4—36no—
298C2×C4—36no—
299C2×C4—412no—
300C2×C4—63no—
301C2×C4—66no—
302C2×C4—63no—
303C2×C4—1212no—
304C2×C4C2×C3×C4×D316yes—
305C2×C4C2×C3×C4×D316yes—
306C2×C3—28no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—23no—
310C2×C3—21no—
311C2×C3—23no—
312C2×C3—64no—
313C2×C3C2×C42×D311yescharacteristic
314C2×C3#548613yes—
315C2×C3#550413yes—
316C2×C3C22×C3×C4211yescharacteristic
317C2×C3C3×#51813yes—
318C2×C3C3×#52013yes—
319D3—24no—
320C22—26no—
321C22—26no—
322C22—34no—
323C22—66no—
324C22—66no—
325C22C3×C42×D314yes—
326C22C22×C3×C4×D313yes—
327C22C3×#100413yes—
328C22C3×#100711yescharacteristic
329C4—26no—
330C4—26no—
331C4—34no—
332C4—66no—
333C4—66no—
334C3—21no—
335C3#7332211yescharacteristic, direct product factor
336C3C3×#227511yescharacteristic, complemented
337C2—24no—
338C2—64no—
339C2C2×C3×C42×D311yescharacteristic
340C2C3×#548613yes—
341C2C3×#550413yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC2×C3C22×C3×C42
FrattiniC23C23×C3×D3
FittingC2×C32×#95C2
socleC23×C32C24
prime radicalC22×C3×#67C22
cosocleC23×C3C24×C3
hypercenterC2×C3×#95D3
hypocenterC3C3×#2275
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#95

Complements

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

NH ≅ G/Ntype
2. C2×C32×#95C2semidirect
3. C3×C4×#217C2semidirect
4. C3×#4666C2semidirect
5. C3×D3×#95C2semidirect
6. C3×#4649C2semidirect
7. C3×#4648C2semidirect
9. #73322C3direct (N × H)
14. C3×C4×#67C4semidirect
31. C2×C3×#95D3semidirect
32. C2×C3×#95C2×C3semidirect
33. C4×#217C2×C3semidirect
34. #4666C2×C3semidirect
35. D3×#95C2×C3semidirect
36. #4649C2×C3semidirect
37. #4648C2×C3semidirect
41. C22×C32×C4C2×C4semidirect
49. C2×C3×C4×D3C2×C4semidirect
50. C3×#217C2×C4semidirect
51. C3×(C2×D3)⋊C4C2×C4semidirect
52. C3×Dic3⋊C4C2×C4semidirect
54. C3×C4×Dic3C2×C4semidirect
70. C4×#67C3×C4semidirect
97. C3×#67C42semidirect
111. C2×#95C3×D3semidirect
129. C22×C3×C4C4×D3semidirect
130. C22×C3×C4C2×C3×C4semidirect
140. C2×C4×D3C2×C3×C4semidirect
146. #217C2×C3×C4semidirect
147. (C2×D3)⋊C4C2×C3×C4semidirect
148. Dic3⋊C4C2×C3×C4semidirect
150. C4×Dic3C2×C3×C4semidirect
219. #67C3×C42semidirect
241. C22×C4C3×C4×D3semidirect
285. C32#2275semidirect
335. C3#73322direct (N × H)
336. C3C3×#2275semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#672
C2×C32×#952
C3×C4×#2172
C3×#46662
C3×D3×#952
C3×#46492
C3×#46482
C3×#22753
#733223