Finite Group Databasebeta

← C3×#70370 #11784121

Subgroups — C3×#70370 #11784121

Order 1152.

669 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#948C211yesmaximal, characteristic
2C2×C32×#104C211yesmaximal, characteristic
3C3×#5233C211yesmaximal, characteristic
4C3×#5122C211yesmaximal, characteristic, complemented
5C2×C3×#803C211yesmaximal, characteristic, complemented
6C3×#4558C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#1765—31nomaximal
9#70370C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×C42C2211yescharacteristic
13C3×#948—22no—
14C3×#948C414yescomplemented
15C3×#926C2211yescharacteristic
16C2×C3×#208—21no—
17C2×C3×#208C2211yescharacteristic
18C22×C3×C3⋊C8—21no—
19C22×C3×C3⋊C8C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C32×#104C414yescomplemented
22C3×#835C412yescomplemented
23C3×#832C412yescomplemented
24C3×#815—21no—
25C3×C8×Dic3—21no—
26C3×#803—22no—
27C2×C3×#131—31no—
28C2×#948C2×C311yescharacteristic
29C2×C3×#104—21no—
30C2×C3×#104D311yescharacteristic
31C2×C3×#104C2×C311yescharacteristic
32C3×#416—31no—
33C3×#405—31no—
34C2×C3×#97—31no—
35#5233C2×C311yescharacteristic
36#5122C2×C311yescharacteristic, complemented
37C2×#803C2×C311yescharacteristic, complemented
38C3×#341—31no—
39C3×#339—31no—
40#4558C2×C311yescharacteristic, complemented
41#4523C2×C311yescharacteristic, complemented
42C32×#41—21no—
43C32×#41—21no—
44C32×#41C2×C412yes—
45C32×#41C2×C412yes—
46C2×C32×Q8C2×C411yescharacteristic
47C2×C32×D4—21no—
48C2×C32×D4C2×C411yescharacteristic
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4C2311yescharacteristic
52C32×M16—21no—
53C32×M16C2×C412yes—
54C2×C32×C8—21no—
55C32×C42—21no—
56C32×C42C2×C412yes—
57C3×#217—21no—
58C3×C4⋊Dic3C2×C412yes—
59C3×C4×Dic3—21no—
60C3×#208—21no—
61C3×#208—24no—
62C3×#208C2×C412yes—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—24no—
67C2×C3×C3⋊C8C2×C412yes—
68C2×C3×C3⋊C8C2×C412yes—
69C2×C3×C3⋊C8D411yescharacteristic
70C2×C3×C3⋊C8D411yescharacteristic
71C2×C3×C3⋊C8D412yes—
72#1765—31noSylow
73C2×C3×#41—21no—
74C2×C3×#41C2×D311yescharacteristic
75C2×C3×#41C22×C311yescharacteristic
76C3×#131—34no—
77C3×#131—62no—
78C2×C3×M16—21no—
79C2×C3×M16—31no—
80C2×C3×M16—61no—
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×M16C22×C311yescharacteristic
83C22×C3×C8—31no—
84C22×C3×C8—31no—
85C22×C3×C8—61no—
86C3×#117—31no—
87C2×C3×C42—21no—
88C2×C3×C42C2×D311yescharacteristic
89C2×C3×C42C22×C311yescharacteristic
90#948—22no—
91#948C3×C414yescomplemented
92#926C22×C311yescharacteristic
93C2×#208—21no—
94C2×#208C22×C311yescharacteristic
95C22×C3⋊C8—21no—
96C22×C3⋊C8C22×C311yescharacteristic
97C22×C3⋊C8C22×C311yescharacteristic
98C3×#104—24no—
99C3×#104Dic314yescomplemented
100C3×#104C3×C414yescomplemented
101C3×#103—32no—
102C3×#102—32no—
103C3×#97—62no—
104C3×#97—61no—
105C3×C4×C8—61no—
106#835C3×C412yescomplemented
107#832C3×C412yescomplemented
108#815—21no—
109C8×Dic3—21no—
110#803—22no—
111C23×C32—21no—
112C23×C32#4111yescharacteristic
113C32×Q8—21no—
114C32×Q8C4212yes—
115C32×D4—22no—
116C32×D4—22no—
117C32×D4—21no—
118C32×D4C4212yes—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—42no—
124C2×C32×C4C22×C411yescharacteristic
125C2×C32×C4C22×C411yescharacteristic
126C2×C32×C4C22×C411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4#4111yescharacteristic
130C2×C3×Dic3—22no—
131C32×C8—22no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8C22⋊C412yes—
135C3×C3⋊C8C22⋊C412yes—
136C2×#131—31no—
137C2×#104C3×D311yescharacteristic
138#416—31no—
139#405—31no—
140C2×#97—31no—
141#341—31no—
142#339—31no—
143C3×#41—21no—
144C3×#41—21no—
145C3×#41—21no—
146C3×#41—21no—
147C3×#41—22no—
148C3×#41—22no—
149C3×#41—41no—
150C3×#41—41no—
151C3×#41C4×D312yes—
152C3×#41C2×Dic312yes—
153C3×#41C2×C3×C412yes—
154C3×#41C2×C3×C412yes—
155C2×C3×Q8—21no—
156C2×C3×Q8C4×D311yescharacteristic
157C2×C3×Q8C2×C3×C411yescharacteristic
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—41no—
162C2×C3×D4C4×D311yescharacteristic
163C2×C3×D4C2×C3×C411yescharacteristic
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—41no—
170C22×C3×C4—41no—
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C23×C311yescharacteristic
173C3×M16—21no—
174C3×M16—21no—
175C3×M16—22no—
176C3×M16—32no—
177C3×M16—41no—
178C3×M16—64no—
179C3×M16—61no—
180C3×M16C2×Dic312yes—
181C3×M16C2×C3×C412yes—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—32no—
185C2×C3×C8—32no—
186C2×C3×C8—31no—
187C2×C3×C8—31no—
188C2×C3×C8—32no—
189C2×C3×C8—41no—
190C2×C3×C8—64no—
191C2×C3×C8—61no—
192C2×C3×C8—61no—
193C2×C3×C8—61no—
194C3×C4⋊C4—32no—
195C3×C22⋊C4—61no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—22no—
199C3×C42—41no—
200C3×C42—61no—
201C3×C42C2×Dic312yes—
202C3×C42C2×C3×C412yes—
203#217—21no—
204C4⋊Dic3C2×C3×C412yes—
205C4×Dic3—21no—
206#208—21no—
207#208—24no—
208#208C2×C3×C412yes—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—24no—
213C2×C3⋊C8C2×C3×C412yes—
214C2×C3⋊C8C2×C3×C412yes—
215C2×C3⋊C8C3×D412yes—
216C2×C3⋊C8C3×D411yescharacteristic
217C2×C3⋊C8C3×D411yescharacteristic
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—42no—
221C22×C32#11711yescharacteristic
222C22×C32C4×D411yescharacteristic
223C22×C32C4×D411yescharacteristic
224C32×C4—22no—
225C32×C4—42no—
226C32×C4C2×C4211yescharacteristic
227C32×C4C2×C22⋊C411yescharacteristic
228C32×C4C4×D411yescharacteristic
229C32×C4C4×D411yescharacteristic
230C3×Dic3—42no—
231C2×#41C2×C3×D311yescharacteristic
232#131—34no—
233#131—62no—
234C2×M16—31no—
235C2×M16—61no—
236C2×M16C2×C3×D311yescharacteristic
237C22×C8—31no—
238C22×C8—31no—
239C22×C8—61no—
240#117—31no—
241C2×C42C2×C3×D311yescharacteristic
242#104C3×Dic314yescomplemented
243#103—32no—
244#102—32no—
245#97—62no—
246#97—61no—
247C4×C8—61no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—41no—
252C23×C3#23711yescharacteristic
253C23×C3C3×#4111yescharacteristic
254C3×Q8—21no—
255C3×Q8—21no—
256C3×Q8—22no—
257C3×Q8—41no—
258C3×Q8C4×Dic312yes—
259C3×Q8C3×C4212yes—
260C3×D4—22no—
261C3×D4—22no—
262C3×D4—22no—
263C3×D4—22no—
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—22no—
267C3×D4—41no—
268C3×D4—42no—
269C3×D4—42no—
270C3×D4C4×Dic312yes—
271C3×D4C3×C4212yes—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—44no—
293C2×C3×C4—62no—
294C2×C3×C4C2×C4×D311yescharacteristic
295C2×C3×C4C2×C4×D311yescharacteristic
296C2×C3×C4D3×D411yescharacteristic
297C2×C3×C4D3×D411yescharacteristic
298C2×C3×C4#23711yescharacteristic
299C2×C3×C4C22×Dic311yescharacteristic
300C2×C3×C4C22×C3×C411yescharacteristic
301C2×C3×C4C22×C3×C411yescharacteristic
302C2×C3×C4C22×C3×C411yescharacteristic
303C2×C3×C4C2×C3×D411yescharacteristic
304C2×C3×C4C2×C3×D411yescharacteristic
305C2×C3×C4C3×#4111yescharacteristic
306C2×Dic3—22no—
307C3×C8—22no—
308C3×C8—22no—
309C3×C8—32no—
310C3×C8—32no—
311C3×C8—42no—
312C3×C8—62no—
313C3×C8—62no—
314C3⋊C8—22no—
315C3⋊C8—22no—
316C3⋊C8C3×C22⋊C412yes—
317C3⋊C8C3×C22⋊C412yes—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—42no—
321C2×C32C4×C22⋊C411yescharacteristic
322C2×C32#44412yes—
323#41—21no—
324#41—21no—
325#41C3×C4×D312yes—
326#41C2×C3×Dic312yes—
327C2×Q8C3×C4×D311yescharacteristic
328C2×D4—21no—
329C2×D4C3×C4×D311yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4C22×C3×D311yescharacteristic
333M16—21no—
334M16—32no—
335M16—64no—
336M16—61no—
337M16C2×C3×Dic312yes—
338C2×C8—21no—
339C2×C8—32no—
340C2×C8—31no—
341C2×C8—31no—
342C2×C8—32no—
343C2×C8—32no—
344C2×C8—64no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8—61no—
348C4⋊C4—32no—
349C22⋊C4—61no—
350C42—21no—
351C42—61no—
352C42C2×C3×Dic312yes—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—82no—
365C22×C3#87711yescharacteristic
366C22×C3#67⋊C411yescharacteristic
367C22×C3D4×Dic311yescharacteristic
368C22×C3C3×#11711yescharacteristic
369C22×C3C3×C4×D411yescharacteristic
370C22×C3C3×C4×D411yescharacteristic
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—44no—
381C3×C4—122no—
382C3×C4D3×C22⋊C411yescharacteristic
383C3×C4#67⋊C411yescharacteristic
384C3×C4C2×C4×Dic311yescharacteristic
385C3×C4D4×Dic311yescharacteristic
386C3×C4C2×C3×C4211yescharacteristic
387C3×C4C2×C3×C22⋊C411yescharacteristic
388C3×C4C3×C4×D411yescharacteristic
389C3×C4C3×C4×D411yescharacteristic
390Dic3—42no—
391C32#176511yescharacteristic, complemented, Sylow
392C23—21no—
393C23C3×#23711yescharacteristic
394Q8—21no—
395Q8C3×C4×Dic312yes—
396D4—22no—
397D4—22no—
398D4—21no—
399D4C3×C4×Dic312yes—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—62no—
406C2×C4C2×C3×C4×D311yescharacteristic
407C2×C4C2×C3×C4×D311yescharacteristic
408C2×C4C3×D3×D411yescharacteristic
409C2×C4C3×D3×D411yescharacteristic
410C2×C4C3×#23711yescharacteristic
411C2×C4C22×C3×Dic311yescharacteristic
412C8—22no—
413C8—32no—
414C8—32no—
415C8—62no—
416C8—62no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—42no—
426C2×C3—42no—
427C2×C3—82no—
428C2×C3#482312yes—
429C2×C3Dic3×C22⋊C411yescharacteristic
430C2×C3C3×C4×C22⋊C411yescharacteristic
431C2×C3C3×#44412yes—
432C22—22no—
433C22—22no—
434C22—42no—
435C22C3×#87711yescharacteristic
436C22C3×#67⋊C411yescharacteristic
437C22C3×D4×Dic311yescharacteristic
438C4—22no—
439C4—42no—
440C4—122no—
441C4C3×D3×C22⋊C411yescharacteristic
442C4C3×#67⋊C411yescharacteristic
443C4C2×C3×C4×Dic311yescharacteristic
444C4C3×D4×Dic311yescharacteristic
445C3—21no—
446C3#7037011yescharacteristic, direct product factor
447C3C3×#176511yescharacteristic, complemented
448C2—21no—
449C2—21no—
450C2—42no—
451C2C3×#482312yes—
452C2C3×Dic3×C22⋊C411yescharacteristic
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC3×C4C2×C3×C42
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#104C2
socleC22×C32C4×D4
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#104D3
hypocenterC3C3×#1765
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#104

Complements

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

NH ≅ G/Ntype
4. C3×#5122C2semidirect
5. C2×C3×#803C2semidirect
6. C3×#4558C2semidirect
7. C3×#4523C2semidirect
9. #70370C3direct (N × H)
14. C3×#948C4semidirect
21. C32×#104C4semidirect
22. C3×#835C4semidirect
23. C3×#832C4semidirect
36. #5122C2×C3semidirect
37. C2×#803C2×C3semidirect
40. #4558C2×C3semidirect
41. #4523C2×C3semidirect
91. #948C3×C4semidirect
99. C3×#104Dic3semidirect
100. C3×#104C3×C4semidirect
106. #835C3×C4semidirect
107. #832C3×C4semidirect
242. #104C3×Dic3semidirect
391. C32#1765semidirect
446. C3#70370direct (N × H)
447. C3C3×#1765semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9482
C2×C32×#1042
C3×#52332
C3×#51222
C2×C3×#8032
C3×#45582
C3×#45232
C3×#17653
#703703