Finite Group Databasebeta

← C3×#70399 #11788365

Subgroups — C3×#70399 #11788365

Order 1152.

685 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#951C211yesmaximal, characteristic
2C3×#5819C211yesmaximal, characteristic
3C32×#421C211yesmaximal, characteristic
4C2×C3×#824C211yesmaximal, characteristic, complemented
5C3×#5004C211yesmaximal, characteristic, complemented
6C3×C3⋊#338C211yesmaximal, characteristic, complemented
7C3×#4533C211yesmaximal, characteristic, complemented
8C3×#1902—31nomaximal
9#70399C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#951—22no—
15C3×#951—22no—
16C3×#948—22no—
17C3×#948—22no—
18C2×C3×#216C412yescomplemented
19C2×C3×#214C412yescomplemented
20C2×C3×#208—22no—
21C2×C3×#208C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C32×#104—22no—
25C3×#824—22no—
26C3×#810—22no—
27C2×C3×#137—31no—
28C3×#568—31no—
29C2×#951C2×C311yescharacteristic
30#5819C2×C311yescharacteristic
31C3×#421—21no—
32C3×#421D311yescharacteristic
33C3×#421C2×C311yescharacteristic
34C3×#419—31no—
35C2×C3×#101—31no—
36C2×#824C2×C311yescharacteristic, complemented
37#5004C2×C311yescharacteristic, complemented
38C3×#344—31no—
39C3×#338—31no—
40C3⋊#338C2×C311yescharacteristic, complemented
41#4533C2×C311yescharacteristic, complemented
42C32×#41—21no—
43C32×#41—21no—
44C32×#41D412yes—
45C32×#41D412yes—
46C2×C32×Q8C2×C411yescharacteristic
47C2×C32×D4—21no—
48C2×C32×D4C2×C411yescharacteristic
49C22×C32×C4—21no—
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6C2×C412yes—
58C32×M16—21no—
59C32×M16—21no—
60C2×C32×C8—21no—
61C32×C4⋊C4—21no—
62C32×C22⋊C4—21no—
63C32×C42—21no—
64C3×#216—24no—
65C3×#214—24no—
66C3×#208—24no—
67C3×#208—21no—
68C3×#208—21no—
69C3×#208—21no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208D412yes—
73C2×C3×C3⋊C8—24no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8C2×C412yes—
76C2×C3×C3⋊C8D412yes—
77#1902—31noSylow
78C2×C3×#41—21no—
79C2×C3×#41C2×D311yescharacteristic
80C2×C3×#41C22×C311yescharacteristic
81C22×C3×Q8—31no—
82C22×Dic6C22×C311yescharacteristic
83C3×#137—62no—
84C3×#137—62no—
85C2×C3×Q16—32no—
86C2×C3×SD16—32no—
87C3×#131—62no—
88C3×#131—62no—
89C2×C3×M16—21no—
90C2×C3×M16—31no—
91C2×C3×M16—31no—
92C2×C3×M16—31no—
93C2×C3×M16—62no—
94C2×C3×M16C2×D311yescharacteristic
95C2×C3×M16C22×C311yescharacteristic
96C3×#117—21no—
97C3×#117C2×D311yescharacteristic
98C3×#117C22×C311yescharacteristic
99#951—22no—
100#951—22no—
101#948—22no—
102#948—22no—
103C2×#216C3×C412yescomplemented
104C2×#214C3×C412yescomplemented
105C2×#208—22no—
106C2×#208C22×C311yescharacteristic
107C2×#208C22×C311yescharacteristic
108C2×#208C22×C311yescharacteristic
109C3×#104—24no—
110C3×#104—22no—
111C3×#104—22no—
112C3×#103—62no—
113C3×#101—62no—
114#824—22no—
115#810—22no—
116C23×C32—21no—
117C23×C32#4111yescharacteristic
118C32×Q8—21no—
119C32×Q8C22⋊C412yes—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4—21no—
123C32×D4C22⋊C412yes—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—41no—
128C2×C32×C4C22×C411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4#4111yescharacteristic
134C2×C3×Dic3—22no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—41no—
137C3×Dic6—22no—
138C3×Dic6—22no—
139C3×Dic6—41no—
140C3×Dic6—41no—
141C32×C8—41no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—22no—
145C2×#137—31no—
146#568—31no—
147#421C3×D311yescharacteristic
148#419—31no—
149C2×#101—31no—
150#344—31no—
151#338—31no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41—21no—
155C3×#41—21no—
156C3×#41—22no—
157C3×#41—22no—
158C3×#41—41no—
159C3×#41—41no—
160C3×#41D1212yes—
161C3×#41#6712yes—
162C3×#41C3×D412yes—
163C3×#41C3×D412yes—
164C2×C3×Q8—21no—
165C2×C3×Q8—32no—
166C2×C3×Q8—61no—
167C2×C3×Q8—61no—
168C2×C3×Q8—61no—
169C2×C3×Q8—61no—
170C2×C3×Q8—61no—
171C2×C3×Q8C4×D311yescharacteristic
172C2×C3×Q8C2×C3×C411yescharacteristic
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—41no—
177C2×C3×D4C4×D311yescharacteristic
178C2×C3×D4C2×C3×C411yescharacteristic
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—41no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C23×C311yescharacteristic
186C22×Dic3—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6C2×C3×C412yes—
193C3×Q16—64no—
194C3×SD16—64no—
195C3×M16—22no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—21no—
199C3×M16—21no—
200C3×M16—32no—
201C3×M16—41no—
202C3×M16—64no—
203C3×M16—61no—
204C3×M16—61no—
205C3×M16—61no—
206C3×M16—61no—
207C3×M16—61no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—32no—
211C2×C3×C8—32no—
212C2×C3×C8—41no—
213C2×C3×C8—64no—
214C2×C3×C8—61no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—41no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—41no—
221C3×C42—22no—
222C3×C42—21no—
223C3×C42—21no—
224#216—24no—
225#214—24no—
226#208—24no—
227#208—21no—
228#208—21no—
229#208—21no—
230#208—21no—
231#208—21no—
232#208C3×D412yes—
233C2×C3⋊C8—24no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8C2×C3×C412yes—
236C2×C3⋊C8C3×D412yes—
237C22×C32—22no—
238C22×C32—41no—
239C22×C32—81no—
240C22×C32C4×D411yescharacteristic
241C22×C32#12111yescharacteristic
242C22×C32#12311yescharacteristic
243C32×C4—22no—
244C32×C4—81no—
245C32×C4C2×C22⋊C411yescharacteristic
246C32×C4C4×D411yescharacteristic
247C32×C4#12011yescharacteristic
248C32×C4#12111yescharacteristic
249C3×Dic3—42no—
250C2×#41C2×C3×D311yescharacteristic
251C22×Q8—31no—
252#137—62no—
253#137—62no—
254C2×Q16—32no—
255C2×SD16—32no—
256#131—62no—
257#131—62no—
258C2×M16—31no—
259C2×M16—31no—
260C2×M16—31no—
261C2×M16—62no—
262C2×M16C2×C3×D311yescharacteristic
263#117C2×C3×D311yescharacteristic
264#104—22no—
265#103—62no—
266#101—62no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—41no—
271C23×C3#23711yescharacteristic
272C23×C3C3×#4111yescharacteristic
273C3×Q8—21no—
274C3×Q8—21no—
275C3×Q8—22no—
276C3×Q8—41no—
277C3×Q8—62no—
278C3×Q8—62no—
279C3×Q8—121no—
280C3×Q8—121no—
281C3×Q8(C2×D3)⋊C412yes—
282C3×Q8C3×C22⋊C412yes—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—22no—
286C3×D4—22no—
287C3×D4—21no—
288C3×D4—21no—
289C3×D4—22no—
290C3×D4—41no—
291C3×D4—42no—
292C3×D4—42no—
293C3×D4(C2×D3)⋊C412yes—
294C3×D4C3×C22⋊C412yes—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—62no—
314C2×C3×C4—121no—
315C2×C3×C4—121no—
316C2×C3×C4C2×C4×D311yescharacteristic
317C2×C3×C4C2×D1211yescharacteristic
318C2×C3×C4D3×D411yescharacteristic
319C2×C3×C4D3×D411yescharacteristic
320C2×C3×C4#23711yescharacteristic
321C2×C3×C4C2×#6711yescharacteristic
322C2×C3×C4C22×C3×C411yescharacteristic
323C2×C3×C4C2×C3×D411yescharacteristic
324C2×C3×C4C2×C3×D411yescharacteristic
325C2×C3×C4C2×C3×D411yescharacteristic
326C2×C3×C4C2×C3×D411yescharacteristic
327C2×C3×C4C3×#4111yescharacteristic
328C2×Dic3—22no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331Dic6—22no—
332Dic6—22no—
333Dic6—41no—
334Dic6—41no—
335C3×C8—42no—
336C3×C8—41no—
337C3×C8—41no—
338C3×C8—62no—
339C3×C8—62no—
340C3×C8—62no—
341C3⋊C8—22no—
342C3⋊C8—22no—
343C3⋊C8—22no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—42no—
348C2×C32#38611yescharacteristic
349#41—21no—
350#41—21no—
351#41C3×D1212yes—
352#41C3×#6712yes—
353C2×Q8—32no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8C3×C4×D311yescharacteristic
360C2×D4—21no—
361C2×D4C3×C4×D311yescharacteristic
362C22×C4—21no—
363C22×C4—61no—
364C22×C4C22×C3×D311yescharacteristic
365Q16—64no—
366SD16—64no—
367M16—21no—
368M16—21no—
369M16—32no—
370M16—64no—
371M16—61no—
372M16—61no—
373M16—61no—
374M16—61no—
375M16—61no—
376C2×C8—21no—
377C2×C8—32no—
378C2×C8—32no—
379C2×C8—64no—
380C2×C8—61no—
381C4⋊C4—21no—
382C22⋊C4—21no—
383C42—21no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—42no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—82no—
396C22×C3#67⋊C411yescharacteristic
397C22×C3#88611yescharacteristic
398C22×C3#93911yescharacteristic
399C22×C3C3×C4×D411yescharacteristic
400C22×C3C3×#12111yescharacteristic
401C22×C3C3×#12311yescharacteristic
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—42no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4—82no—
412C3×C4—122no—
413C3×C4#67⋊C411yescharacteristic
414C3×C4#88211yescharacteristic
415C3×C4C2×(C2×D3)⋊C411yescharacteristic
416C3×C4#93911yescharacteristic
417C3×C4C2×C3×C22⋊C411yescharacteristic
418C3×C4C3×C4×D411yescharacteristic
419C3×C4C3×#12011yescharacteristic
420C3×C4C3×#12111yescharacteristic
421Dic3—42no—
422C32#190211yescharacteristic, complemented, Sylow
423C23—21no—
424C23C3×#23711yescharacteristic
425Q8—21no—
426Q8—62no—
427Q8—62no—
428Q8—121no—
429Q8—121no—
430Q8C3×(C2×D3)⋊C412yes—
431D4—22no—
432D4—22no—
433D4—21no—
434D4C3×(C2×D3)⋊C412yes—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—62no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4C2×C3×C4×D311yescharacteristic
443C2×C4C2×C3×D1211yescharacteristic
444C2×C4C3×D3×D411yescharacteristic
445C2×C4C3×D3×D411yescharacteristic
446C2×C4C3×#23711yescharacteristic
447C2×C4C2×C3×#6711yescharacteristic
448C8—41no—
449C8—62no—
450C8—62no—
451C8—62no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—42no—
463C2×C3—42no—
464C2×C3—82no—
465C2×C3#495811yescharacteristic
466C2×C3C3×#38611yescharacteristic
467C22—22no—
468C22—41no—
469C22—81no—
470C22C3×#67⋊C411yescharacteristic
471C22C3×#88611yescharacteristic
472C22C3×#93911yescharacteristic
473C4—22no—
474C4—81no—
475C4—122no—
476C4C3×#67⋊C411yescharacteristic
477C4C3×#88211yescharacteristic
478C4C2×C3×(C2×D3)⋊C411yescharacteristic
479C4C3×#93911yescharacteristic
480C3—21no—
481C3#7039911yescharacteristic, direct product factor
482C3C3×#190211yescharacteristic, complemented
483C2—21no—
484C2—21no—
485C2—21no—
486C2—42no—
487C2C3×#495811yescharacteristic
subgroupG/H
centerC2×C3#4958
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#421C2
socleC2×C32#386
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#421D3
hypocenterC3C3×#1902
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#421

Complements

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

NH ≅ G/Ntype
4. C2×C3×#824C2semidirect
5. C3×#5004C2semidirect
6. C3×C3⋊#338C2semidirect
7. C3×#4533C2semidirect
9. #70399C3direct (N × H)
18. C2×C3×#216C4semidirect
19. C2×C3×#214C4semidirect
36. C2×#824C2×C3semidirect
37. #5004C2×C3semidirect
40. C3⋊#338C2×C3semidirect
41. #4533C2×C3semidirect
103. C2×#216C3×C4semidirect
104. C2×#214C3×C4semidirect
422. C32#1902semidirect
481. C3#70399direct (N × H)
482. C3C3×#1902semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9512
C3×#58192
C32×#4212
C2×C3×#8242
C3×#50042
C3×C3⋊#3382
C3×#45332
C3×#19023
#703993