Finite Group Databasebeta

← C3×#32964 #10651571

Subgroups — C3×#32964 #10651571

Order 768.

1206 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×#475C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic
5C3×#1865C212yesmaximal, complemented
6C3×#1863C212yesmaximal, complemented
7C3×#1854C212yesmaximal
8C3×#1805C211yesmaximal, characteristic, complemented
9C3×#1805C211yesmaximal, characteristic, complemented
10C3×#1798C211yesmaximal, characteristic
11C3×#1787C212yesmaximal
12#32964C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C3×#519—24no—
16C3×#519—24no—
17C2×C3×C4×Q8—21no—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×#117—21no—
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#477—24no—
28C3×#475—24no—
29C2×C3×#107C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yes—
43C3×#382C2211yescharacteristic
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#519C2×C311yescharacteristic
49C2×#477C2×C311yescharacteristic, complemented
50C2×#475C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic
52#1865C2×C312yescomplemented
53#1863C2×C312yescomplemented
54#1854C2×C312yes—
55#1805C2×C311yescharacteristic, complemented
56#1805C2×C311yescharacteristic, complemented
57#1798C2×C311yescharacteristic
58#1787C2×C312yes—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—21no—
64C22×C3×Q8—21no—
65C22×C3×Q8—22no—
66C22×C3×Q8Q814yescomplemented
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×C4×Q8—28no—
76C3×C4×Q8—28no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D414yes—
84C3×C4×Q8D414yes—
85C3×#117—28no—
86C3×#117—28no—
87C3×#117—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—24no—
115C3×#106—24no—
116C3×C4⋊C8—22no—
117C3×C4⋊C8—22no—
118C3×#103—24no—
119C3×#103—24no—
120C3×#103—24no—
121C3×#103—24no—
122C3×C22⋊C8—24no—
123C3×#95—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—22no—
127C23×Q8C22×C311yescharacteristic
128C2×C4⋊Q8C22×C311yescharacteristic
129#519—24no—
130#519—24no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×Q8C22×C311yescharacteristic
136C2×C4×Q8C22×C311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C2×#117C22×C311yescharacteristic
140C2×#117C22×C311yescharacteristic
141#477—24no—
142#475—24no—
143C2×#107C22×C312yes—
144C2×#106C22×C312yes—
145C2×C4⋊C8C22×C311yescharacteristic
146C2×C4⋊C8C22×C311yescharacteristic
147C2×#103C22×C312yes—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×C22⋊C8C22×C312yes—
152#400—21no—
153#399—21no—
154#395—21no—
155#393—21no—
156#389C22×C312yes—
157#382C22×C311yescharacteristic
158#380C22×C312yes—
159#340C22×C312yes—
160#340C22×C312yes—
161#336C22×C312yes—
162C24×C3C2×D411yescharacteristic
163C2×C3×Q8—24no—
164C2×C3×Q8—28no—
165C2×C3×Q8—28no—
166C2×C3×Q8—24no—
167C2×C3×Q8—28no—
168C2×C3×Q8—28no—
169C2×C3×Q8—24no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—42no—
177C2×C3×Q8—82no—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8C2×Q812yes—
181C2×C3×Q8C2×Q812yes—
182C2×C3×Q8#4112yes—
183C2×C3×Q8#4112yes—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×Q812yes—
203C22×C3×C4#4112yes—
204C22×C3×C4C2411yescharacteristic
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—82no—
225C3×C4⋊C4C2×D412yes—
226C3×C4⋊C4C2×D412yes—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—82no—
231C3×C42—24no—
232C3×C42—24no—
233C3×C42—24no—
234C3×C42—24no—
235C3×C42—24no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—22no—
247C22×Q8C3×Q814yescomplemented
248C22×Q8C23×C311yescharacteristic
249C22×Q8C23×C311yescharacteristic
250C23×C4—21no—
251C23×C4C23×C311yescharacteristic
252C22×C8C23×C312yes—
253C22×C8C23×C312yes—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4×Q8—28no—
257C4×Q8—28no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8C3×D414yes—
265C4×Q8C3×D414yes—
266#117—28no—
267#117—28no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C3×D411yescharacteristic
279C2×C4⋊C4C3×D411yescharacteristic
280C2×C4⋊C4C23×C312yes—
281C2×C4⋊C4C23×C312yes—
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4C3×D411yescharacteristic
287C2×C22⋊C4C3×D411yescharacteristic
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#107—24no—
296#106—24no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299#103—24no—
300#103—24no—
301#103—24no—
302#103—24no—
303C22⋊C8—24no—
304#95—22no—
305#95—21no—
306#95—21no—
307#95—22no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3C22⋊Q812yes—
313C23×C3C22×D411yescharacteristic
314C3×Q8—216no—
315C3×Q8—22no—
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—22no—
321C3×Q8—84no—
322C3×Q8C22⋊Q814yes—
323C3×Q8C22⋊Q814yes—
324C2×C3×C4—216no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4#12012yes—
354C2×C3×C4#12012yes—
355C2×C3×C4C22⋊Q812yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×Q811yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C3×C8—42no—
363C3×C8—42no—
364C24C2×C3×D411yescharacteristic
365C2×Q8—24no—
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—24no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—24no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8—82no—
380C2×Q8C2×C3×D412yes—
381C2×Q8C2×C3×D412yes—
382C2×Q8C2×C3×Q812yes—
383C2×Q8C2×C3×Q812yes—
384C2×Q8C3×#4112yes—
385C2×Q8C3×#4112yes—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×Q812yes—
405C22×C4C3×#4112yes—
406C22×C4C24×C311yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—82no—
427C4⋊C4C2×C3×D412yes—
428C4⋊C4C2×C3×D412yes—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—82no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—21no—
439C42—21no—
440C42C2×C3×D412yes—
441C42C2×C3×D412yes—
442C22×C3—24no—
443C22×C3—22no—
444C22×C3—42no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3C2×#12011yescharacteristic
448C22×C3C2×C22⋊Q811yescharacteristic
449C22×C3#52511yescharacteristic
450C22×C3#57712yes—
451C22×C3#57812yes—
452C3×C4—24no—
453C3×C4—24no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—41no—
458C3×C4—42no—
459C3×C4—41no—
460C3×C4—42no—
461C3×C4—82no—
462C3×C4—82no—
463C3×C4C2×C22⋊Q811yescharacteristic
464C3×C4C2×C22⋊Q811yescharacteristic
465C3×C4#52511yescharacteristic
466C3×C4#52511yescharacteristic
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23C3×C22⋊Q812yes—
472C23C22×C3×D411yescharacteristic
473Q8—216no—
474Q8—22no—
475Q8—22no—
476Q8—22no—
477Q8—22no—
478Q8—22no—
479Q8—22no—
480Q8—84no—
481Q8C3×C22⋊Q814yes—
482Q8C3×C22⋊Q814yes—
483C2×C4—216no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4C3×#12012yes—
513C2×C4C3×#12012yes—
514C2×C4C3×C22⋊Q812yes—
515C2×C4C22×C3×D411yescharacteristic
516C2×C4C22×C3×D411yescharacteristic
517C2×C4C22×C3×Q811yescharacteristic
518C2×C4C2×C3×#4111yescharacteristic
519C2×C4C2×C3×#4111yescharacteristic
520C2×C4C2×C3×#4111yescharacteristic
521C8—42no—
522C8—42no—
523C2×C3—42no—
524C2×C3#284811yescharacteristic
525C2×C3#301011yescharacteristic
526C2×C3#301111yescharacteristic
527C2×C3#308212yes—
528C2×C3#312112yes—
529C22—24no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—42no—
534C22C2×C3×#12011yescharacteristic
535C22C2×C3×C22⋊Q811yescharacteristic
536C22C3×#52511yescharacteristic
537C22C3×#57712yes—
538C22C3×#57812yes—
539C4—22no—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—22no—
544C4—41no—
545C4—42no—
546C4—41no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4C2×C3×C22⋊Q811yescharacteristic
551C4C2×C3×C22⋊Q811yescharacteristic
552C4C3×#52511yescharacteristic
553C4C3×#52511yescharacteristic
554C3#3296411yescharacteristic, direct product factor, Sylow
555C2—42no—
556C2C3×#284811yescharacteristic
557C2C3×#301011yescharacteristic
558C2C3×#301111yescharacteristic
559C2C3×#308212yes—
560C2C3×#312112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#477C2semidirect
3. C2×C3×#475C2semidirect
5. C3×#1865C2semidirect
6. C3×#1863C2semidirect
8. C3×#1805C2semidirect
9. C3×#1805C2semidirect
12. #32964C3direct (N × H)
49. C2×#477C2×C3semidirect
50. C2×#475C2×C3semidirect
52. #1865C2×C3semidirect
53. #1863C2×C3semidirect
55. #1805C2×C3semidirect
56. #1805C2×C3semidirect
66. C22×C3×Q8Q8semidirect
247. C22×Q8C3×Q8semidirect
554. C3#32964direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C2×C3×#4772
C2×C3×#4752
C3×#26282
C3×#18652
C3×#18632
C3×#18542
C3×#18052
C3×#17982
C3×#17872
#329643