Finite Group Databasebeta

← C3×#70374 #11791202

Subgroups — C3×#70374 #11791202

Order 1152.

669 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#950C211yesmaximal, characteristic
2C2×C32×#104C211yesmaximal, characteristic, complemented
3C3×#5234C211yesmaximal, characteristic, complemented
4C3×#5129C211yesmaximal, characteristic, complemented
5C2×C3×#822C211yesmaximal, characteristic, complemented
6C2×C3×#805C211yesmaximal, characteristic, complemented
7C2×C3×#803C211yesmaximal, characteristic, complemented
8C3×#1975—31nomaximal
9#70374C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C3×#950—22no—
15C3×#950—22no—
16C2×C3×#216C411yescharacteristic, complemented
17C2×C3×#215C411yescharacteristic, complemented
18C2×C3×#214C411yescharacteristic, complemented
19C2×C3×#213C411yescharacteristic, complemented
20C2×C3×#208C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C22×C3×C3⋊C8C2211yescharacteristic
23C32×#104—22no—
24C3×#830—22no—
25C3×#822—22no—
26C3×(C2×D3)⋊C8—22no—
27C3×#805—22no—
28C3×#803—22no—
29C2×C3×#135—31no—
30C2×#950C2×C311yescharacteristic
31C2×C3×#108—31no—
32C2×C3×#104—21no—
33C2×C3×#104—31no—
34C2×C3×#104D311yescharacteristic, complemented
35C2×C3×#104C2×C311yescharacteristic, complemented
36C3×#408—31no—
37C3×#408—31no—
38C2×C3×#97—31no—
39#5234C2×C311yescharacteristic, complemented
40#5129C2×C311yescharacteristic, complemented
41C2×#822C2×C311yescharacteristic, complemented
42C2×#805C2×C311yescharacteristic, complemented
43C2×#803C2×C311yescharacteristic, complemented
44C32×#41—21no—
45C32×#41—21no—
46C32×#41—21no—
47C32×#41—21no—
48C2×C32×Q8C2×C411yescharacteristic
49C2×C32×D4—21no—
50C2×C32×D4C2×C411yescharacteristic
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4C2311yescharacteristic
54C2×C3×#67—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235—21no—
58C3×#235—21no—
59C2×C3×D12C2×C411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×Dic6C2×C411yescharacteristic
62C32×M16—21no—
63C32×M16—21no—
64C2×C32×C8—21no—
65C32×C42—21no—
66C32×C42D412yes—
67C3×#216—22no—
68C3×#215—22no—
69C3×#214—22no—
70C3×#213—22no—
71C3×#208—21no—
72C3×#208—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8C2×C411yescharacteristic
77C2×C3×C3⋊C8C2×C411yescharacteristic
78C2×C3×C3⋊C8D412yes—
79C2×C3×C3⋊C8D412yes—
80C2×C3×C3⋊C8D412yes—
81#1975—31noSylow
82C2×C3×#41—21no—
83C2×C3×#41—31no—
84C2×C3×#41C2×D311yescharacteristic
85C2×C3×#41C22×C311yescharacteristic
86C2×#235C22×C311yescharacteristic
87C3×#135—62no—
88C3×#135—62no—
89C2×C3×Q16—31no—
90C2×C3×SD16—31no—
91C2×C3×SD16—31no—
92C2×C3×D8—31no—
93C2×C3×M16—21no—
94C2×C3×M16—31no—
95C2×C3×M16C2×D311yescharacteristic
96C2×C3×M16C22×C311yescharacteristic
97C22×C3×C8—31no—
98C22×C3×C8—31no—
99C2×C3×C42—21no—
100C2×C3×C42C2×D311yescharacteristic
101C2×C3×C42C22×C311yescharacteristic
102#950—22no—
103#950—22no—
104C2×#216C3×C411yescharacteristic, complemented
105C2×#215C3×C411yescharacteristic, complemented
106C2×#214C3×C411yescharacteristic, complemented
107C2×#213C3×C411yescharacteristic, complemented
108C2×#208C22×C311yescharacteristic
109C22×C3⋊C8C22×C311yescharacteristic
110C22×C3⋊C8C22×C311yescharacteristic
111C3×#108—62no—
112C3×#104—24no—
113C3×#104—22no—
114C3×#104—22no—
115C3×#104—62no—
116C3×C22⋊C8—62no—
117C3×C22⋊C8—62no—
118C3×#97—62no—
119#830—22no—
120#822—22no—
121(C2×D3)⋊C8—22no—
122#805—22no—
123#803—22no—
124C23×C32—21no—
125C23×C32#4111yescharacteristic
126C22×C3×D3—21no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—41no—
132C32×D4—41no—
133C2×C32×C4—21no—
134C2×C32×C4—22no—
135C2×C32×C4—42no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4C22×C411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4#4111yescharacteristic
144C3×#67—41no—
145C3×#67—41no—
146C2×C3×Dic3—21no—
147C3×D12—21no—
148C3×D12—21no—
149C3×C4×D3—41no—
150C3×C4×D3—41no—
151C3×Dic6—21no—
152C3×Dic6—21no—
153C32×C8—41no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—41no—
156C3×C3⋊C8C22⋊C412yes—
157C3×C3⋊C8C22⋊C412yes—
158C2×#135—31no—
159C2×#108—31no—
160C2×#104—31no—
161C2×#104C3×D311yescharacteristic, complemented
162#408—31no—
163#408—31no—
164C2×#97—31no—
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—21no—
168C3×#41—21no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C3×#41—21no—
174C3×#41—21no—
175C3×#41—41no—
176C3×#41—41no—
177C3×#41—61no—
178C3×#41—61no—
179C3×#41—61no—
180C3×#41—61no—
181C2×C3×Q8—21no—
182C2×C3×Q8—31no—
183C2×C3×Q8C4×D311yescharacteristic
184C2×C3×Q8C2×C3×C411yescharacteristic
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—31no—
189C2×C3×D4—41no—
190C2×C3×D4—61no—
191C2×C3×D4C4×D311yescharacteristic
192C2×C3×D4C2×C3×C411yescharacteristic
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—41no—
199C22×C3×C4—41no—
200C22×C3×C4—61no—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C23×C311yescharacteristic
203C2×#67—21no—
204#235—21no—
205#235—21no—
206#235—21no—
207#235—21no—
208C2×D12C2×C3×C411yescharacteristic
209C2×C4×D3—21no—
210C2×Dic6C2×C3×C411yescharacteristic
211C3×Q16—62no—
212C3×SD16—62no—
213C3×SD16—62no—
214C3×D8—62no—
215C3×M16—22no—
216C3×M16—21no—
217C3×M16—21no—
218C3×M16—21no—
219C3×M16—21no—
220C3×M16—41no—
221C3×M16—61no—
222C3×M16—61no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—32no—
228C2×C3×C8—32no—
229C2×C3×C8—32no—
230C2×C3×C8—41no—
231C2×C3×C8—61no—
232C2×C3×C8—61no—
233C2×C3×C8—61no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—22no—
237C3×C42—41no—
238C3×C42#6712yes—
239C3×C42C3×D412yes—
240#216—22no—
241#215—22no—
242#214—22no—
243#213—22no—
244#208—21no—
245#208—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8C2×C3×C411yescharacteristic
250C2×C3⋊C8C2×C3×C411yescharacteristic
251C2×C3⋊C8C3×D412yes—
252C2×C3⋊C8C3×D412yes—
253C2×C3⋊C8C3×D412yes—
254C22×C32—22no—
255C22×C32—41no—
256C22×C32—42no—
257C22×C32C4×D411yescharacteristic
258C22×C32#12111yescharacteristic
259C22×C32#12411yescharacteristic
260C2×C3×D3—41no—
261C2×C3×D3—42no—
262C32×C4—42no—
263C32×C4—41no—
264C32×C4C2×C22⋊C411yescharacteristic
265C32×C4C4×D411yescharacteristic
266C32×C4#12111yescharacteristic
267C32×C4#12711yescharacteristic
268C3×Dic3—41no—
269C2×#41—31no—
270C2×#41C2×C3×D311yescharacteristic
271#135—62no—
272#135—62no—
273C2×Q16—31no—
274C2×SD16—31no—
275C2×SD16—31no—
276C2×D8—31no—
277C2×M16—31no—
278C2×M16C2×C3×D311yescharacteristic
279C22×C8—31no—
280C22×C8—31no—
281C2×C42C2×C3×D311yescharacteristic
282#108—62no—
283#104—22no—
284#104—62no—
285C22⋊C8—62no—
286C22⋊C8—62no—
287#97—62no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—41no—
292C23×C3—61no—
293C23×C3#23711yescharacteristic
294C23×C3C3×#4111yescharacteristic
295C22×D3—21no—
296C3×Q8—22no—
297C3×Q8—21no—
298C3×Q8—21no—
299C3×Q8—21no—
300C3×Q8—21no—
301C3×Q8—41no—
302C3×Q8—61no—
303C3×Q8—61no—
304C3×D4—22no—
305C3×D4—21no—
306C3×D4—21no—
307C3×D4—21no—
308C3×D4—21no—
309C3×D4—41no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—61no—
317C3×D4—61no—
318C3×D4—121no—
319C3×D4—121no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4C2×C4×D311yescharacteristic
345C2×C3×C4#23511yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C22×C3×C411yescharacteristic
351C2×C3×C4C2×C3×D411yescharacteristic
352C2×C3×C4C2×C3×D411yescharacteristic
353C2×C3×C4C2×C3×D411yescharacteristic
354C2×C3×C4C2×C3×D411yescharacteristic
355C2×C3×C4C3×#4111yescharacteristic
356#67—41no—
357#67—41no—
358C2×Dic3—21no—
359D12—21no—
360D12—21no—
361C4×D3—41no—
362C4×D3—41no—
363Dic6—21no—
364Dic6—21no—
365C3×C8—32no—
366C3×C8—32no—
367C3×C8—42no—
368C3×C8—41no—
369C3×C8—41no—
370C3×C8—62no—
371C3×C8—121no—
372C3⋊C8—22no—
373C3⋊C8—41no—
374C3⋊C8C3×C22⋊C412yes—
375C3⋊C8C3×C22⋊C412yes—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—81no—
379C2×C32#39011yescharacteristic
380C2×C32#44412yes—
381C3×D3—81no—
382#41—21no—
383#41—21no—
384#41—21no—
385#41—21no—
386#41—61no—
387#41—61no—
388#41—61no—
389#41—61no—
390C2×Q8—31no—
391C2×Q8C3×C4×D311yescharacteristic
392C2×D4—21no—
393C2×D4—31no—
394C2×D4—61no—
395C2×D4C3×C4×D311yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—61no—
399C22×C4C22×C3×D311yescharacteristic
400Q16—62no—
401SD16—62no—
402SD16—62no—
403D8—62no—
404M16—21no—
405M16—21no—
406M16—61no—
407M16—61no—
408C2×C8—21no—
409C2×C8—31no—
410C2×C8—31no—
411C2×C8—32no—
412C2×C8—32no—
413C2×C8—32no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C42—21no—
418C42C3×#6712yes—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—82no—
431C22×C3—121no—
432C22×C3—122no—
433C22×C3#67⋊C411yescharacteristic
434C22×C3#88511yescharacteristic
435C22×C3#93911yescharacteristic
436C22×C3C3×C4×D411yescharacteristic
437C22×C3C3×#12111yescharacteristic
438C22×C3C3×#12411yescharacteristic
439C2×D3—41no—
440C2×D3—42no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—121no—
452C3×C4D3×C22⋊C411yescharacteristic
453C3×C4#88411yescharacteristic
454C3×C4C4×#6711yescharacteristic
455C3×C4#94011yescharacteristic
456C3×C4C2×C3×C22⋊C411yescharacteristic
457C3×C4C3×C4×D411yescharacteristic
458C3×C4C3×#12111yescharacteristic
459C3×C4C3×#12711yescharacteristic
460Dic3—41no—
461C32#197511yescharacteristic, complemented, Sylow
462C23—21no—
463C23—61no—
464C23C3×#23711yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—61no—
468Q8—61no—
469D4—21no—
470D4—21no—
471D4—41no—
472D4—41no—
473D4—61no—
474D4—61no—
475D4—121no—
476D4—121no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—61no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4C2×C3×C4×D311yescharacteristic
486C2×C4C3×#23511yescharacteristic
487C2×C4C3×D3×D411yescharacteristic
488C2×C4C3×D3×D411yescharacteristic
489C2×C4C3×D3×D411yescharacteristic
490C2×C4C2×C3×#6711yescharacteristic
491C8—32no—
492C8—32no—
493C8—41no—
494C8—62no—
495C8—121no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—82no—
505C2×C3—81no—
506C2×C3—81no—
507C2×C3—241no—
508C2×C3#482312yes—
509C2×C3#495711yescharacteristic
510C2×C3C3×#39011yescharacteristic
511C2×C3C3×#44412yes—
512D3—81no—
513C22—22no—
514C22—41no—
515C22—42no—
516C22—121no—
517C22—122no—
518C22C3×#67⋊C411yescharacteristic
519C22C3×#88511yescharacteristic
520C22C3×#93911yescharacteristic
521C4—42no—
522C4—41no—
523C4—121no—
524C4C3×D3×C22⋊C411yescharacteristic
525C4C3×#88411yescharacteristic
526C4C3×C4×#6711yescharacteristic
527C4C3×#94011yescharacteristic
528C3—21no—
529C3#7037411yescharacteristic, direct product factor
530C3C3×#197511yescharacteristic, complemented
531C2—21no—
532C2—21no—
533C2—81no—
534C2—241no—
535C2C3×#482312yes—
536C2C3×#495711yescharacteristic
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#104C2
socleC22×C32C4×D4
prime radicalC2×C3×#950C2
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#104D3
hypocenterC3C3×#1975
LCSG ▷ C2×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
2. C2×C32×#104C2semidirect
3. C3×#5234C2semidirect
4. C3×#5129C2semidirect
5. C2×C3×#822C2semidirect
6. C2×C3×#805C2semidirect
7. C2×C3×#803C2semidirect
9. #70374C3direct (N × H)
16. C2×C3×#216C4semidirect
17. C2×C3×#215C4semidirect
18. C2×C3×#214C4semidirect
19. C2×C3×#213C4semidirect
34. C2×C3×#104D3semidirect
35. C2×C3×#104C2×C3semidirect
39. #5234C2×C3semidirect
40. #5129C2×C3semidirect
41. C2×#822C2×C3semidirect
42. C2×#805C2×C3semidirect
43. C2×#803C2×C3semidirect
104. C2×#216C3×C4semidirect
105. C2×#215C3×C4semidirect
106. C2×#214C3×C4semidirect
107. C2×#213C3×C4semidirect
161. C2×#104C3×D3semidirect
461. C32#1975semidirect
529. C3#70374direct (N × H)
530. C3C3×#1975semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9502
C2×C32×#1042
C3×#52342
C3×#51292
C2×C3×#8222
C2×C3×#8052
C2×C3×#8032
C3×#19753
#703743