Finite Group Databasebeta

← C3×#17591 #10636198

Subgroups — C3×#17591 #10636198

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3444C211yesmaximal, characteristic
2C2×C3×#432C211yesmaximal, characteristic, complemented
3C3×#2283C211yesmaximal, characteristic, complemented
4C3×#1516C214yesmaximal, complemented
5C3×#1512C214yesmaximal, complemented
6C3×#1506C212yesmaximal, complemented
7C3×#1505C212yesmaximal, complemented
8#17591C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C411yescharacteristic, complemented
10C3×#533C411yescharacteristic, complemented
11C3×#533C2211yescharacteristic
12C3×#532C411yescharacteristic, complemented
13C3×#532C411yescharacteristic, complemented
14C3×#532C2211yescharacteristic
15C3×#529—21no—
16C3×#529—21no—
17C3×#529—21no—
18C3×#529—21no—
19C3×#529C412yescomplemented
20C3×#529C2212yes—
21C3×#525—21no—
22C3×#525—21no—
23C3×#525C2212yes—
24C3×C4×#41—21no—
25C3×C4×#41—21no—
26C3×C4×#41C412yescomplemented
27C22×C3×C42C2211yescharacteristic
28C3×#432—21no—
29C3×#432—21no—
30C3×#432C2212yes—
31C3×#432C2212yes—
32C2×C3×C4⋊C8—21no—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×#97—21no—
35C3×#383—21no—
36C3×#382C2212yes—
37C3×#381C2212yes—
38C3×#380—21no—
39C3×C4×C4⋊C4—21no—
40C3×C4×C22⋊C4—21no—
41C3×#376C2212yes—
42C3×#328C2218yes—
43C3×#327C2218yes—
44#3444C2×C311yescharacteristic
45C2×#432C2×C311yescharacteristic, complemented
46#2283C2×C311yescharacteristic, complemented
47#1516C2×C314yescomplemented
48#1512C2×C314yescomplemented
49#1506C2×C312yescomplemented
50#1505C2×C312yescomplemented
51C2×C3×#41—21no—
52C2×C3×#41—21no—
53C2×C3×#41C2×C412yes—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C22×C3×C8—22no—
58C3×C4⋊Q8C2×C411yescharacteristic
59C3×C4⋊Q8C2×C411yescharacteristic
60C3×C4⋊Q8C2×C411yescharacteristic
61C3×#127C2×C411yescharacteristic
62C3×#126—22no—
63C3×#126—22no—
64C3×#125—21no—
65C3×#125—21no—
66C3×#125C2×C412yes—
67C3×#124—21no—
68C3×#124—21no—
69C3×#124C2×C412yes—
70C3×#123—22no—
71C3×#123—22no—
72C3×#123—22no—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8—21no—
75C3×C22⋊Q8—21no—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8C2×C412yes—
78C3×C22⋊Q8C2312yes—
79C3×#121—21no—
80C3×#121—21no—
81C3×#121—21no—
82C3×#121—21no—
83C3×#121C2×C412yes—
84C3×#121C2312yes—
85C3×#120—22no—
86C3×C4×Q8—21no—
87C3×C4×Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8C2×C412yes—
91C3×C4×Q8C2×C412yes—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4C2×C412yes—
103C3×C4×D4C2×C412yes—
104C3×#117—21no—
105C3×#117—21no—
106C3×#117—22no—
107C3×#117—22no—
108C3×#117—22no—
109C3×#117C2×C412yes—
110C2×C3×C4⋊C4—22no—
111C2×C3×C22⋊C4—22no—
112C2×C3×C42—22no—
113C2×C3×C42—22no—
114C2×C3×C42C2×C411yescharacteristic
115C2×C3×C42C2×C411yescharacteristic
116C2×C3×C42C2×C411yescharacteristic
117C2×C3×C42C2×C411yescharacteristic
118C2×C3×C42D411yescharacteristic
119C2×C3×C42D411yescharacteristic
120C2×C3×C42D411yescharacteristic
121C2×C3×C42D411yescharacteristic
122C2×C3×C42D411yescharacteristic
123C2×C3×C42D411yescharacteristic
124C2×C3×C42C2311yescharacteristic
125C2×C3×C42C2311yescharacteristic
126C3×C4⋊C8—22no—
127C3×C4⋊C8—22no—
128C3×C22⋊C8—22no—
129C3×C22⋊C8C2314yes—
130C3×#97—22no—
131C3×#97—22no—
132C3×#95—22no—
133C3×#95—22no—
134C3×#95—22no—
135C3×#95C2314yes—
136#533C3×C411yescharacteristic, complemented
137#533C3×C411yescharacteristic, complemented
138#533C22×C311yescharacteristic
139#532C3×C411yescharacteristic, complemented
140#532C3×C411yescharacteristic, complemented
141#532C22×C311yescharacteristic
142#529—21no—
143#529—21no—
144#529—21no—
145#529—21no—
146#529C3×C412yescomplemented
147#529C22×C312yes—
148#525—21no—
149#525—21no—
150#525C22×C312yes—
151C4×#41—21no—
152C4×#41—21no—
153C4×#41C3×C412yescomplemented
154C22×C42C22×C311yescharacteristic
155#432—21no—
156#432—21no—
157#432C22×C312yes—
158#432C22×C312yes—
159C2×C4⋊C8—21no—
160C2×C22⋊C8C22×C312yes—
161C2×#97—21no—
162#383—21no—
163#382C22×C312yes—
164#381C22×C312yes—
165#380—21no—
166C4×C4⋊C4—21no—
167C4×C22⋊C4—21no—
168#376C22×C312yes—
169#328C22×C318yes—
170#327C22×C318yes—
171C24×C3C2×D411yescharacteristic
172C3×#41—44no—
173C3×#41—44no—
174C3×#41—44no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8C22×C412yes—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4C22×C412yes—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—24no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4C22⋊C411yescharacteristic
192C22×C3×C4C22⋊C411yescharacteristic
193C22×C3×C4C22⋊C411yescharacteristic
194C22×C3×C4C22⋊C411yescharacteristic
195C22×C3×C4C22⋊C411yescharacteristic
196C22×C3×C4C22⋊C411yescharacteristic
197C22×C3×C4C22⋊C411yescharacteristic
198C22×C3×C4C22⋊C411yescharacteristic
199C22×C3×C4C22×C411yescharacteristic
200C22×C3×C4C22×C411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×C3×C8—24no—
210C2×C3×C8—44no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—44no—
222C3×C4⋊C4C22×C412yes—
223C3×C4⋊C4C22×C412yes—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—44no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—44no—
238C3×C42C22⋊C411yescharacteristic
239C3×C42C22⋊C411yescharacteristic
240C3×C42C22⋊C411yescharacteristic
241C3×C42C22⋊C411yescharacteristic
242C3×C42C22⋊C411yescharacteristic
243C3×C42C22⋊C411yescharacteristic
244C3×C42C22⋊C411yescharacteristic
245C3×C42C22⋊C411yescharacteristic
246C3×C42C22×C411yescharacteristic
247C3×C42C22×C411yescharacteristic
248C3×C42C22×C411yescharacteristic
249C3×C42C22×C411yescharacteristic
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C2×#41—21no—
255C2×#41—21no—
256C2×#41C2×C3×C412yes—
257C23×C4C3×D411yescharacteristic
258C23×C4C3×D411yescharacteristic
259C23×C4C23×C311yescharacteristic
260C22×C8—22no—
261C4⋊Q8C2×C3×C411yescharacteristic
262C4⋊Q8C2×C3×C411yescharacteristic
263C4⋊Q8C2×C3×C411yescharacteristic
264#127C2×C3×C411yescharacteristic
265#126—22no—
266#126—22no—
267#125—21no—
268#125—21no—
269#125C2×C3×C412yes—
270#124—21no—
271#124—21no—
272#124C2×C3×C412yes—
273#123—22no—
274#123—22no—
275#123—22no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8C2×C3×C412yes—
281C22⋊Q8C23×C312yes—
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286#121C2×C3×C412yes—
287#121C23×C312yes—
288#120—22no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8C2×C3×C412yes—
294C4×Q8C2×C3×C412yes—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4C2×C3×C412yes—
306C4×D4C2×C3×C412yes—
307#117—21no—
308#117—21no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117C2×C3×C412yes—
313C2×C4⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42C2×C3×C411yescharacteristic
318C2×C42C2×C3×C411yescharacteristic
319C2×C42C2×C3×C411yescharacteristic
320C2×C42C2×C3×C411yescharacteristic
321C2×C42C3×D411yescharacteristic
322C2×C42C3×D411yescharacteristic
323C2×C42C3×D411yescharacteristic
324C2×C42C3×D411yescharacteristic
325C2×C42C3×D411yescharacteristic
326C2×C42C3×D411yescharacteristic
327C2×C42C23×C311yescharacteristic
328C2×C42C23×C311yescharacteristic
329C4⋊C8—22no—
330C4⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8C23×C314yes—
333#97—22no—
334#97—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95C23×C314yes—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—41no—
343C23×C3C2×C22⋊C411yescharacteristic
344C23×C3C2×C22⋊C411yescharacteristic
345C23×C3C22×D411yescharacteristic
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—44no—
352C3×D4—44no—
353C3×D4—44no—
354C3×D4—42no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—44no—
377C2×C3×C4—44no—
378C2×C3×C4—44no—
379C2×C3×C4—44no—
380C2×C3×C4—44no—
381C2×C3×C4—44no—
382C2×C3×C4C2×C22⋊C411yescharacteristic
383C2×C3×C4C2×C22⋊C411yescharacteristic
384C2×C3×C4C2×C22⋊C411yescharacteristic
385C2×C3×C4C2×C22⋊C411yescharacteristic
386C2×C3×C4C2×C22⋊C411yescharacteristic
387C2×C3×C4C2×C22⋊C411yescharacteristic
388C2×C3×C4C2×C22⋊C411yescharacteristic
389C2×C3×C4C2×C22⋊C411yescharacteristic
390C2×C3×C4C2×C22⋊C411yescharacteristic
391C2×C3×C4C2×C22⋊C411yescharacteristic
392C2×C3×C4C23×C411yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C3×C8—44no—
395C24C2×C3×D411yescharacteristic
396#41—44no—
397#41—44no—
398#41—44no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C22×C3×C412yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C22×C3×C412yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4C3×C22⋊C411yescharacteristic
416C22×C4C3×C22⋊C411yescharacteristic
417C22×C4C3×C22⋊C411yescharacteristic
418C22×C4C3×C22⋊C411yescharacteristic
419C22×C4C3×C22⋊C411yescharacteristic
420C22×C4C3×C22⋊C411yescharacteristic
421C22×C4C3×C22⋊C411yescharacteristic
422C22×C4C3×C22⋊C411yescharacteristic
423C22×C4C22×C3×C411yescharacteristic
424C22×C4C22×C3×C411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C24×C311yescharacteristic
433C2×C8—24no—
434C2×C8—44no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—44no—
446C4⋊C4C22×C3×C412yes—
447C4⋊C4C22×C3×C412yes—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—44no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—44no—
462C42C3×C22⋊C411yescharacteristic
463C42C3×C22⋊C411yescharacteristic
464C42C3×C22⋊C411yescharacteristic
465C42C3×C22⋊C411yescharacteristic
466C42C3×C22⋊C411yescharacteristic
467C42C3×C22⋊C411yescharacteristic
468C42C3×C22⋊C411yescharacteristic
469C42C3×C22⋊C411yescharacteristic
470C42C22×C3×C411yescharacteristic
471C42C22×C3×C411yescharacteristic
472C42C22×C3×C411yescharacteristic
473C42C22×C3×C411yescharacteristic
474C42C2×C3×D411yescharacteristic
475C42C2×C3×D411yescharacteristic
476C42C2×C3×D411yescharacteristic
477C42C2×C3×D411yescharacteristic
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—44no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3#41012yes—
490C22×C3C22×C22⋊C411yescharacteristic
491C22×C3#57512yes—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—84no—
507C3×C4#42112yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23C2×C3×C22⋊C411yescharacteristic
513C23C2×C3×C22⋊C411yescharacteristic
514C23C22×C3×D411yescharacteristic
515Q8—42no—
516Q8—42no—
517Q8—42no—
518D4—42no—
519D4—42no—
520D4—44no—
521D4—44no—
522D4—44no—
523D4—42no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4C2×C3×C22⋊C411yescharacteristic
552C2×C4C2×C3×C22⋊C411yescharacteristic
553C2×C4C2×C3×C22⋊C411yescharacteristic
554C2×C4C2×C3×C22⋊C411yescharacteristic
555C2×C4C2×C3×C22⋊C411yescharacteristic
556C2×C4C2×C3×C22⋊C411yescharacteristic
557C2×C4C2×C3×C22⋊C411yescharacteristic
558C2×C4C2×C3×C22⋊C411yescharacteristic
559C2×C4C2×C3×C22⋊C411yescharacteristic
560C2×C4C2×C3×C22⋊C411yescharacteristic
561C2×C4C23×C3×C411yescharacteristic
562C2×C4C22×C3×D411yescharacteristic
563C8—44no—
564C2×C3—22no—
565C2×C3—22no—
566C2×C3—41no—
567C2×C3—82no—
568C2×C3C2×#41011yescharacteristic
569C2×C3#289611yescharacteristic
570C2×C3C2×#42111yescharacteristic
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22C3×#41012yes—
583C22C22×C3×C22⋊C411yescharacteristic
584C22C3×#57512yes—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—84no—
600C4C3×#42112yes—
601C3#1759111yescharacteristic, direct product factor, Sylow
602C2—22no—
603C2—22no—
604C2—41no—
605C2—82no—
606C2C2×C3×#41011yescharacteristic
607C2C3×#289611yescharacteristic
608C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3444C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#432C2semidirect
3. C3×#2283C2semidirect
4. C3×#1516C2semidirect
5. C3×#1512C2semidirect
6. C3×#1506C2semidirect
7. C3×#1505C2semidirect
8. #17591C3direct (N × H)
9. C3×#533C4semidirect
10. C3×#533C4semidirect
12. C3×#532C4semidirect
13. C3×#532C4semidirect
19. C3×#529C4semidirect
26. C3×C4×#41C4semidirect
45. C2×#432C2×C3semidirect
46. #2283C2×C3semidirect
47. #1516C2×C3semidirect
48. #1512C2×C3semidirect
49. #1506C2×C3semidirect
50. #1505C2×C3semidirect
136. #533C3×C4semidirect
137. #533C3×C4semidirect
139. #532C3×C4semidirect
140. #532C3×C4semidirect
146. #529C3×C4semidirect
153. C4×#41C3×C4semidirect
601. C3#17591direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34442
C2×C3×#4322
C3×#22832
C3×#15162
C3×#15122
C3×#15062
C3×#15052
#175913