Finite Group Databasebeta

← C3×#29604 #10648211

Subgroups — C3×#29604 #10648211

Order 768.

1460 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic, complemented
3C2×C3×#450C211yesmaximal, characteristic, complemented
4C3×#2616C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1786C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic, complemented
12#29604C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C3×#535—24no—
17C3×#535—24no—
18C2×C3×#127—21no—
19C2×C3×#124—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yescomplemented
22C2×C3×#121C2212yes—
23C2×C3×#121C2212yes—
24C2×C3×#120—22no—
25C2×C3×#117C2211yescharacteristic
26C3×#485—24no—
27C3×#450—24no—
28C2×C3×#107C2211yescharacteristic, complemented
29C2×C3×#106C2211yescharacteristic, complemented
30C2×C3×#103C2211yescharacteristic, complemented
31C2×C3×#103C2211yescharacteristic, complemented
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2212yescomplemented
38C2×C3×C22⋊C8C2212yescomplemented
39C3×#395—21no—
40C3×#394—21no—
41C3×#394—22no—
42C3×#380C2211yescharacteristic
43C3×#380C2212yescomplemented
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46C2×#535C2×C311yescharacteristic
47C2×#485C2×C311yescharacteristic, complemented
48C2×#450C2×C311yescharacteristic, complemented
49#2616C2×C311yescharacteristic, complemented
50#2034C2×C312yescomplemented
51#2033C2×C312yescomplemented
52#2000C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1786C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—42no—
61C2×C3×#41—42no—
62C22×C3×Q8D411yescharacteristic
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—22no—
65C22×C3×D4—22no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4—42no—
69C22×C3×D4—42no—
70C22×C3×D4D412yes—
71C22×C3×D4D411yescharacteristic
72C22×C3×D4D414yes—
73C22×C3×D4C2312yes—
74C23×C3×C4C2312yes—
75C23×C3×C4C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—42no—
78C2×C3×SD16—42no—
79C22×C3×C8C2312yes—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#127—28no—
83C3×#124—28no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—42no—
87C3×C22⋊Q8—42no—
88C3×#121—24no—
89C3×#121—24no—
90C3×#121—24no—
91C3×#121—24no—
92C3×#120—216no—
93C3×#117—22no—
94C3×#117D414yes—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C42—21no—
109C3×#107—22no—
110C3×#106—22no—
111C3×#103—22no—
112C3×#103—22no—
113C3×#102—24no—
114C3×#102—42no—
115C3×#102—42no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—42no—
119C3×C22⋊C8—42no—
120C3×#95—21no—
121C3×#95—22no—
122C22×#41C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×SD16C22×C312yes—
125#535—24no—
126#535—24no—
127C2×#127—21no—
128C2×#124—21no—
129C2×C22⋊Q8C22×C311yescharacteristic
130C2×C22⋊Q8C22×C312yescomplemented
131C2×#121C22×C312yes—
132C2×#121C22×C312yes—
133C2×#120—22no—
134C2×#117C22×C311yescharacteristic
135#485—24no—
136#450—24no—
137C2×#107C22×C311yescharacteristic, complemented
138C2×#106C22×C311yescharacteristic, complemented
139C2×#103C22×C311yescharacteristic, complemented
140C2×#103C22×C311yescharacteristic, complemented
141C2×#102C22×C312yes—
142C2×#102C22×C312yes—
143C2×#102C22×C312yes—
144C2×C22⋊C8C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yescomplemented
147C2×C22⋊C8C22×C312yescomplemented
148#395—21no—
149#394—21no—
150#394—22no—
151#380C22×C311yescharacteristic
152#380C22×C312yescomplemented
153#340C22×C312yescomplemented
154#340C22×C312yescomplemented
155C24×C3—22no—
156C24×C3—24no—
157C24×C3—44no—
158C24×C3—81no—
159C24×C3—82no—
160C24×C3—81no—
161C24×C3C2×D412yes—
162C24×C3C2×D411yescharacteristic
163C3×#41—48no—
164C3×#41—44no—
165C3×#41—44no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—42no—
172C2×C3×D4—24no—
173C2×C3×D4—28no—
174C2×C3×D4—28no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—48no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—82no—
190C2×C3×D4—82no—
191C2×C3×D4—82no—
192C2×C3×D4—82no—
193C2×C3×D4C2×D414yes—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2411yescharacteristic
212C3×SD16—48no—
213C2×C3×C8—22no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—44no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4C2×D412yes—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—81no—
236C3×C22⋊C4—82no—
237C3×C42—24no—
238C25—21no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—42no—
242C2×#41—42no—
243C22×Q8C3×D411yescharacteristic
244C22×Q8C23×C311yescharacteristic
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—42no—
250C22×D4—42no—
251C22×D4C3×D412yes—
252C22×D4C3×D411yescharacteristic
253C22×D4C3×D414yes—
254C22×D4C23×C312yes—
255C23×C4C23×C312yes—
256C23×C4C23×C311yescharacteristic
257C2×SD16—24no—
258C2×SD16—42no—
259C2×SD16—42no—
260C22×C8C23×C312yes—
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263#127—28no—
264#124—28no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269#121—24no—
270#121—24no—
271#121—24no—
272#121—24no—
273#120—216no—
274#117—22no—
275#117C3×D414yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C312yes—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C42—21no—
290#107—22no—
291#106—22no—
292#103—22no—
293#103—22no—
294#102—24no—
295#102—42no—
296#102—42no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301#95—21no—
302#95—22no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—21no—
307C23×C3—24no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—48no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—84no—
323C23×C3—84no—
324C23×C3—88no—
325C23×C3—88no—
326C23×C3#12012yes—
327C23×C3C22×D411yescharacteristic
328C3×Q8—44no—
329C3×Q8—42no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—48no—
335C3×D4—44no—
336C3×D4—48no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—88no—
340C3×D4—88no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4#12012yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4H(5,𝔽2)12yes—
369C2×C3×C4H(5,𝔽2)12yes—
370C3×C8—42no—
371C3×C8—41no—
372C3×C8—41no—
373C24—22no—
374C24—24no—
375C24—44no—
376C24—81no—
377C24—82no—
378C24—81no—
379C24C2×C3×D412yes—
380C24C2×C3×D411yescharacteristic
381#41—48no—
382#41—44no—
383#41—44no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4C2×C3×D414yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4C2×C3×D412yes—
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D412yes—
428C22×C4C2×C3×D412yes—
429C22×C4C24×C311yescharacteristic
430SD16—48no—
431C2×C8—22no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4C2×C3×D412yes—
447C4⋊C4C2×C3×D412yes—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—81no—
454C22⋊C4—82no—
455C42—24no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—44no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—44no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—816no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3C2×#12011yescharacteristic
477C22×C3#53412yes—
478C22×C3#57611yescharacteristic
479C22×C3#57712yes—
480C22×C3#57711yescharacteristic
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—81no—
486C3×C4—82no—
487C3×C4—81no—
488C3×C4#53412yes—
489C3×C4#53511yescharacteristic
490C3×C4#53511yescharacteristic
491C23—21no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—84no—
511C23—84no—
512C23—88no—
513C23—88no—
514C23C3×#12012yes—
515C23C22×C3×D411yescharacteristic
516Q8—44no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520D4—24no—
521D4—24no—
522D4—48no—
523D4—44no—
524D4—48no—
525D4—42no—
526D4—42no—
527D4—88no—
528D4—88no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4C3×#12012yes—
554C2×C4C22×C3×D411yescharacteristic
555C2×C4C22×C3×D411yescharacteristic
556C2×C4C3×H(5,𝔽2)12yes—
557C2×C4C3×H(5,𝔽2)12yes—
558C8—42no—
559C8—41no—
560C8—41no—
561C2×C3—42no—
562C2×C3—82no—
563C2×C3—84no—
564C2×C3#261311yescharacteristic
565C2×C3#300911yescharacteristic
566C2×C3#301211yescharacteristic
567C2×C3#319812yes—
568C2×C3#320012yes—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—816no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22C2×C3×#12011yescharacteristic
590C22C3×#53412yes—
591C22C3×#57611yescharacteristic
592C22C3×#57712yes—
593C22C3×#57711yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—81no—
600C4—81no—
601C4C3×#53412yes—
602C4C3×#53511yescharacteristic
603C4C3×#53511yescharacteristic
604C3#2960411yescharacteristic, direct product factor, Sylow
605C2—42no—
606C2—82no—
607C2—84no—
608C2C3×#261311yescharacteristic
609C2C3×#300911yescharacteristic
610C2C3×#301211yescharacteristic
611C2C3×#319812yes—
612C2C3×#320012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#485C2semidirect
3. C2×C3×#450C2semidirect
4. C3×#2616C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#2000C2semidirect
8. C3×#2000C2semidirect
9. C3×#1825C2semidirect
10. C3×#1786C2semidirect
11. C3×#1780C2semidirect
12. #29604C3direct (N × H)
21. C2×C3×C22⋊Q8C22semidirect
28. C2×C3×#107C22semidirect
29. C2×C3×#106C22semidirect
30. C2×C3×#103C22semidirect
31. C2×C3×#103C22semidirect
37. C2×C3×C22⋊C8C22semidirect
38. C2×C3×C22⋊C8C22semidirect
43. C3×#380C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
47. C2×#485C2×C3semidirect
48. C2×#450C2×C3semidirect
49. #2616C2×C3semidirect
50. #2034C2×C3semidirect
51. #2033C2×C3semidirect
52. #2000C2×C3semidirect
53. #2000C2×C3semidirect
54. #1825C2×C3semidirect
55. #1786C2×C3semidirect
56. #1780C2×C3semidirect
130. C2×C22⋊Q8C22×C3semidirect
137. C2×#107C22×C3semidirect
138. C2×#106C22×C3semidirect
139. C2×#103C22×C3semidirect
140. C2×#103C22×C3semidirect
146. C2×C22⋊C8C22×C3semidirect
147. C2×C22⋊C8C22×C3semidirect
152. #380C22×C3semidirect
153. #340C22×C3semidirect
154. #340C22×C3semidirect
604. C3#29604direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×C3×#4852
C2×C3×#4502
C3×#26162
C3×#20342
C3×#20332
C3×#20002
C3×#18252
C3×#17862
C3×#17802
#296043