Finite Group Databasebeta

← C3×#70300 #11785609

Subgroups — C3×#70300 #11785609

Order 1152.

609 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#948C211yesmaximal, characteristic
2C32×#417C211yesmaximal, characteristic
3C3×#5232C211yesmaximal, characteristic
4C2×C3×#822C211yesmaximal, characteristic, complemented
5C2×C3×#807C211yesmaximal, characteristic, complemented
6C3×#4554C211yesmaximal, characteristic, complemented
7C3×#4530C211yesmaximal, characteristic, complemented
8C3×#1816—31nomaximal
9#70300C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C3×#948—22no—
14C3×#948C414yescomplemented
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×#208—21no—
17C2×C3×#208C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C22×C3×C3⋊C8—21no—
20C22×C3×C3⋊C8C2211yescharacteristic
21C32×#103—21no—
22C32×#102—21no—
23C3×#835—21no—
24C3×#832—21no—
25C3×#822—22no—
26C3×#807—21no—
27C3×#807—21no—
28C2×C3×#131—31no—
29C2×#948C2×C311yescharacteristic
30C2×C3×#108—31no—
31C2×C3×#107—31no—
32C3×#417—21no—
33C3×#417—31no—
34C3×#417D311yescharacteristic
35C3×#417C2×C311yescharacteristic
36#5232C2×C311yescharacteristic
37C2×#822C2×C311yescharacteristic, complemented
38C2×#807C2×C311yescharacteristic, complemented
39C3×#343—31no—
40C3×#343—31no—
41#4554C2×C311yescharacteristic, complemented
42#4530C2×C311yescharacteristic, complemented
43C32×#41—21no—
44C32×#41—21no—
45C32×#41C2×C412yes—
46C32×#41Q812yes—
47C2×C32×Q8D411yescharacteristic
48C2×C32×D4—21no—
49C2×C32×D4D411yescharacteristic
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3—21no—
54C32×M16—21no—
55C32×M16—21no—
56C2×C32×C8—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×#208—21no—
62C3×#208—24no—
63C3×#208—21no—
64C3×#208—21no—
65C3×#208C2×C412yes—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—24no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8C2×C412yes—
70C2×C3×C3⋊C8D411yescharacteristic
71C2×C3×C3⋊C8D411yescharacteristic
72C2×C3×C3⋊C8D412yes—
73#1816—31noSylow
74C2×C3×#41—21no—
75C2×C3×#41C2×D311yescharacteristic
76C2×C3×#41C22×C311yescharacteristic
77C3×#131—34no—
78C3×#131—62no—
79C2×C3×M16—21no—
80C2×C3×M16—31no—
81C2×C3×M16—31no—
82C2×C3×M16—61no—
83C2×C3×M16C2×D311yescharacteristic
84C2×C3×M16C22×C311yescharacteristic
85C22×C3×C8—31no—
86C22×C3×C8—61no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—31no—
89C2×C3×C4⋊C4C2×D311yescharacteristic
90C2×C3×C4⋊C4C22×C311yescharacteristic
91#948—22no—
92#948C3×C414yescomplemented
93C2×C4⋊Dic3C22×C311yescharacteristic
94C2×#208—21no—
95C2×#208C22×C311yescharacteristic
96C2×#208C22×C311yescharacteristic
97C22×C3⋊C8—21no—
98C22×C3⋊C8C22×C311yescharacteristic
99C3×#108—62no—
100C3×#107—61no—
101C3×#107—61no—
102C3×#103—22no—
103C3×#103—21no—
104C3×#103—21no—
105C3×#103—61no—
106C3×#102—22no—
107C3×#102—21no—
108C3×#102—21no—
109C3×#102—61no—
110#835—21no—
111#832—21no—
112#822—22no—
113#807—21no—
114#807—21no—
115C23×C32—21no—
116C23×C32#4111yescharacteristic
117C32×Q8—21no—
118C32×Q8C4⋊C412yes—
119C32×D4—22no—
120C32×D4—22no—
121C32×D4—21no—
122C32×D4C4⋊C412yes—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—81no—
128C2×C32×C4C22×C411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×Q811yescharacteristic
133C2×C32×C4#4111yescharacteristic
134C2×C3×Dic3—41no—
135C2×C3×Dic3—81no—
136C32×C8—41no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—41no—
139C3×C3⋊C8C22⋊C412yes—
140C3×C3⋊C8C22⋊C412yes—
141C2×#131—31no—
142C2×#108—31no—
143C2×#107—31no—
144#417—31no—
145#417C3×D311yescharacteristic
146#343—31no—
147#343—31no—
148C3×#41—21no—
149C3×#41—21no—
150C3×#41—21no—
151C3×#41—21no—
152C3×#41—22no—
153C3×#41—22no—
154C3×#41—41no—
155C3×#41—41no—
156C3×#41Dic612yes—
157C3×#41C4×D312yes—
158C3×#41C2×C3×C412yes—
159C3×#41C3×Q812yes—
160C2×C3×Q8—21no—
161C2×C3×Q8#6711yescharacteristic
162C2×C3×Q8C3×D411yescharacteristic
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—41no—
167C2×C3×D4#6711yescharacteristic
168C2×C3×D4C3×D411yescharacteristic
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—41no—
175C22×C3×C4—41no—
176C22×C3×C4—61no—
177C22×C3×C4C22×D311yescharacteristic
178C22×C3×C4C23×C311yescharacteristic
179C22×Dic3—21no—
180C3×M16—21no—
181C3×M16—21no—
182C3×M16—21no—
183C3×M16—21no—
184C3×M16—32no—
185C3×M16—41no—
186C3×M16—41no—
187C3×M16—64no—
188C3×M16—61no—
189C3×M16—61no—
190C3×M16—61no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—32no—
195C2×C3×C8—31no—
196C2×C3×C8—31no—
197C2×C3×C8—32no—
198C2×C3×C8—64no—
199C2×C3×C8—61no—
200C2×C3×C8—61no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—61no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211#208—21no—
212#208—24no—
213#208—21no—
214#208—21no—
215#208C2×C3×C412yes—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—24no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8C2×C3×C412yes—
220C2×C3⋊C8C3×D411yescharacteristic
221C2×C3⋊C8C3×D411yescharacteristic
222C2×C3⋊C8C3×D412yes—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32#11711yescharacteristic
229C22×C32#12111yescharacteristic
230C22×C32C22⋊Q811yescharacteristic
231C32×C4—22no—
232C32×C4—81no—
233C32×C4C2×C22⋊C411yescharacteristic
234C32×C4C2×C4⋊C411yescharacteristic
235C32×C4#12111yescharacteristic
236C32×C4C22⋊Q811yescharacteristic
237C3×Dic3—81no—
238C2×#41C2×C3×D311yescharacteristic
239#131—34no—
240#131—62no—
241C2×M16—31no—
242C2×M16—31no—
243C2×M16—61no—
244C2×M16C2×C3×D311yescharacteristic
245C22×C8—31no—
246C22×C8—61no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4C2×C3×D311yescharacteristic
249#108—62no—
250#107—61no—
251#107—61no—
252#103—21no—
253#103—61no—
254#102—21no—
255#102—61no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—41no—
260C23×C3#23711yescharacteristic
261C23×C3C3×#4111yescharacteristic
262C3×Q8—21no—
263C3×Q8—21no—
264C3×Q8—22no—
265C3×Q8—41no—
266C3×Q8Dic3⋊C412yes—
267C3×Q8C3×C4⋊C412yes—
268C3×D4—22no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—22no—
272C3×D4—21no—
273C3×D4—21no—
274C3×D4—22no—
275C3×D4—41no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4Dic3⋊C412yes—
279C3×D4C3×C4⋊C412yes—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—82no—
299C2×C3×C4—81no—
300C2×C3×C4—81no—
301C2×C3×C4—121no—
302C2×C3×C4—241no—
303C2×C3×C4C2×Dic611yescharacteristic
304C2×C3×C4C2×C4×D311yescharacteristic
305C2×C3×C4D3×D411yescharacteristic
306C2×C3×C4D3×D411yescharacteristic
307C2×C3×C4#23711yescharacteristic
308C2×C3×C4C2×#6711yescharacteristic
309C2×C3×C4C22×C3×C411yescharacteristic
310C2×C3×C4C2×C3×D411yescharacteristic
311C2×C3×C4C2×C3×D411yescharacteristic
312C2×C3×C4C2×C3×D411yescharacteristic
313C2×C3×C4C2×C3×Q811yescharacteristic
314C2×C3×C4C3×#4111yescharacteristic
315C2×Dic3—41no—
316C2×Dic3—81no—
317C3×C8—32no—
318C3×C8—32no—
319C3×C8—42no—
320C3×C8—41no—
321C3×C8—41no—
322C3×C8—62no—
323C3×C8—121no—
324C3⋊C8—22no—
325C3⋊C8—41no—
326C3⋊C8C3×C22⋊C412yes—
327C3⋊C8C3×C22⋊C412yes—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—42no—
331C2×C32#38011yescharacteristic
332C2×C32#47211yescharacteristic
333C2×C32#47311yescharacteristic
334#41—21no—
335#41—21no—
336#41C3×Dic612yes—
337#41C3×C4×D312yes—
338C2×Q8C3×#6711yescharacteristic
339C2×D4—21no—
340C2×D4C3×#6711yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—61no—
344C22×C4C22×C3×D311yescharacteristic
345M16—21no—
346M16—21no—
347M16—32no—
348M16—64no—
349M16—61no—
350M16—61no—
351M16—61no—
352C2×C8—21no—
353C2×C8—32no—
354C2×C8—31no—
355C2×C8—31no—
356C2×C8—32no—
357C2×C8—64no—
358C2×C8—61no—
359C2×C8—61no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—61no—
363C4⋊C4—61no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—42no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3#87711yescharacteristic
383C22×C3#87811yescharacteristic
384C22×C3#93911yescharacteristic
385C22×C3C3×#11711yescharacteristic
386C22×C3C3×#12111yescharacteristic
387C22×C3C3×C22⋊Q811yescharacteristic
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—42no—
395C3×C4—82no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4—241no—
399C3×C4#87811yescharacteristic
400C3×C4D3×C22⋊C411yescharacteristic
401C3×C4C2×Dic3⋊C411yescharacteristic
402C3×C4#93911yescharacteristic
403C3×C4C2×C3×C22⋊C411yescharacteristic
404C3×C4C2×C3×C4⋊C411yescharacteristic
405C3×C4C3×#12111yescharacteristic
406C3×C4C3×C22⋊Q811yescharacteristic
407Dic3—81no—
408C32#181611yescharacteristic, complemented, Sylow
409C23—21no—
410C23C3×#23711yescharacteristic
411Q8—21no—
412Q8C3×Dic3⋊C412yes—
413D4—22no—
414D4—22no—
415D4—21no—
416D4C3×Dic3⋊C412yes—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—121no—
423C2×C4—241no—
424C2×C4C2×C3×Dic611yescharacteristic
425C2×C4C2×C3×C4×D311yescharacteristic
426C2×C4C3×D3×D411yescharacteristic
427C2×C4C3×D3×D411yescharacteristic
428C2×C4C3×#23711yescharacteristic
429C2×C4C2×C3×#6711yescharacteristic
430C8—32no—
431C8—32no—
432C8—41no—
433C8—62no—
434C8—121no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—42no—
445C2×C3—42no—
446C2×C3—82no—
447C2×C3#494211yescharacteristic
448C2×C3#519911yescharacteristic
449C2×C3#520411yescharacteristic
450C2×C3C3×#38011yescharacteristic
451C2×C3C3×#47211yescharacteristic
452C2×C3C3×#47311yescharacteristic
453C22—21no—
454C22—21no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22C3×#87711yescharacteristic
459C22C3×#87811yescharacteristic
460C22C3×#93911yescharacteristic
461C4—22no—
462C4—81no—
463C4—241no—
464C4C3×#87811yescharacteristic
465C4C3×D3×C22⋊C411yescharacteristic
466C4C2×C3×Dic3⋊C411yescharacteristic
467C4C3×#93911yescharacteristic
468C3—21no—
469C3#7030011yescharacteristic, direct product factor
470C3C3×#181611yescharacteristic, complemented
471C2—21no—
472C2—21no—
473C2—42no—
474C2C3×#494211yescharacteristic
475C2C3×#519911yescharacteristic
476C2C3×#520411yescharacteristic
subgroupG/H
centerC22×C3#939
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#417C2
socleC22×C32#121
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#417D3
hypocenterC3C3×#1816
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#417

Complements

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

NH ≅ G/Ntype
4. C2×C3×#822C2semidirect
5. C2×C3×#807C2semidirect
6. C3×#4554C2semidirect
7. C3×#4530C2semidirect
9. #70300C3direct (N × H)
14. C3×#948C4semidirect
37. C2×#822C2×C3semidirect
38. C2×#807C2×C3semidirect
41. #4554C2×C3semidirect
42. #4530C2×C3semidirect
92. #948C3×C4semidirect
408. C32#1816semidirect
469. C3#70300direct (N × H)
470. C3C3×#1816semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9482
C32×#4172
C3×#52322
C2×C3×#8222
C2×C3×#8072
C3×#45542
C3×#45302
C3×#18163
#703003