Finite Group Databasebeta

← C2×C3×#5574 #11900073

Subgroups — C2×C3×#5574 #11900073

Order 1152.

3429 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#239C211yesmaximal, characteristic
2C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
3C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
4C3×#5574C2116yesmaximal, direct product factor
5C2×C3×#893C213yesmaximal, complemented
6C2×C3×D3×C4⋊C4C213yesmaximal, complemented
7C2×C3×C4×D12C213yesmaximal, complemented
8C2×C3×#872C213yesmaximal, complemented
9C2×C3×#520—31nomaximal
10C2×#5574C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C2×C3×#239C4116yescomplemented
13C2×C3×#239C2218yes—
14C22×C3×D12C2213yes—
15C22×C3×C4×D3C2216yes—
16C22×C3×C4×D3C2213yes—
17C32×C4×Q8C2218yescomplemented
18C2×C32×C4⋊C4C2213yes—
19C2×C32×C42C2213yes—
20C3×Q8×Dic3C2218yescomplemented
21C2×C3×(C2×D3)⋊C4C2213yes—
22C2×C3×(C2×D3)⋊C4C2213yes—
23C2×C3×C4⋊Dic3C2213yes—
24C2×C3×Dic3⋊C4C2213yes—
25C2×C3×Dic3⋊C4C2211yescharacteristic
26C2×C3×C4×Dic3C2213yes—
27C3×#893C22124yescomplemented
28C3×D3×C4⋊C4C22124yescomplemented
29C3×C4×D12C22124yescomplemented
30C3×#872C22124yescomplemented
31C22×C3×#41—31no—
32C22×#239C2×C311yescharacteristic
33C3×#520—316no—
34C2×C3×C4×Q8—21no—
35C2×C3×C4×Q8—31no—
36C2×C3×C4×Q8D311yescharacteristic, complemented
37C2×C3×C4×Q8C2×C311yescharacteristic, complemented
38C2×C3×C4×D4—33no—
39C2×C3×C4×D4—33no—
40C2×C3×#117—33no—
41C22×C3×C4⋊C4—33no—
42C2×Q8×Dic3C2×C311yescharacteristic, complemented
43#5574C2×C3116yesdirect product factor
44C2×#893C2×C313yescomplemented
45C2×D3×C4⋊C4C2×C313yescomplemented
46C2×C4×D12C2×C313yescomplemented
47C2×#872C2×C313yescomplemented
48C23×C3×D3C2313yes—
49C2×C32×Q8C2×C418yescomplemented
50C2×C32×Q8C2314yes—
51C22×C32×C4C2311yescharacteristic
52C22×C32×C4C2313yes—
53C22×C32×C4C2313yes—
54C22×C3×Dic3C2311yescharacteristic
55C22×C3×Dic3C2313yes—
56C22×C3×Dic3C2311yescharacteristic
57C3×#239C2×C4164yescomplemented
58C2×C3×D12C2×C4124yescomplemented
59C2×C3×D12C23112yes—
60C2×C3×C4×D3—412no—
61C2×C3×C4×D3C2×C4124yescomplemented
62C2×C3×C4×D3C23124yes—
63C2×C3×C4×D3C23112yes—
64C32×C4⋊C4C23112yes—
65C32×C42C23112yes—
66C3×(C2×D3)⋊C4C23112yes—
67C3×(C2×D3)⋊C4C23112yes—
68C3×C4⋊Dic3C23112yes—
69C3×Dic3⋊C4C23112yes—
70C3×Dic3⋊C4C2314yes—
71C3×C4×Dic3C23112yes—
72C2×#520—31noSylow
73C2×C3×#41—316no—
74C2×C3×#41—38no—
75C22×C3×Q8—21no—
76C22×C3×Q8C2×D311yescharacteristic
77C22×C3×Q8C22×C311yescharacteristic
78C22×C3×D4—33no—
79C23×C3×C4—33no—
80C23×C3×C4—36no—
81C2×#239C3×C4116yescomplemented
82C2×#239C22×C318yes—
83C22×D12C22×C313yes—
84C22×C4×D3C22×C316yes—
85C22×C4×D3C22×C313yes—
86C3×C4×Q8—28no—
87C3×C4×Q8—38no—
88C3×C4×Q8C2×D318yescomplemented
89C3×C4×Q8C22×C318yescomplemented
90C3×C4×D4—324no—
91C3×C4×D4—324no—
92C3×#117—324no—
93C2×C3×C4⋊C4—23no—
94C2×C3×C4⋊C4—33no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C4⋊C4—33no—
97C2×C3×C4⋊C4—324no—
98C2×C3×C4⋊C4C2×D313yes—
99C2×C3×C4⋊C4C22×C313yes—
100C2×C3×C22⋊C4—33no—
101C2×C3×C22⋊C4—33no—
102C2×C3×C42—23no—
103C2×C3×C42—33no—
104C2×C3×C42C2×D313yes—
105C2×C3×C42C22×C313yes—
106Q8×Dic3C22×C318yescomplemented
107C2×(C2×D3)⋊C4C22×C313yes—
108C2×(C2×D3)⋊C4C22×C313yes—
109C2×C4⋊Dic3C22×C313yes—
110C2×Dic3⋊C4C22×C311yescharacteristic
111C2×Dic3⋊C4C22×C313yes—
112C2×C4×Dic3C22×C313yes—
113#893C22×C3124yescomplemented
114D3×C4⋊C4C22×C3124yescomplemented
115C4×D12C22×C3124yescomplemented
116#872C22×C3124yescomplemented
117C23×C32C2411yescharacteristic
118C22×C3×D3—26no—
119C22×C3×D3—26no—
120C22×C3×D3C22×C4112yes—
121C22×C3×D3C2416yes—
122C32×Q8C22×C4116yescomplemented
123C2×C32×C4—22no—
124C2×C32×C4—26no—
125C2×C32×C4C22×C4112yes—
126C2×C32×C4C2412yes—
127C2×C32×C4C2416yes—
128C2×C32×C4C2416yes—
129C2×C3×Dic3—26no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3C22×C414yes—
132C2×C3×Dic3C2412yes—
133C2×C3×Dic3C2416yes—
134C2×C3×Dic3C2412yes—
135C3×D12C22×C4148yescomplemented
136C3×C4×D3—424no—
137C3×C4×D3C22×C4148yescomplemented
138C22×#41—31no—
139#520—316no—
140C2×C4×Q8—31no—
141C2×C4×Q8C3×D311yescharacteristic, complemented
142C2×C4×D4—33no—
143C2×C4×D4—33no—
144C2×#117—33no—
145C22×C4⋊C4—33no—
146C24×C3—33no—
147C23×D3C23×C313yes—
148C3×#41—364no—
149C2×C3×Q8—28no—
150C2×C3×Q8—24no—
151C2×C3×Q8C4×D318yescomplemented
152C2×C3×Q8C2×C3×C418yescomplemented
153C2×C3×Q8C22×D314yes—
154C2×C3×Q8C23×C314yes—
155C2×C3×D4—324no—
156C2×C3×D4—312no—
157C22×C3×C4—21no—
158C22×C3×C4—23no—
159C22×C3×C4—23no—
160C22×C3×C4—31no—
161C22×C3×C4—324no—
162C22×C3×C4—312no—
163C22×C3×C4—31no—
164C22×C3×C4—33no—
165C22×C3×C4—324no—
166C22×C3×C4—1212no—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C22×D313yes—
169C22×C3×C4C22×D313yes—
170C22×C3×C4C23×C311yescharacteristic
171C22×C3×C4C23×C313yes—
172C22×C3×C4C23×C313yes—
173C22×Dic3C23×C311yescharacteristic
174C22×Dic3C23×C313yes—
175C22×Dic3C23×C311yescharacteristic
176#239C2×C3×C4164yescomplemented
177C2×D12C2×C3×C4124yescomplemented
178C2×D12C23×C3112yes—
179C2×C4×D3—412no—
180C2×C4×D3C2×C3×C4124yescomplemented
181C2×C4×D3C23×C3124yes—
182C2×C4×D3C23×C3112yes—
183C3×C4⋊C4—212no—
184C3×C4⋊C4—312no—
185C3×C4⋊C4—34no—
186C3×C4⋊C4—312no—
187C3×C4⋊C4C22×D3112yes—
188C3×C4⋊C4C23×C3112yes—
189C3×C22⋊C4—312no—
190C3×C22⋊C4—312no—
191C3×C42—212no—
192C3×C42—312no—
193C3×C42C22×D3112yes—
194C3×C42C23×C3112yes—
195(C2×D3)⋊C4C23×C3112yes—
196(C2×D3)⋊C4C23×C3112yes—
197C4⋊Dic3C23×C3112yes—
198Dic3⋊C4C23×C314yes—
199Dic3⋊C4C23×C3112yes—
200C4×Dic3C23×C3112yes—
201C22×C32C23×C412yes—
202C22×C32H(5,𝔽2)12yes—
203C22×C322−1+412yes—
204C22×C32C2511yescharacteristic
205C2×C3×D3—224no—
206C2×C3×D3—26no—
207C2×C3×D3—26no—
208C2×C3×D3C23×C4112yes—
209C32×C4—22no—
210C32×C4—26no—
211C32×C4C23×C4112yes—
212C3×Dic3—26no—
213C3×Dic3—22no—
214C3×Dic3C23×C414yes—
215C2×#41—316no—
216C2×#41—38no—
217C22×Q8C2×C3×D311yescharacteristic
218C22×D4—33no—
219C23×C4—33no—
220C23×C4—36no—
221C4×Q8—38no—
222C4×Q8C2×C3×D318yescomplemented
223C4×D4—324no—
224C4×D4—324no—
225#117—324no—
226C2×C4⋊C4—33no—
227C2×C4⋊C4—33no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—324no—
230C2×C4⋊C4C2×C3×D313yes—
231C2×C22⋊C4—33no—
232C2×C22⋊C4—33no—
233C2×C42—33no—
234C2×C42C2×C3×D313yes—
235C23×C3—21no—
236C23×C3—312no—
237C23×C3—36no—
238C23×C3—66no—
239C23×C3—66no—
240C23×C3C23×D311yescharacteristic
241C23×C3C24×C311yescharacteristic
242C22×D3—26no—
243C22×D3—26no—
244C22×D3C22×C3×C4112yes—
245C22×D3C24×C316yes—
246C3×Q8—216no—
247C3×Q8C2×C4×D3116yescomplemented
248C3×Q8C22×C3×C4116yescomplemented
249C3×D4—348no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—26no—
253C2×C3×C4—26no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—26no—
257C2×C3×C4—26no—
258C2×C3×C4—212no—
259C2×C3×C4—34no—
260C2×C3×C4—32no—
261C2×C3×C4—348no—
262C2×C3×C4—32no—
263C2×C3×C4—36no—
264C2×C3×C4—46no—
265C2×C3×C4—62no—
266C2×C3×C4—66no—
267C2×C3×C4—1224no—
268C2×C3×C4C2×C4×D3112yes—
269C2×C3×C4C22×C3×C4112yes—
270C2×C3×C4C23×D312yes—
271C2×C3×C4C23×D316yes—
272C2×C3×C4C23×D316yes—
273C2×C3×C4C24×C312yes—
274C2×C3×C4C24×C316yes—
275C2×C3×C4C24×C316yes—
276C2×Dic3—26no—
277C2×Dic3—22no—
278C2×Dic3C22×C3×C414yes—
279C2×Dic3C24×C312yes—
280C2×Dic3C24×C316yes—
281C2×Dic3C24×C312yes—
282D12C22×C3×C4148yescomplemented
283C4×D3—424no—
284C4×D3C22×C3×C4148yescomplemented
285C2×C32#52014yescomplemented
286C2×C32C24×C411yescharacteristic
287C2×C32C2×H(5,𝔽2)11yescharacteristic
288C2×C32C2×2−1+411yescharacteristic
289C3×D3—212no—
290C24—33no—
291#41—364no—
292C2×Q8C3×C4×D318yescomplemented
293C2×Q8C22×C3×D314yes—
294C2×D4—312no—
295C2×D4—324no—
296C22×C4—31no—
297C22×C4—312no—
298C22×C4—324no—
299C22×C4—31no—
300C22×C4—33no—
301C22×C4—324no—
302C22×C4—1212no—
303C22×C4C22×C3×D311yescharacteristic
304C22×C4C22×C3×D313yes—
305C22×C4C22×C3×D313yes—
306C4⋊C4—312no—
307C4⋊C4—312no—
308C4⋊C4—34no—
309C4⋊C4C22×C3×D3112yes—
310C22⋊C4—312no—
311C22⋊C4—312no—
312C42—312no—
313C42C22×C3×D3112yes—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—21no—
318C22×C3—312no—
319C22×C3—66no—
320C22×C3—66no—
321C22×C3—624no—
322C22×C3C22×C4×D312yes—
323C22×C3#100712yes—
324C22×C3#100912yes—
325C22×C3C23×C3×C412yes—
326C22×C3C3×H(5,𝔽2)12yes—
327C22×C3C3×2−1+412yes—
328C22×C3C24×D311yescharacteristic
329C22×C3C25×C311yescharacteristic
330C2×D3—26no—
331C2×D3—26no—
332C2×D3—224no—
333C2×D3C23×C3×C4112yes—
334C3×C4—24no—
335C3×C4—22no—
336C3×C4—26no—
337C3×C4—26no—
338C3×C4—22no—
339C3×C4—212no—
340C3×C4—34no—
341C3×C4—46no—
342C3×C4—62no—
343C3×C4—66no—
344C3×C4C22×C4×D3112yes—
345C3×C4C23×C3×C4112yes—
346Dic3—26no—
347Dic3—22no—
348Dic3C23×C3×C414yes—
349C32C2×#52011yescharacteristic, complemented, Sylow
350C23—312no—
351C23—36no—
352C23—66no—
353C23—66no—
354C23C23×C3×D311yescharacteristic
355Q8C2×C3×C4×D3116yescomplemented
356D4—348no—
357C2×C4—22no—
358C2×C4—26no—
359C2×C4—34no—
360C2×C4—32no—
361C2×C4—348no—
362C2×C4—32no—
363C2×C4—36no—
364C2×C4—62no—
365C2×C4—66no—
366C2×C4—1224no—
367C2×C4C2×C3×C4×D3112yes—
368C2×C4C23×C3×D312yes—
369C2×C4C23×C3×D316yes—
370C2×C4C23×C3×D316yes—
371C2×C3—24no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—612no—
376C2×C3#557414yesdirect product factor
377C2×C3C3×#52014yescomplemented
378C2×C3C23×C4×D311yescharacteristic
379C2×C3C2×#100711yescharacteristic
380C2×C3C2×#100911yescharacteristic
381C2×C3C24×C3×C411yescharacteristic
382C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
383C2×C3C2×C3×2−1+411yescharacteristic
384D3—212no—
385C22—312no—
386C22—624no—
387C22—66no—
388C22—66no—
389C22C22×C3×C4×D312yes—
390C22C3×#100712yes—
391C22C3×#100912yes—
392C22C24×C3×D311yescharacteristic
393C4—22no—
394C4—26no—
395C4—34no—
396C4—62no—
397C4—66no—
398C4C22×C3×C4×D3112yes—
399C3—21no—
400C3C2×#557411yescharacteristic, direct product factor
401C3C2×C3×#52011yescharacteristic, complemented
402C2—612no—
403C2C3×#557414yesdirect product factor
404C2C23×C3×C4×D311yescharacteristic
405C2C2×C3×#100711yescharacteristic
406C2C2×C3×#100911yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC2×C3C24×C3×C4
FrattiniC22C24×C3×D3
FittingC2×C32×C4×Q8C2
socleC23×C32C24
prime radicalC22×C3×#239C2
cosocleC22×C3C25×C3
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×#520
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×Q8C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
4. C3×#5574C2direct (N × H)
5. C2×C3×#893C2semidirect
6. C2×C3×D3×C4⋊C4C2semidirect
7. C2×C3×C4×D12C2semidirect
8. C2×C3×#872C2semidirect
10. C2×#5574C3direct (N × H)
12. C2×C3×#239C4semidirect
17. C32×C4×Q8C22semidirect
20. C3×Q8×Dic3C22semidirect
27. C3×#893C22semidirect
28. C3×D3×C4⋊C4C22semidirect
29. C3×C4×D12C22semidirect
30. C3×#872C22semidirect
36. C2×C3×C4×Q8D3semidirect
37. C2×C3×C4×Q8C2×C3semidirect
42. C2×Q8×Dic3C2×C3semidirect
43. #5574C2×C3direct (N × H)
44. C2×#893C2×C3semidirect
45. C2×D3×C4⋊C4C2×C3semidirect
46. C2×C4×D12C2×C3semidirect
47. C2×#872C2×C3semidirect
49. C2×C32×Q8C2×C4semidirect
57. C3×#239C2×C4semidirect
58. C2×C3×D12C2×C4semidirect
61. C2×C3×C4×D3C2×C4semidirect
81. C2×#239C3×C4semidirect
88. C3×C4×Q8C2×D3semidirect
89. C3×C4×Q8C22×C3semidirect
106. Q8×Dic3C22×C3semidirect
113. #893C22×C3semidirect
114. D3×C4⋊C4C22×C3semidirect
115. C4×D12C22×C3semidirect
116. #872C22×C3semidirect
122. C32×Q8C22×C4semidirect
135. C3×D12C22×C4semidirect
137. C3×C4×D3C22×C4semidirect
141. C2×C4×Q8C3×D3semidirect
151. C2×C3×Q8C4×D3semidirect
152. C2×C3×Q8C2×C3×C4semidirect
176. #239C2×C3×C4semidirect
177. C2×D12C2×C3×C4semidirect
180. C2×C4×D3C2×C3×C4semidirect
222. C4×Q8C2×C3×D3semidirect
247. C3×Q8C2×C4×D3semidirect
248. C3×Q8C22×C3×C4semidirect
282. D12C22×C3×C4semidirect
284. C4×D3C22×C3×C4semidirect
285. C2×C32#520semidirect
292. C2×Q8C3×C4×D3semidirect
349. C32C2×#520semidirect
355. Q8C2×C3×C4×D3semidirect
376. C2×C3#5574direct (N × H)
377. C2×C3C3×#520semidirect
400. C3C2×#5574direct (N × H)
401. C3C2×C3×#520semidirect
403. C2C3×#5574direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C2×C32×C4×Q82
C2×C3×Q8×Dic32
C3×#55742
C2×C3×#8932
C2×C3×D3×C4⋊C42
C2×C3×C4×D122
C2×C3×#8722
C2×C3×#5203
C2×#55743