Finite Group Databasebeta

← C2×C3×#3329 #10679756

Subgroups — C2×C3×#3329 #10679756

Order 768.

2146 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic
2C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
3C3×#3329C2116yesmaximal, direct product factor
4C2×C3×#487C211yesmaximal, characteristic, complemented
5C2×C3×#485C212yesmaximal, complemented
6C2×C3×#477C211yesmaximal, characteristic, complemented
7C2×C3×#468C212yesmaximal, complemented
8C2×C3×#461C211yesmaximal, characteristic, complemented
9C2×C3×#448C212yesmaximal, complemented
10C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
11C2×C3×#436C211yesmaximal, characteristic, complemented
12C2×C3×#419C212yesmaximal, complemented
13C2×#3329C311yesmaximal, characteristic, direct product factor, Sylow
14C23×C3×Q8C2212yes—
15C22×C3×SD16C2211yescharacteristic
16C22×C3×SD16C2212yes—
17C22×C3×M16C2212yes—
18C3×#555—24no—
19C3×#555C2218yes—
20C3×D4×Q8—24no—
21C3×D4×Q8C2218yescomplemented
22C2×C3×C4⋊Q8—21no—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×#124—22no—
26C2×C3×C22⋊Q8—22no—
27C2×C3×C22⋊Q8C2212yes—
28C2×C3×#121—21no—
29C2×C3×#121C2212yes—
30C2×C3×C4×Q8C2211yescharacteristic
31C2×C3×C4×D4—21no—
32C2×C3×C4×D4C2211yescharacteristic
33C2×C3×C4×D4C2211yescharacteristic
34C2×C3×#117—21no—
35C2×C3×#117C2212yes—
36C3×#487C2218yescomplemented
37C3×#485C22116yescomplemented
38C3×#477C2218yescomplemented
39C3×#468C22116yescomplemented
40C3×#461C2218yescomplemented
41C3×#448C22116yescomplemented
42C3×C4×SD16C2218yescomplemented
43C3×#436C2218yescomplemented
44C2×C3×#107C2212yes—
45C2×C3×#106C2211yescharacteristic
46C2×C3×C4⋊C8C2211yescharacteristic
47C3×#419C22116yescomplemented
48C2×C3×#103C2212yes—
49C2×C3×#103C2212yes—
50C2×C3×#103C2211yescharacteristic
51C2×C3×#103C2212yes—
52C2×C3×#103C2212yes—
53C2×C3×#102C2211yescharacteristic
54C2×C3×C22⋊C8C2212yes—
55C2×C3×C4×C8C2211yescharacteristic
56C2×#555C2×C311yescharacteristic
57C2×D4×Q8C2×C311yescharacteristic, complemented
58#3329C2×C3116yesdirect product factor
59C2×#487C2×C311yescharacteristic, complemented
60C2×#485C2×C312yescomplemented
61C2×#477C2×C311yescharacteristic, complemented
62C2×#468C2×C312yescomplemented
63C2×#461C2×C311yescharacteristic, complemented
64C2×#448C2×C312yescomplemented
65C2×C4×SD16C2×C311yescharacteristic, complemented
66C2×#436C2×C311yescharacteristic, complemented
67C2×#419C2×C312yescomplemented
68C22×C3×Q8—22no—
69C22×C3×Q8—24no—
70C22×C3×Q8—21no—
71C22×C3×Q8—28no—
72C22×C3×Q8—44no—
73C22×C3×Q8C2312yes—
74C22×C3×Q8C2318yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4—22no—
77C22×C3×D4D411yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—22no—
80C23×C3×C4—21no—
81C23×C3×C4C2312yes—
82C2×C3×SD16—24no—
83C2×C3×SD16—44no—
84C2×C3×SD16C2314yes—
85C2×C3×SD16C2318yes—
86C2×C3×M16—44no—
87C2×C3×M16C2318yes—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2312yes—
90C22×C3×C8C2311yescharacteristic
91C3×C4⋊Q8—28no—
92C3×C4⋊Q8—22no—
93C3×C4⋊Q8—22no—
94C3×C4⋊Q8C2314yes—
95C3×C4⋊Q8C2314yes—
96C3×#124—216no—
97C3×C22⋊Q8—24no—
98C3×C22⋊Q8—216no—
99C3×C22⋊Q8C2318yes—
100C3×#121—28no—
101C3×#121—24no—
102C3×#121C2318yes—
103C3×C4×Q8—22no—
104C3×C4×Q8C2314yes—
105C3×C4×D4—28no—
106C3×C4×D4—22no—
107C3×C4×D4D414yes—
108C3×C4×D4C2314yes—
109C3×C4×D4C2314yes—
110C3×#117—24no—
111C3×#117—28no—
112C3×#117C2318yes—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—21no—
115C2×C3×C4⋊C4—21no—
116C2×C3×C4⋊C4—21no—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4D411yescharacteristic
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C22⋊C4—22no—
123C2×C3×C22⋊C4—22no—
124C2×C3×C22⋊C4—22no—
125C2×C3×C22⋊C4—21no—
126C2×C3×C22⋊C4D412yes—
127C2×C3×C42—22no—
128C2×C3×C42—21no—
129C2×C3×C42C2311yescharacteristic
130C3×#107C2318yes—
131C3×#106C2314yes—
132C3×C4⋊C8C2314yes—
133C3×#103C2314yes—
134C3×#103C2318yes—
135C3×#103C2318yes—
136C3×#103C2318yes—
137C3×#103C2318yes—
138C3×#102C2314yes—
139C3×C22⋊C8C2318yes—
140C3×C4×C8C2314yes—
141C23×Q8C22×C312yes—
142C22×SD16C22×C311yescharacteristic
143C22×SD16C22×C312yes—
144C22×M16C22×C312yes—
145#555—24no—
146#555C22×C318yes—
147D4×Q8—24no—
148D4×Q8C22×C318yescomplemented
149C2×C4⋊Q8—21no—
150C2×C4⋊Q8C22×C311yescharacteristic
151C2×C4⋊Q8C22×C311yescharacteristic
152C2×#124—22no—
153C2×C22⋊Q8—22no—
154C2×C22⋊Q8C22×C312yes—
155C2×#121—21no—
156C2×#121C22×C312yes—
157C2×C4×Q8C22×C311yescharacteristic
158C2×C4×D4—21no—
159C2×C4×D4C22×C311yescharacteristic
160C2×C4×D4C22×C311yescharacteristic
161C2×#117—21no—
162C2×#117C22×C312yes—
163#487C22×C318yescomplemented
164#485C22×C3116yescomplemented
165#477C22×C318yescomplemented
166#468C22×C3116yescomplemented
167#461C22×C318yescomplemented
168#448C22×C3116yescomplemented
169C4×SD16C22×C318yescomplemented
170#436C22×C318yescomplemented
171C2×#107C22×C312yes—
172C2×#106C22×C311yescharacteristic
173C2×C4⋊C8C22×C311yescharacteristic
174#419C22×C3116yescomplemented
175C2×#103C22×C312yes—
176C2×#103C22×C312yes—
177C2×#103C22×C312yes—
178C2×#103C22×C312yes—
179C2×#103C22×C311yescharacteristic
180C2×#102C22×C311yescharacteristic
181C2×C22⋊C8C22×C312yes—
182C2×C4×C8C22×C311yescharacteristic
183C24×C3—21no—
184C24×C3C2×D412yes—
185C2×C3×Q8—21no—
186C2×C3×Q8—28no—
187C2×C3×Q8—22no—
188C2×C3×Q8—24no—
189C2×C3×Q8—22no—
190C2×C3×Q8—216no—
191C2×C3×Q8—24no—
192C2×C3×Q8—24no—
193C2×C3×Q8—48no—
194C2×C3×Q8—48no—
195C2×C3×Q8—416no—
196C2×C3×Q8—82no—
197C2×C3×Q8#4114yes—
198C2×C3×Q8C2414yes—
199C2×C3×Q8C2412yes—
200C2×C3×D4—28no—
201C2×C3×D4—24no—
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4—84no—
206C2×C3×D4C2×D414yes—
207C2×C3×D4C2412yes—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—28no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—24no—
217C22×C3×C4—48no—
218C22×C3×C4—44no—
219C22×C3×C4C2×D414yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2414yes—
224C22×C3×C4C2411yescharacteristic
225C3×SD16—28no—
226C3×SD16—48no—
227C3×M16—48no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—24no—
231C2×C3×C8C2412yes—
232C2×C3×C8C2412yes—
233C2×C3×C8C2414yes—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—28no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D414yes—
243C3×C4⋊C4C2412yes—
244C3×C4⋊C4C2412yes—
245C3×C4⋊C4C2414yes—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—28no—
248C3×C22⋊C4—28no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4C2×D418yes—
251C3×C42—28no—
252C3×C42—24no—
253C3×C42C2×D412yes—
254C3×C42C2412yes—
255C22×Q8—22no—
256C22×Q8—24no—
257C22×Q8—21no—
258C22×Q8—28no—
259C22×Q8—44no—
260C22×Q8C23×C312yes—
261C22×Q8C23×C318yes—
262C22×Q8C23×C311yescharacteristic
263C22×D4—22no—
264C22×D4C3×D411yescharacteristic
265C22×D4C23×C311yescharacteristic
266C23×C4—22no—
267C23×C4—21no—
268C23×C4C23×C312yes—
269C2×SD16—24no—
270C2×SD16—44no—
271C2×SD16C23×C314yes—
272C2×SD16C23×C318yes—
273C2×M16—44no—
274C2×M16C23×C318yes—
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C312yes—
277C22×C8C23×C311yescharacteristic
278C4⋊Q8—28no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8C23×C314yes—
282C4⋊Q8C23×C314yes—
283#124—216no—
284C22⋊Q8—24no—
285C22⋊Q8—216no—
286C22⋊Q8C23×C318yes—
287#121—28no—
288#121—24no—
289#121C23×C318yes—
290C4×Q8—22no—
291C4×Q8C23×C314yes—
292C4×D4—28no—
293C4×D4—22no—
294C4×D4C3×D414yes—
295C4×D4C23×C314yes—
296C4×D4C23×C314yes—
297#117—24no—
298#117—28no—
299#117C23×C318yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C3×D411yescharacteristic
306C2×C4⋊C4C23×C311yescharacteristic
307C2×C4⋊C4C23×C312yes—
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4C3×D412yes—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42C23×C311yescharacteristic
317#107C23×C318yes—
318#106C23×C314yes—
319C4⋊C8C23×C314yes—
320#103C23×C314yes—
321#103C23×C318yes—
322#103C23×C318yes—
323#103C23×C318yes—
324#103C23×C318yes—
325#102C23×C314yes—
326C22⋊C8C23×C318yes—
327C4×C8C23×C314yes—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—24no—
331C23×C3—22no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3C22×D414yes—
336C23×C3C22×D411yescharacteristic
337C3×Q8—24no—
338C3×Q8—22no—
339C3×Q8—24no—
340C3×Q8—24no—
341C3×Q8—416no—
342C3×Q8—44no—
343C3×Q8—84no—
344C3×Q8C2×#4114yes—
345C3×D4—28no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—42no—
349C3×D4—88no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—28no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—416no—
362C2×C3×C4—44no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—44no—
366C2×C3×C4—48no—
367C2×C3×C4#13714yes—
368C2×C3×C4C22×D414yes—
369C2×C3×C4C22×D412yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D412yes—
372C2×C3×C4C2×#4112yes—
373C2×C3×C4H(5,𝔽2)12yes—
374C2×C3×C4C2511yescharacteristic
375C3×C8—22no—
376C3×C8—22no—
377C3×C8—24no—
378C24—21no—
379C24C2×C3×D412yes—
380C2×Q8—21no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—216no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—416no—
391C2×Q8—82no—
392C2×Q8C3×#4114yes—
393C2×Q8C24×C314yes—
394C2×Q8C24×C312yes—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—84no—
401C2×D4C2×C3×D414yes—
402C2×D4C24×C312yes—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D414yes—
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C24×C314yes—
419C22×C4C24×C311yescharacteristic
420SD16—28no—
421SD16—48no—
422M16—48no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8C24×C312yes—
427C2×C8C24×C312yes—
428C2×C8C24×C314yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—28no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4C2×C3×D414yes—
438C4⋊C4C24×C312yes—
439C4⋊C4C24×C312yes—
440C4⋊C4C24×C314yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—24no—
445C22⋊C4C2×C3×D418yes—
446C42—28no—
447C42—24no—
448C42C2×C3×D412yes—
449C42C24×C312yes—
450C22×C3—24no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3—48no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—84no—
458C22×C3#54612yes—
459C22×C3C2×#13712yes—
460C22×C3#57712yes—
461C22×C3C23×D411yescharacteristic
462C3×C4—24no—
463C3×C4—24no—
464C3×C4—22no—
465C3×C4—44no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4#54612yes—
470C3×C4C2×#13714yes—
471C3×C4C22×#4111yescharacteristic
472C3×C4C2×H(5,𝔽2)11yescharacteristic
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23C22×C3×D414yes—
481C23C22×C3×D411yescharacteristic
482Q8—22no—
483Q8—24no—
484Q8—24no—
485Q8—24no—
486Q8—416no—
487Q8—44no—
488Q8—84no—
489Q8C2×C3×#4114yes—
490D4—28no—
491D4—22no—
492D4—22no—
493D4—42no—
494D4—88no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—416no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4C3×#13714yes—
513C2×C4C22×C3×D414yes—
514C2×C4C22×C3×D412yes—
515C2×C4C22×C3×D412yes—
516C2×C4C22×C3×D411yescharacteristic
517C2×C4C2×C3×#4112yes—
518C2×C4C3×H(5,𝔽2)12yes—
519C2×C4C25×C311yescharacteristic
520C8—22no—
521C8—22no—
522C8—24no—
523C2×C3—44no—
524C2×C3—82no—
525C2×C3#332914yesdirect product factor
526C2×C3C2×#54611yescharacteristic
527C2×C3C22×#13711yescharacteristic
528C2×C3C2×#57711yescharacteristic
529C22—24no—
530C22—24no—
531C22—24no—
532C22—48no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22C3×#54612yes—
538C22C2×C3×#13712yes—
539C22C3×#57712yes—
540C22C23×C3×D411yescharacteristic
541C4—24no—
542C4—24no—
543C4—22no—
544C4—44no—
545C4—42no—
546C4—44no—
547C4—42no—
548C4C3×#54612yes—
549C4C2×C3×#13714yes—
550C4C22×C3×#4111yescharacteristic
551C4C2×C3×H(5,𝔽2)11yescharacteristic
552C3C2×#332911yescharacteristic, direct product factor, Sylow
553C2—44no—
554C2—82no—
555C2C3×#332914yesdirect product factor
556C2C2×C3×#54611yescharacteristic
557C2C22×C3×#13711yescharacteristic
558C2C2×C3×#57711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#555C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×D4×Q8C2semidirect
3. C3×#3329C2direct (N × H)
4. C2×C3×#487C2semidirect
5. C2×C3×#485C2semidirect
6. C2×C3×#477C2semidirect
7. C2×C3×#468C2semidirect
8. C2×C3×#461C2semidirect
9. C2×C3×#448C2semidirect
10. C2×C3×C4×SD16C2semidirect
11. C2×C3×#436C2semidirect
12. C2×C3×#419C2semidirect
13. C2×#3329C3direct (N × H)
21. C3×D4×Q8C22semidirect
36. C3×#487C22semidirect
37. C3×#485C22semidirect
38. C3×#477C22semidirect
39. C3×#468C22semidirect
40. C3×#461C22semidirect
41. C3×#448C22semidirect
42. C3×C4×SD16C22semidirect
43. C3×#436C22semidirect
47. C3×#419C22semidirect
57. C2×D4×Q8C2×C3semidirect
58. #3329C2×C3direct (N × H)
59. C2×#487C2×C3semidirect
60. C2×#485C2×C3semidirect
61. C2×#477C2×C3semidirect
62. C2×#468C2×C3semidirect
63. C2×#461C2×C3semidirect
64. C2×#448C2×C3semidirect
65. C2×C4×SD16C2×C3semidirect
66. C2×#436C2×C3semidirect
67. C2×#419C2×C3semidirect
148. D4×Q8C22×C3semidirect
163. #487C22×C3semidirect
164. #485C22×C3semidirect
165. #477C22×C3semidirect
166. #468C22×C3semidirect
167. #461C22×C3semidirect
168. #448C22×C3semidirect
169. C4×SD16C22×C3semidirect
170. #436C22×C3semidirect
174. #419C22×C3semidirect
525. C2×C3#3329direct (N × H)
552. C3C2×#3329direct (N × H)
555. C2C3×#3329direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×C3×D4×Q82
C3×#33292
C2×C3×#4872
C2×C3×#4852
C2×C3×#4772
C2×C3×#4682
C2×C3×#4612
C2×C3×#4482
C2×C3×C4×SD162
C2×C3×#4362
C2×C3×#4192
C2×#33293