Finite Group Databasebeta

← C2×C3×#5007 #11868238

Subgroups — C2×C3×#5007 #11868238

Order 1152.

1173 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×C3×#869C211yesmaximal, characteristic, complemented
3C2×C3×#832C212yesmaximal
4C3×#5007C218yesmaximal, direct product factor
5C2×C3×#807C212yesmaximal, complemented
6C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C2×C3×#474—31nomaximal
8C2×#5007C311yesmaximal, characteristic, direct product factor
9C22×C32×D4C2211yescharacteristic
10C23×C32×C4—21no—
11C22×C3×Dic6—21no—
12C32×C4×D4—22no—
13C32×C4×D4C2214yes—
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C32×C22⋊C4—21no—
16C2×C32×C42C2211yescharacteristic
17C2×C3×C4⋊Dic3—21no—
18C2×C3×C4⋊Dic3C2212yes—
19C22×C3×C3⋊C8C2212yes—
20C3×#869—22no—
21C3×#869C2214yescomplemented
22C3×#832C2218yes—
23C3×#807C2218yescomplemented
24C3×C4⋊(C3⋊C8)C2214yescomplemented
25C2×C3×C4⋊Q8—31no—
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4D311yescharacteristic
28C2×C3×C4×D4C2×C311yescharacteristic
29C2×#869C2×C311yescharacteristic, complemented
30C3×#474—38no—
31C2×C3×#107—32no—
32C2×C3×C4⋊C8—31no—
33C2×C3×#102—32no—
34C2×#832C2×C312yes—
35#5007C2×C318yesdirect product factor
36C2×#807C2×C312yescomplemented
37C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
38C24×C32—21no—
39C2×C32×D4—21no—
40C2×C32×D4—22no—
41C2×C32×D4Q814yescomplemented
42C2×C32×D4C2312yes—
43C22×C32×C4—21no—
44C22×C32×C4—24no—
45C22×C32×C4—44no—
46C22×C32×C4D411yescharacteristic
47C22×C32×C4D411yescharacteristic
48C22×C32×C4C2311yescharacteristic
49C22×C3×Dic3—22no—
50C2×C3×Dic6—24no—
51C2×C3×Dic6—44no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2312yes—
54C32×C22⋊C4—24no—
55C32×C42D412yes—
56C32×C42C2312yes—
57C3×C4⋊Dic3—24no—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2314yes—
60C2×C3×C3⋊C8—24no—
61C2×C3×C3⋊C8C2314yes—
62C2×#474—31noSylow
63C22×C3×Q8—61no—
64C22×C3×D4—21no—
65C22×C3×D4C2×D311yescharacteristic
66C22×C3×D4C22×C311yescharacteristic
67C23×C3×C4—22no—
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C22×Dic6—21no—
71C22×C3×C8—32no—
72C3×C4⋊Q8—34no—
73C3×C4⋊Q8—62no—
74C3×C4×D4—24no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—24no—
78C3×C4×D4C2×D314yes—
79C3×C4×D4C22×C314yes—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—32no—
82C2×C3×C4⋊C4—61no—
83C2×C3×C4⋊C4C2×D311yescharacteristic
84C2×C3×C4⋊C4C22×C311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×C3×C42C2×D311yescharacteristic
90C2×C3×C42C22×C311yescharacteristic
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C22×C312yes—
93C22×C3⋊C8C22×C312yes—
94#869—22no—
95#869C22×C314yescomplemented
96C3×#107—38no—
97C3×C4⋊C8—34no—
98C3×#102—38no—
99#832C22×C318yes—
100#807C22×C318yescomplemented
101C4⋊(C3⋊C8)C22×C314yescomplemented
102C23×C32—24no—
103C23×C32—22no—
104C23×C32—42no—
105C23×C32—42no—
106C23×C32C2×D411yescharacteristic
107C32×D4—24no—
108C32×D4—22no—
109C32×D4C2×Q814yescomplemented
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—42no—
113C2×C32×C4—48no—
114C2×C32×C4D814yes—
115C2×C32×C4C2×D412yes—
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D412yes—
118C2×C32×C4C2×Q812yes—
119C2×C32×C4#4112yes—
120C2×C32×C4C2411yescharacteristic
121C2×C3×Dic3—24no—
122C2×C3×Dic3—44no—
123C3×Dic6—48no—
124C3×C3⋊C8—24no—
125C2×C4⋊Q8—31no—
126C2×C4×D4C3×D311yescharacteristic
127#474—38no—
128C2×#107—32no—
129C2×C4⋊C8—31no—
130C2×#102—32no—
131C24×C3—22no—
132C24×C3—21no—
133C24×C3—21no—
134C2×C3×Q8—64no—
135C2×C3×Q8—124no—
136C2×C3×D4—22no—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4Dic614yescomplemented
145C2×C3×D4C3×Q814yescomplemented
146C2×C3×D4C22×D312yes—
147C2×C3×D4C23×C312yes—
148C22×C3×C4—22no—
149C22×C3×C4—28no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—48no—
158C22×C3×C4—44no—
159C22×C3×C4—44no—
160C22×C3×C4—62no—
161C22×C3×C4#6711yescharacteristic
162C22×C3×C4#6711yescharacteristic
163C22×C3×C4C3×D411yescharacteristic
164C22×C3×C4C3×D411yescharacteristic
165C22×C3×C4C22×D311yescharacteristic
166C22×C3×C4C23×C311yescharacteristic
167C22×Dic3—22no—
168C2×Dic6—24no—
169C2×Dic6—44no—
170C2×C3×C8—34no—
171C2×C3×C8—64no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—34no—
177C3×C4⋊C4—62no—
178C3×C4⋊C4—64no—
179C3×C4⋊C4C22×D312yes—
180C3×C4⋊C4C23×C312yes—
181C3×C22⋊C4—28no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42#6712yes—
187C3×C42C3×D412yes—
188C3×C42C22×D312yes—
189C3×C42C23×C312yes—
190C4⋊Dic3—22no—
191C4⋊Dic3—24no—
192C4⋊Dic3C23×C314yes—
193C2×C3⋊C8—24no—
194C2×C3⋊C8C23×C314yes—
195C22×C32—24no—
196C22×C32—42no—
197C22×C32—42no—
198C22×C32—48no—
199C22×C32C22⋊Q812yes—
200C22×C32C2×D812yes—
201C22×C32#13712yes—
202C22×C32C22×D411yescharacteristic
203C32×C4—22no—
204C32×C4—42no—
205C32×C4C22⋊Q812yes—
206C32×C4C2×D814yes—
207C32×C4C22×Q811yescharacteristic
208C32×C4C2×#4111yescharacteristic
209C3×Dic3—44no—
210C22×Q8—61no—
211C22×D4C2×C3×D311yescharacteristic
212C23×C4—21no—
213C22×C8—32no—
214C4⋊Q8—34no—
215C4⋊Q8—62no—
216C4×D4—22no—
217C4×D4C2×C3×D314yes—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4C2×C3×D311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C42C2×C3×D311yescharacteristic
223#107—38no—
224C4⋊C8—34no—
225#102—38no—
226C23×C3—24no—
227C23×C3—28no—
228C23×C3—24no—
229C23×C3—24no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—21no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3C2×#6711yescharacteristic
240C23×C3C2×C3×D411yescharacteristic
241C3×Q8—128no—
242C3×D4—28no—
243C3×D4—24no—
244C3×D4—24no—
245C3×D4—24no—
246C3×D4—22no—
247C3×D4—22no—
248C3×D4—24no—
249C3×D4C2×Dic614yescomplemented
250C3×D4C2×C3×Q814yescomplemented
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—42no—
264C2×C3×C4—416no—
265C2×C3×C4—44no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—48no—
269C2×C3×C4—48no—
270C2×C3×C4—64no—
271C2×C3×C4—124no—
272C2×C3×C4#21314yes—
273C2×C3×C4C3×D814yes—
274C2×C3×C4C2×Dic612yes—
275C2×C3×C4#23512yes—
276C2×C3×C4C2×#6712yes—
277C2×C3×C4C2×#6711yescharacteristic
278C2×C3×C4C2×#6712yes—
279C2×C3×C4C2×C3×D412yes—
280C2×C3×C4C2×C3×D411yescharacteristic
281C2×C3×C4C2×C3×D412yes—
282C2×C3×C4C2×C3×Q812yes—
283C2×C3×C4C3×#4112yes—
284C2×C3×C4C23×D311yescharacteristic
285C2×C3×C4C24×C311yescharacteristic
286C2×Dic3—24no—
287C2×Dic3—44no—
288Dic6—48no—
289C3×C8—64no—
290C3⋊C8—24no—
291C2×C32—44no—
292C2×C32#47414yescomplemented
293C2×C32C2×C22⋊Q811yescharacteristic
294C2×C32C22×D811yescharacteristic
295C2×C32C2×#13711yescharacteristic
296C24—21no—
297C2×Q8—64no—
298C2×Q8—124no—
299C2×D4—22no—
300C2×D4—21no—
301C2×D4C3×Dic614yescomplemented
302C2×D4C22×C3×D312yes—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—44no—
306C22×C4—62no—
307C22×C4C3×#6711yescharacteristic
308C22×C4C3×#6711yescharacteristic
309C22×C4C22×C3×D311yescharacteristic
310C2×C8—34no—
311C2×C8—64no—
312C4⋊C4—21no—
313C4⋊C4—34no—
314C4⋊C4—62no—
315C4⋊C4—64no—
316C4⋊C4C22×C3×D312yes—
317C22⋊C4—24no—
318C42C3×#6712yes—
319C42C22×C3×D312yes—
320C22×C3—28no—
321C22×C3—24no—
322C22×C3—24no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—21no—
327C22×C3—44no—
328C22×C3—44no—
329C22×C3—416no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—48no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—48no—
336C22×C3#92412yes—
337C22×C3C2×#21312yes—
338C22×C3#95112yes—
339C22×C3C3×C22⋊Q812yes—
340C22×C3C2×C3×D812yes—
341C22×C3C3×#13712yes—
342C22×C3C22×#6711yescharacteristic
343C22×C3C22×C3×D411yescharacteristic
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—24no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—42no—
351C3×C4—44no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—124no—
355C3×C4#92412yes—
356C3×C4C2×#21314yes—
357C3×C4C3×C22⋊Q812yes—
358C3×C4C2×C3×D814yes—
359C3×C4C22×Dic611yescharacteristic
360C3×C4C2×#23511yescharacteristic
361C3×C4C22×C3×Q811yescharacteristic
362C3×C4C2×C3×#4111yescharacteristic
363Dic3—44no—
364C32C2×#47411yescharacteristic, complemented, Sylow
365C23—24no—
366C23—22no—
367C23—42no—
368C23—42no—
369C23C2×C3×#6711yescharacteristic
370Q8—128no—
371D4—24no—
372D4—22no—
373D4C2×C3×Dic614yescomplemented
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—64no—
379C2×C4—124no—
380C2×C4C3×#21314yes—
381C2×C4C2×C3×Dic612yes—
382C2×C4C3×#23512yes—
383C2×C4C2×C3×#6712yes—
384C2×C4C2×C3×#6711yescharacteristic
385C2×C4C2×C3×#6712yes—
386C2×C4C23×C3×D311yescharacteristic
387C8—64no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—24no—
392C2×C3—48no—
393C2×C3—44no—
394C2×C3—44no—
395C2×C3#500714yesdirect product factor
396C2×C3C3×#47414yescomplemented
397C2×C3C2×#92411yescharacteristic
398C2×C3C22×#21311yescharacteristic
399C2×C3C2×#95111yescharacteristic
400C2×C3C2×C3×C22⋊Q811yescharacteristic
401C2×C3C22×C3×D811yescharacteristic
402C2×C3C2×C3×#13711yescharacteristic
403C22—24no—
404C22—42no—
405C22—42no—
406C22—48no—
407C22C3×#92412yes—
408C22C2×C3×#21312yes—
409C22C3×#95112yes—
410C22C22×C3×#6711yescharacteristic
411C4—22no—
412C4—42no—
413C4—124no—
414C4C3×#92412yes—
415C4C2×C3×#21314yes—
416C4C22×C3×Dic611yescharacteristic
417C4C2×C3×#23511yescharacteristic
418C3—21no—
419C3C2×#500711yescharacteristic, direct product factor
420C3C2×C3×#47411yescharacteristic, complemented
421C2—44no—
422C2C3×#500714yesdirect product factor
423C2C2×C3×#92411yescharacteristic
424C2C22×C3×#21311yescharacteristic
425C2C2×C3×#95111yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×#474
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#869C2semidirect
4. C3×#5007C2direct (N × H)
5. C2×C3×#807C2semidirect
6. C2×C3×C4⋊(C3⋊C8)C2semidirect
8. C2×#5007C3direct (N × H)
21. C3×#869C22semidirect
23. C3×#807C22semidirect
24. C3×C4⋊(C3⋊C8)C22semidirect
29. C2×#869C2×C3semidirect
35. #5007C2×C3direct (N × H)
36. C2×#807C2×C3semidirect
37. C2×C4⋊(C3⋊C8)C2×C3semidirect
41. C2×C32×D4Q8semidirect
95. #869C22×C3semidirect
100. #807C22×C3semidirect
101. C4⋊(C3⋊C8)C22×C3semidirect
109. C32×D4C2×Q8semidirect
144. C2×C3×D4Dic6semidirect
145. C2×C3×D4C3×Q8semidirect
249. C3×D4C2×Dic6semidirect
250. C3×D4C2×C3×Q8semidirect
292. C2×C32#474semidirect
301. C2×D4C3×Dic6semidirect
364. C32C2×#474semidirect
373. D4C2×C3×Dic6semidirect
395. C2×C3#5007direct (N × H)
396. C2×C3C3×#474semidirect
419. C3C2×#5007direct (N × H)
420. C3C2×C3×#474semidirect
422. C2C3×#5007direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C2×C3×#8692
C2×C3×#8322
C3×#50072
C2×C3×#8072
C2×C3×C4⋊(C3⋊C8)2
C2×C3×#4743
C2×#50073