Finite Group Databasebeta

← C2×C3×#2709 #10670738

Subgroups — C2×C3×#2709 #10670738

Order 768.

3542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#127C211yesmaximal, characteristic
2C22×C3×#123C211yesmaximal, characteristic, complemented
3C3×#2709C2116yesmaximal, direct product factor
4C2×C3×#394C212yesmaximal, complemented
5C2×C3×#394C212yesmaximal, complemented
6C2×C3×#394C212yesmaximal, complemented
7C2×C3×#387C212yesmaximal, complemented
8C2×C3×#386C212yesmaximal, complemented
9C2×C3×#386C212yesmaximal, complemented
10C2×C3×#386C211yesmaximal, characteristic, complemented
11C2×#2709C311yesmaximal, characteristic, direct product factor, Sylow
12C23×C3×D4C2212yes—
13C23×C3×D4C2212yes—
14C23×C3×D4C2211yescharacteristic
15C23×C3×D4C2211yescharacteristic
16C24×C3×C4C2211yescharacteristic
17C24×C3×C4C2211yescharacteristic
18C2×C3×#127—28no—
19C2×C3×#127C2218yes—
20C2×C3×#123—28no—
21C2×C3×#123C2218yescomplemented
22C22×C3×C4⋊C4C2212yes—
23C22×C3×C4⋊C4C2212yes—
24C22×C3×C22⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2212yes—
26C22×C3×C22⋊C4C2212yes—
27C22×C3×C22⋊C4C2212yes—
28C22×C3×C22⋊C4C2212yes—
29C22×C3×C22⋊C4C2211yescharacteristic
30C22×C3×C22⋊C4C2212yes—
31C22×C3×C22⋊C4C2212yes—
32C22×C3×C42C2211yescharacteristic
33C3×#394C22116yescomplemented
34C3×#394C22116yescomplemented
35C3×#394C22116yescomplemented
36C3×#387C22116yescomplemented
37C3×#386C22116yescomplemented
38C3×#386C22116yescomplemented
39C3×#386C2218yescomplemented
40C2×C3×#95C2212yes—
41C2×C3×#95C2212yes—
42C2×C3×#95C2212yes—
43C2×C3×#95C2212yes—
44C22×#127C2×C311yescharacteristic
45C22×#123C2×C311yescharacteristic, complemented
46#2709C2×C3116yesdirect product factor
47C2×#394C2×C312yescomplemented
48C2×#394C2×C312yescomplemented
49C2×#394C2×C312yescomplemented
50C2×#387C2×C312yescomplemented
51C2×#386C2×C312yescomplemented
52C2×#386C2×C312yescomplemented
53C2×#386C2×C311yescharacteristic, complemented
54C25×C3C2312yes—
55C25×C3C2311yescharacteristic
56C25×C3C2311yescharacteristic
57C22×C3×D4—28no—
58C22×C3×D4—28no—
59C22×C3×D4—24no—
60C22×C3×D4—28no—
61C22×C3×D4—28no—
62C22×C3×D4—42no—
63C22×C3×D4—42no—
64C22×C3×D4—44no—
65C22×C3×D4—44no—
66C22×C3×D4—42no—
67C22×C3×D4—42no—
68C22×C3×D4D418yes—
69C22×C3×D4C2318yes—
70C22×C3×D4C2318yes—
71C22×C3×D4C2314yes—
72C22×C3×D4C2314yes—
73C23×C3×C4—24no—
74C23×C3×C4—42no—
75C23×C3×C4—42no—
76C23×C3×C4—42no—
77C23×C3×C4—44no—
78C23×C3×C4C2312yes—
79C23×C3×C4C2312yes—
80C23×C3×C4C2312yes—
81C23×C3×C4C2312yes—
82C23×C3×C4C2311yescharacteristic
83C23×C3×C4C2312yes—
84C23×C3×C4C2314yes—
85C23×C3×C4C2314yes—
86C3×#127—232no—
87C3×#123—232no—
88C2×C3×C4⋊C4—28no—
89C2×C3×C4⋊C4—28no—
90C2×C3×C4⋊C4C2318yes—
91C2×C3×C4⋊C4C2318yes—
92C2×C3×C22⋊C4—24no—
93C2×C3×C22⋊C4—28no—
94C2×C3×C22⋊C4—42no—
95C2×C3×C22⋊C4—44no—
96C2×C3×C22⋊C4—44no—
97C2×C3×C22⋊C4—44no—
98C2×C3×C22⋊C4—44no—
99C2×C3×C22⋊C4—44no—
100C2×C3×C22⋊C4C2318yes—
101C2×C3×C22⋊C4C2314yes—
102C2×C3×C22⋊C4C2318yes—
103C2×C3×C22⋊C4C2318yes—
104C2×C3×C22⋊C4C2318yes—
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2318yes—
107C2×C3×C22⋊C4C2318yes—
108C2×C3×C42—24no—
109C2×C3×C42C2314yes—
110C3×#95C2318yes—
111C3×#95C2318yes—
112C3×#95C2318yes—
113C3×#95C2318yes—
114C23×D4C22×C312yes—
115C23×D4C22×C312yes—
116C23×D4C22×C311yescharacteristic
117C23×D4C22×C311yescharacteristic
118C24×C4C22×C311yescharacteristic
119C24×C4C22×C311yescharacteristic
120C2×#127—28no—
121C2×#127C22×C318yes—
122C2×#123—28no—
123C2×#123C22×C318yescomplemented
124C22×C4⋊C4C22×C312yes—
125C22×C4⋊C4C22×C312yes—
126C22×C22⋊C4C22×C311yescharacteristic
127C22×C22⋊C4C22×C312yes—
128C22×C22⋊C4C22×C312yes—
129C22×C22⋊C4C22×C312yes—
130C22×C22⋊C4C22×C311yescharacteristic
131C22×C22⋊C4C22×C312yes—
132C22×C22⋊C4C22×C312yes—
133C22×C22⋊C4C22×C312yes—
134C22×C42C22×C311yescharacteristic
135#394C22×C3116yescomplemented
136#394C22×C3116yescomplemented
137#394C22×C3116yescomplemented
138#387C22×C3116yescomplemented
139#386C22×C3116yescomplemented
140#386C22×C3116yescomplemented
141#386C22×C318yescomplemented
142C2×#95C22×C312yes—
143C2×#95C22×C312yes—
144C2×#95C22×C312yes—
145C2×#95C22×C312yes—
146C24×C3—24no—
147C24×C3—24no—
148C24×C3—22no—
149C24×C3—24no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—24no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—24no—
157C24×C3—24no—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—24no—
161C24×C3C2×D414yes—
162C24×C3C2×D414yes—
163C24×C3C2414yes—
164C24×C3C2412yes—
165C24×C3C2412yes—
166C24×C3C2411yescharacteristic
167C2×C3×D4—216no—
168C2×C3×D4—216no—
169C2×C3×D4—28no—
170C2×C3×D4—216no—
171C2×C3×D4—216no—
172C2×C3×D4—416no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—48no—
176C2×C3×D4—48no—
177C2×C3×D4—44no—
178C2×C3×D4—44no—
179C2×C3×D4—432no—
180C2×C3×D4—432no—
181C2×C3×D4—416no—
182C2×C3×D4C2×D4116yes—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—28no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—44no—
201C22×C3×C4—44no—
202C22×C3×C4—44no—
203C22×C3×C4—48no—
204C22×C3×C4—416no—
205C22×C3×C4—416no—
206C22×C3×C4C2×D414yes—
207C22×C3×C4#4118yes—
208C22×C3×C4C2414yes—
209C22×C3×C4C2414yes—
210C22×C3×C4C2414yes—
211C22×C3×C4C2414yes—
212C22×C3×C4C2412yes—
213C22×C3×C4C2414yes—
214C3×C4⋊C4—216no—
215C3×C4⋊C4—216no—
216C3×C22⋊C4—28no—
217C3×C22⋊C4—216no—
218C3×C22⋊C4—44no—
219C3×C22⋊C4—48no—
220C3×C22⋊C4—48no—
221C3×C22⋊C4—48no—
222C3×C22⋊C4—48no—
223C3×C22⋊C4—48no—
224C3×C42—28no—
225C25C23×C312yes—
226C25C23×C311yescharacteristic
227C25C23×C311yescharacteristic
228C22×D4—28no—
229C22×D4—28no—
230C22×D4—24no—
231C22×D4—28no—
232C22×D4—28no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4C3×D418yes—
240C22×D4C23×C318yes—
241C22×D4C23×C318yes—
242C22×D4C23×C314yes—
243C22×D4C23×C314yes—
244C23×C4—24no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—44no—
249C23×C4C23×C312yes—
250C23×C4C23×C312yes—
251C23×C4C23×C312yes—
252C23×C4C23×C312yes—
253C23×C4C23×C311yescharacteristic
254C23×C4C23×C312yes—
255C23×C4C23×C314yes—
256C23×C4C23×C314yes—
257#127—232no—
258#123—232no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4C23×C318yes—
262C2×C4⋊C4C23×C318yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C23×C318yes—
272C2×C22⋊C4C23×C314yes—
273C2×C22⋊C4C23×C318yes—
274C2×C22⋊C4C23×C318yes—
275C2×C22⋊C4C23×C318yes—
276C2×C22⋊C4C23×C314yes—
277C2×C22⋊C4C23×C318yes—
278C2×C22⋊C4C23×C318yes—
279C2×C42—24no—
280C2×C42C23×C314yes—
281#95C23×C318yes—
282#95C23×C318yes—
283#95C23×C318yes—
284#95C23×C318yes—
285C23×C3—28no—
286C23×C3—28no—
287C23×C3—216no—
288C23×C3—216no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—22no—
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—44no—
309C23×C3—44no—
310C23×C3—48no—
311C23×C3—48no—
312C23×C3—48no—
313C23×C3—48no—
314C23×C3—48no—
315C23×C3—48no—
316C23×C3—48no—
317C23×C3C22×D414yes—
318C23×C3C22×D414yes—
319C23×C3C22×D412yes—
320C23×C3C2×#4114yes—
321C23×C3H(5,𝔽2)12yes—
322C23×C3H(5,𝔽2)12yes—
323C23×C3H(5,𝔽2)14yes—
324C23×C3C2511yescharacteristic
325C3×D4—416no—
326C3×D4—432no—
327C3×D4—432no—
328C3×D4—416no—
329C2×C3×C4—28no—
330C2×C3×C4—216no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—416no—
348C2×C3×C4—416no—
349C2×C3×C4—48no—
350C2×C3×C4—48no—
351C2×C3×C4—48no—
352C2×C3×C4—48no—
353C2×C3×C4C22×D414yes—
354C2×C3×C4C2×#4118yes—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24C2×C3×D414yes—
371C24C2×C3×D414yes—
372C24C24×C314yes—
373C24C24×C312yes—
374C24C24×C312yes—
375C24C24×C311yescharacteristic
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—28no—
379C2×D4—216no—
380C2×D4—216no—
381C2×D4—416no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—432no—
389C2×D4—432no—
390C2×D4—416no—
391C2×D4C2×C3×D4116yes—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4C2×C3×D414yes—
416C22×C4C3×#4118yes—
417C22×C4C24×C314yes—
418C22×C4C24×C314yes—
419C22×C4C24×C314yes—
420C22×C4C24×C314yes—
421C22×C4C24×C312yes—
422C22×C4C24×C314yes—
423C4⋊C4—216no—
424C4⋊C4—216no—
425C22⋊C4—28no—
426C22⋊C4—216no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C42—28no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—28no—
437C22×C3—28no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—416no—
441C22×C3—416no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—816no—
454C22×C3#53414yes—
455C22×C3#54018yes—
456C22×C3#54618yes—
457C22×C3#55914yes—
458C22×C3#56014yes—
459C22×C3C23×D411yescharacteristic
460C22×C3C22×#4112yes—
461C22×C3C2×H(5,𝔽2)11yescharacteristic
462C22×C3C2×H(5,𝔽2)11yescharacteristic
463C22×C3C2×H(5,𝔽2)12yes—
464C3×C4—24no—
465C3×C4—28no—
466C3×C4—44no—
467C3×C4—44no—
468C3×C4—44no—
469C3×C4—44no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—216no—
481C23—216no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—48no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23C22×C3×D414yes—
503C23C22×C3×D414yes—
504C23C22×C3×D412yes—
505C23C2×C3×#4114yes—
506C23C3×H(5,𝔽2)12yes—
507C23C3×H(5,𝔽2)14yes—
508C23C3×H(5,𝔽2)12yes—
509C23C25×C311yescharacteristic
510D4—416no—
511D4—432no—
512D4—432no—
513D4—416no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—216no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4C22×C3×D414yes—
539C2×C4C2×C3×#4118yes—
540C2×C3—44no—
541C2×C3—44no—
542C2×C3—84no—
543C2×C3#270918yesdirect product factor
544C2×C3C2×#53411yescharacteristic
545C2×C3C2×#54012yes—
546C2×C3C2×#54612yes—
547C2×C3C2×#55911yescharacteristic
548C2×C3C2×#56011yescharacteristic
549C22—24no—
550C22—24no—
551C22—28no—
552C22—28no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—416no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—816no—
569C22C3×#53414yes—
570C22C3×#54018yes—
571C22C3×#54618yes—
572C22C3×#55914yes—
573C22C3×#56014yes—
574C22C23×C3×D411yescharacteristic
575C22C22×C3×#4112yes—
576C22C2×C3×H(5,𝔽2)11yescharacteristic
577C22C2×C3×H(5,𝔽2)11yescharacteristic
578C22C2×C3×H(5,𝔽2)12yes—
579C4—24no—
580C4—28no—
581C4—44no—
582C4—44no—
583C4—44no—
584C4—44no—
585C3C2×#270911yescharacteristic, direct product factor, Sylow
586C2—44no—
587C2—44no—
588C2—84no—
589C2C3×#270918yesdirect product factor
590C2C2×C3×#53411yescharacteristic
591C2C2×C3×#54012yes—
592C2C2×C3×#54612yes—
593C2C2×C3×#55911yescharacteristic
594C2C2×C3×#56011yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC23C25×C3
prime radicalC22×C3×#127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#123C2semidirect
3. C3×#2709C2direct (N × H)
4. C2×C3×#394C2semidirect
5. C2×C3×#394C2semidirect
6. C2×C3×#394C2semidirect
7. C2×C3×#387C2semidirect
8. C2×C3×#386C2semidirect
9. C2×C3×#386C2semidirect
10. C2×C3×#386C2semidirect
11. C2×#2709C3direct (N × H)
21. C2×C3×#123C22semidirect
33. C3×#394C22semidirect
34. C3×#394C22semidirect
35. C3×#394C22semidirect
36. C3×#387C22semidirect
37. C3×#386C22semidirect
38. C3×#386C22semidirect
39. C3×#386C22semidirect
45. C22×#123C2×C3semidirect
46. #2709C2×C3direct (N × H)
47. C2×#394C2×C3semidirect
48. C2×#394C2×C3semidirect
49. C2×#394C2×C3semidirect
50. C2×#387C2×C3semidirect
51. C2×#386C2×C3semidirect
52. C2×#386C2×C3semidirect
53. C2×#386C2×C3semidirect
123. C2×#123C22×C3semidirect
135. #394C22×C3semidirect
136. #394C22×C3semidirect
137. #394C22×C3semidirect
138. #387C22×C3semidirect
139. #386C22×C3semidirect
140. #386C22×C3semidirect
141. #386C22×C3semidirect
543. C2×C3#2709direct (N × H)
585. C3C2×#2709direct (N × H)
589. C2C3×#2709direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1272
C22×C3×#1232
C3×#27092
C2×C3×#3942
C2×C3×#3872
C2×C3×#3862
C2×#27093