Finite Group Databasebeta

← C2×C3×#3097 #10679120

Subgroups — C2×C3×#3097 #10679120

Order 768.

2170 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×#536C211yesmaximal, characteristic
2C2×C3×#520C211yesmaximal, characteristic
3C3×#3097C2116yesmaximal, direct product factor
4C2×C3×#484C212yesmaximal, complemented
5C2×C3×#485C212yesmaximal, complemented
6C2×C3×#485C212yesmaximal, complemented
7C2×C3×#483C212yesmaximal, complemented
8C2×C3×#428C211yesmaximal, characteristic, complemented
9C2×C3×#427C211yesmaximal, characteristic, complemented
10C2×C3×#419C211yesmaximal, characteristic, complemented
11C22×C3×#103C211yesmaximal, characteristic, complemented
12C2×C3×#408C211yesmaximal, characteristic
13C2×#3097C311yesmaximal, characteristic, direct product factor, Sylow
14C22×C3×#41C2211yescharacteristic
15C23×C3×Q8C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C23×C3×C8C2211yescharacteristic
18C3×#536—24no—
19C3×#536C2218yes—
20C2×C3×C4⋊Q8—21no—
21C2×C3×#124—21no—
22C2×C3×#123—22no—
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×C22⋊Q8C2212yes—
25C3×#520—24no—
26C3×#520C2218yes—
27C2×C3×C4×Q8—21no—
28C2×C3×C4×D4—22no—
29C2×C3×C4×D4—21no—
30C2×C3×#117C2211yescharacteristic
31C2×C3×#117C2212yes—
32C2×C3×#117C2211yescharacteristic
33C22×C3×C4⋊C4C2211yescharacteristic
34C22×C3×C4⋊C4C2212yes—
35C3×#485C22116yescomplemented
36C3×#485C22116yescomplemented
37C3×#484C22116yescomplemented
38C3×#483C22116yescomplemented
39C3×#428C2218yescomplemented
40C3×#427C2218yescomplemented
41C2×C3×#107C2211yescharacteristic
42C2×C3×#107C2211yescharacteristic
43C2×C3×#107C2212yes—
44C2×C3×#106C2211yescharacteristic
45C2×C3×#106C2211yescharacteristic
46C2×C3×#106C2212yes—
47C3×#419C2218yescomplemented
48C2×C3×#103C2212yes—
49C2×C3×#103C2212yes—
50C2×C3×#103C2212yes—
51C2×C3×#103C2212yes—
52C2×C3×#103C2218yescomplemented
53C3×#408C2218yes—
54C2×C3×C22⋊C8C2212yes—
55C2×C3×C22⋊C8C2212yes—
56C2×#536C2×C311yescharacteristic
57C2×#520C2×C311yescharacteristic
58#3097C2×C3116yesdirect product factor
59C2×#484C2×C312yescomplemented
60C2×#485C2×C312yescomplemented
61C2×#485C2×C312yescomplemented
62C2×#483C2×C312yescomplemented
63C2×#428C2×C311yescharacteristic, complemented
64C2×#427C2×C311yescharacteristic, complemented
65C2×#419C2×C311yescharacteristic, complemented
66C22×#103C2×C311yescharacteristic, complemented
67C2×#408C2×C311yescharacteristic
68C2×C3×#41—24no—
69C2×C3×#41—24no—
70C2×C3×#41D414yes—
71C2×C3×#41C2314yes—
72C22×C3×Q8—21no—
73C22×C3×Q8—22no—
74C22×C3×Q8—24no—
75C22×C3×Q8—24no—
76C22×C3×Q8D411yescharacteristic
77C22×C3×Q8C2312yes—
78C22×C3×Q8C2314yes—
79C22×C3×D4D412yes—
80C22×C3×D4D411yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4—22no—
83C23×C3×C4—21no—
84C23×C3×C4C2312yes—
85C23×C3×C4C2311yescharacteristic
86C2×C3×M16—24no—
87C2×C3×M16C2314yes—
88C22×C3×C8—24no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2314yes—
92C22×C3×C8C2312yes—
93C3×C4⋊Q8—28no—
94C3×#124—28no—
95C3×#123—216no—
96C3×C22⋊Q8—24no—
97C3×C22⋊Q8—24no—
98C3×C22⋊Q8C2318yes—
99C3×C22⋊Q8C2318yes—
100C3×C4×Q8—28no—
101C3×C4×D4—216no—
102C3×C4×D4—28no—
103C3×#117—24no—
104C3×#117—22no—
105C3×#117—22no—
106C3×#117C2314yes—
107C3×#117C2314yes—
108C3×#117C2318yes—
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4—24no—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C4⋊C4C2314yes—
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2318yes—
122C2×C3×C22⋊C4—22no—
123C2×C3×C22⋊C4—21no—
124C2×C3×C22⋊C4—21no—
125C2×C3×C22⋊C4—22no—
126C2×C3×C22⋊C4—22no—
127C2×C3×C42—22no—
128C2×C3×C42—21no—
129C2×C3×C42—21no—
130C3×#107C2314yes—
131C3×#107C2314yes—
132C3×#107C2318yes—
133C3×#106C2314yes—
134C3×#106C2314yes—
135C3×#106C2318yes—
136C3×#103C2318yes—
137C3×#103C2318yes—
138C3×#103C2318yes—
139C3×#103C2318yes—
140C3×C22⋊C8C2318yes—
141C3×C22⋊C8C2318yes—
142C22×#41C22×C311yescharacteristic
143C23×Q8C22×C311yescharacteristic
144C22×M16C22×C311yescharacteristic
145C23×C8C22×C311yescharacteristic
146#536—24no—
147#536C22×C318yes—
148C2×C4⋊Q8—21no—
149C2×#124—21no—
150C2×#123—22no—
151C2×C22⋊Q8C22×C312yes—
152C2×C22⋊Q8C22×C312yes—
153#520—24no—
154#520C22×C318yes—
155C2×C4×Q8—21no—
156C2×C4×D4—22no—
157C2×C4×D4—21no—
158C2×#117C22×C312yes—
159C2×#117C22×C311yescharacteristic
160C2×#117C22×C311yescharacteristic
161C22×C4⋊C4C22×C312yes—
162C22×C4⋊C4C22×C311yescharacteristic
163#485C22×C3116yescomplemented
164#485C22×C3116yescomplemented
165#484C22×C3116yescomplemented
166#483C22×C3116yescomplemented
167#428C22×C318yescomplemented
168#427C22×C318yescomplemented
169C2×#107C22×C311yescharacteristic
170C2×#107C22×C312yes—
171C2×#107C22×C311yescharacteristic
172C2×#106C22×C311yescharacteristic
173C2×#106C22×C312yes—
174C2×#106C22×C311yescharacteristic
175#419C22×C318yescomplemented
176C2×#103C22×C312yes—
177C2×#103C22×C312yes—
178C2×#103C22×C312yes—
179C2×#103C22×C312yes—
180C2×#103C22×C318yescomplemented
181#408C22×C318yes—
182C2×C22⋊C8C22×C312yes—
183C2×C22⋊C8C22×C312yes—
184C24×C3C2×D412yes—
185C24×C3C2×D411yescharacteristic
186C3×#41—216no—
187C3×#41—28no—
188C3×#41—28no—
189C2×C3×Q8—24no—
190C2×C3×Q8—22no—
191C2×C3×Q8—24no—
192C2×C3×Q8—24no—
193C2×C3×Q8—24no—
194C2×C3×Q8—28no—
195C2×C3×Q8—28no—
196C2×C3×Q8—44no—
197C2×C3×Q8—48no—
198C2×C3×Q8C2×D414yes—
199C2×C3×Q8C2414yes—
200C2×C3×D4—28no—
201C2×C3×D4—24no—
202C2×C3×D4C2×D418yes—
203C2×C3×D4C2×D414yes—
204C22×C3×C4—24no—
205C22×C3×C4—24no—
206C22×C3×C4—24no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—28no—
214C22×C3×C4—24no—
215C22×C3×C4—44no—
216C22×C3×C4—82no—
217C22×C3×C4—84no—
218C22×C3×C4C2×D414yes—
219C22×C3×C4C2×D412yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4114yes—
223C22×C3×C4#4114yes—
224C22×C3×C4C2414yes—
225C22×C3×C4C2412yes—
226C22×C3×C4C2411yescharacteristic
227C3×M16—28no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—28no—
231C2×C3×C8—24no—
232C2×C3×C8C2412yes—
233C2×C3×C8C2414yes—
234C2×C3×C8C2412yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—28no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—28no—
240C3×C4⋊C4—28no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2414yes—
245C3×C4⋊C4C2412yes—
246C3×C4⋊C4C2412yes—
247C3×C4⋊C4C2412yes—
248C3×C4⋊C4C2412yes—
249C3×C22⋊C4—28no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—28no—
253C3×C22⋊C4—28no—
254C3×C42—28no—
255C3×C42—24no—
256C3×C42—24no—
257C2×#41—24no—
258C2×#41—24no—
259C2×#41C3×D414yes—
260C2×#41C23×C314yes—
261C22×Q8—21no—
262C22×Q8—22no—
263C22×Q8—24no—
264C22×Q8—24no—
265C22×Q8C3×D411yescharacteristic
266C22×Q8C23×C312yes—
267C22×Q8C23×C314yes—
268C22×D4C3×D412yes—
269C22×D4C3×D411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—21no—
273C23×C4C23×C312yes—
274C23×C4C23×C311yescharacteristic
275C2×M16—24no—
276C2×M16C23×C314yes—
277C22×C8—24no—
278C22×C8C23×C311yescharacteristic
279C22×C8C23×C311yescharacteristic
280C22×C8C23×C314yes—
281C22×C8C23×C312yes—
282C4⋊Q8—28no—
283#124—28no—
284#123—216no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8C23×C318yes—
288C22⋊Q8C23×C318yes—
289C4×Q8—28no—
290C4×D4—216no—
291C4×D4—28no—
292#117—24no—
293#117—22no—
294#117—22no—
295#117C23×C314yes—
296#117C23×C318yes—
297#117C23×C314yes—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C23×C311yescharacteristic
305C2×C4⋊C4C23×C312yes—
306C2×C4⋊C4C23×C314yes—
307C2×C4⋊C4C23×C318yes—
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C4⋊C4C23×C311yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C42—22no—
317C2×C42—21no—
318C2×C42—21no—
319#107C23×C314yes—
320#107C23×C314yes—
321#107C23×C318yes—
322#106C23×C314yes—
323#106C23×C314yes—
324#106C23×C318yes—
325#103C23×C318yes—
326#103C23×C318yes—
327#103C23×C318yes—
328#103C23×C318yes—
329C22⋊C8C23×C318yes—
330C22⋊C8C23×C318yes—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—24no—
334C23×C3—24no—
335C23×C3—24no—
336C23×C3#12314yes—
337C23×C3C22×D414yes—
338C23×C3C22×D412yes—
339C23×C3C22×D411yescharacteristic
340C3×Q8—28no—
341C3×Q8—24no—
342C3×Q8—24no—
343C3×Q8—44no—
344C3×Q8—48no—
345C3×D4—216no—
346C3×D4—28no—
347C2×C3×C4—24no—
348C2×C3×C4—28no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—48no—
358C2×C3×C4—44no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—84no—
365C2×C3×C4—88no—
366C2×C3×C4#12314yes—
367C2×C3×C4#12318yes—
368C2×C3×C4C22×D414yes—
369C2×C3×C4C22×D412yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4112yes—
372C2×C3×C4C2×#4112yes—
373C2×C3×C4C2×#4114yes—
374C2×C3×C4C2×#4114yes—
375C2×C3×C4C2511yescharacteristic
376C3×C8—22no—
377C3×C8—22no—
378C3×C8—24no—
379C24C2×C3×D412yes—
380C24C2×C3×D411yescharacteristic
381#41—216no—
382#41—28no—
383#41—28no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8C2×C3×D414yes—
394C2×Q8C24×C314yes—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4C2×C3×D418yes—
398C2×D4C2×C3×D414yes—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—44no—
411C22×C4—82no—
412C22×C4—84no—
413C22×C4C2×C3×D412yes—
414C22×C4C2×C3×D414yes—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D412yes—
417C22×C4C3×#4114yes—
418C22×C4C3×#4114yes—
419C22×C4C24×C314yes—
420C22×C4C24×C311yescharacteristic
421C22×C4C24×C312yes—
422M16—28no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—28no—
426C2×C8—24no—
427C2×C8C24×C312yes—
428C2×C8C24×C314yes—
429C2×C8C24×C312yes—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4C24×C314yes—
440C4⋊C4C24×C312yes—
441C4⋊C4C24×C312yes—
442C4⋊C4C24×C312yes—
443C4⋊C4C24×C312yes—
444C22⋊C4—28no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C42—28no—
450C42—24no—
451C42—24no—
452C22×C3—28no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—24no—
458C22×C3—48no—
459C22×C3C2×#12314yes—
460C22×C3C2×#12312yes—
461C22×C3#57712yes—
462C22×C3#57812yes—
463C22×C3C23×D411yescharacteristic
464C3×C4—24no—
465C3×C4—44no—
466C3×C4—42no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4C2×#12312yes—
472C3×C4C2×#12314yes—
473C3×C4C22×#4111yescharacteristic
474C3×C4C22×#4111yescharacteristic
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23C3×#12314yes—
481C23C22×C3×D414yes—
482C23C22×C3×D412yes—
483C23C22×C3×D411yescharacteristic
484Q8—28no—
485Q8—24no—
486Q8—24no—
487Q8—44no—
488Q8—48no—
489D4—216no—
490D4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—84no—
509C2×C4—88no—
510C2×C4C3×#12314yes—
511C2×C4C3×#12318yes—
512C2×C4C22×C3×D414yes—
513C2×C4C22×C3×D412yes—
514C2×C4C22×C3×D411yescharacteristic
515C2×C4C2×C3×#4112yes—
516C2×C4C2×C3×#4112yes—
517C2×C4C2×C3×#4114yes—
518C2×C4C2×C3×#4114yes—
519C2×C4C25×C311yescharacteristic
520C8—22no—
521C8—22no—
522C8—24no—
523C2×C3—24no—
524C2×C3—44no—
525C2×C3#309714yesdirect product factor
526C2×C3C22×#12311yescharacteristic
527C2×C3C2×#57711yescharacteristic
528C2×C3C2×#57811yescharacteristic
529C22—22no—
530C22—22no—
531C22—28no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—48no—
536C22C2×C3×#12314yes—
537C22C2×C3×#12312yes—
538C22C3×#57712yes—
539C22C3×#57812yes—
540C22C23×C3×D411yescharacteristic
541C4—24no—
542C4—44no—
543C4—44no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4C2×C3×#12312yes—
549C4C2×C3×#12314yes—
550C4C22×C3×#4111yescharacteristic
551C4C22×C3×#4111yescharacteristic
552C3C2×#309711yescharacteristic, direct product factor, Sylow
553C2—24no—
554C2—44no—
555C2C3×#309714yesdirect product factor
556C2C22×C3×#12311yescharacteristic
557C2C2×C3×#57711yescharacteristic
558C2C2×C3×#57811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#3097C2direct (N × H)
4. C2×C3×#484C2semidirect
5. C2×C3×#485C2semidirect
6. C2×C3×#485C2semidirect
7. C2×C3×#483C2semidirect
8. C2×C3×#428C2semidirect
9. C2×C3×#427C2semidirect
10. C2×C3×#419C2semidirect
11. C22×C3×#103C2semidirect
13. C2×#3097C3direct (N × H)
35. C3×#485C22semidirect
36. C3×#485C22semidirect
37. C3×#484C22semidirect
38. C3×#483C22semidirect
39. C3×#428C22semidirect
40. C3×#427C22semidirect
47. C3×#419C22semidirect
52. C2×C3×#103C22semidirect
58. #3097C2×C3direct (N × H)
59. C2×#484C2×C3semidirect
60. C2×#485C2×C3semidirect
61. C2×#485C2×C3semidirect
62. C2×#483C2×C3semidirect
63. C2×#428C2×C3semidirect
64. C2×#427C2×C3semidirect
65. C2×#419C2×C3semidirect
66. C22×#103C2×C3semidirect
163. #485C22×C3semidirect
164. #485C22×C3semidirect
165. #484C22×C3semidirect
166. #483C22×C3semidirect
167. #428C22×C3semidirect
168. #427C22×C3semidirect
175. #419C22×C3semidirect
180. C2×#103C22×C3semidirect
525. C2×C3#3097direct (N × H)
552. C3C2×#3097direct (N × H)
555. C2C3×#3097direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#5202
C3×#30972
C2×C3×#4842
C2×C3×#4852
C2×C3×#4832
C2×C3×#4282
C2×C3×#4272
C2×C3×#4192
C22×C3×#1032
C2×C3×#4082
C2×#30973