Finite Group Databasebeta

← #86759⋊C3 #11923261

Subgroups — #86759⋊C3 #11923261

Order 1152.

839 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1007⋊C3C211yesmaximal, characteristic, complemented
2C22×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
5#5817⋊C3C211yesmaximal, characteristic, complemented
6#5576⋊C3C211yesmaximal, characteristic, complemented
7#5572⋊C3C211yesmaximal, characteristic, complemented
8#86759C311yesmaximal, characteristic, complemented
9#3486⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12#1007⋊C3—22no—
13C2×D3×SL(2,𝔽3)—22no—
14C2×D3×SL(2,𝔽3)C2211yescharacteristic
15C2×D3×SL(2,𝔽3)C2211yescharacteristic
16C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
17C3×#937—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
20C2×#1007C2×C311yescharacteristic, complemented
21C22×D3×Q8C2×C311yescharacteristic, complemented
22C2×2−1+4⋊C3—31no—
23C23×SL(2,𝔽3)—31no—
24C3×D4×Q8C2×C311yescharacteristic, complemented
25#5817C2×C311yescharacteristic, complemented
26Q8×#67C2×C311yescharacteristic, complemented
27#5727—31no—
28#5724—31no—
29#5633—31no—
30#5632—31no—
31#5628—31no—
32#5627—31no—
33#5576C2×C311yescharacteristic, complemented
34#5572C2×C311yescharacteristic, complemented
35#5552—31no—
36#5547—31no—
37D4×SL(2,𝔽3)—21no—
38D4×SL(2,𝔽3)—31no—
39D4×SL(2,𝔽3)D311yescharacteristic, complemented
40#548⋊C3—31no—
41#548⋊C3—31no—
42#519⋊C3—31no—
43C23×C3×D3—41no—
44C2×C32×D4—41no—
45C2×C3×#67—41no—
46C2×C3×#67—41no—
47C2×C3×SL(2,𝔽3)—21no—
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)C2311yescharacteristic
50D3×SL(2,𝔽3)—21no—
51D3×SL(2,𝔽3)D412yescomplemented
52C3⋊(#41⋊C3)—21no—
53C3×#217—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×(C2×D3)⋊C4—41no—
56#3486—31noSylow
57C22×C3×Q8C22×C311yescharacteristic
58C22×C3×Q8C22×C311yescharacteristic, complemented
59#1007—22no—
60#1007—62no—
61C2×#239—31no—
62C2×#239C22×C311yescharacteristic, complemented
63C2×D3×Q8—22no—
64C2×D3×Q8—31no—
65C2×D3×Q8—31no—
66C2×D3×Q8—31no—
67C2×D3×Q8—62no—
68C2×D3×Q8C22×C311yescharacteristic
69C2×D3×Q8C22×C311yescharacteristic
70C2×#235—31no—
71C2×#235—31no—
72C22×C4×D3—31no—
73C22×Dic6—31no—
742−1+4⋊C3—62no—
75C2×#41⋊C3—31no—
76C22×SL(2,𝔽3)—21no—
77C22×SL(2,𝔽3)—21no—
78C22×SL(2,𝔽3)—31no—
79C22×SL(2,𝔽3)—31no—
80C22×SL(2,𝔽3)—62no—
81C22×SL(2,𝔽3)C2×D311yescharacteristic
82C22×SL(2,𝔽3)C2×D311yescharacteristic, complemented
83C3×C4⋊Q8—31no—
84C3×C22⋊Q8—31no—
85C3×C22⋊Q8—31no—
86C3×#120—121no—
87C3×C4×Q8C22×C311yescharacteristic, complemented
88C3×C4×D4—31no—
89#947—31no—
90#946—31no—
91#946—31no—
92Q8×Dic3C22×C311yescharacteristic, complemented
93#944—31no—
94#939—31no—
95#937A411yescharacteristic
96#930—31no—
97C4×#67—31no—
98C4×#67—31no—
99#924—31no—
100#897—31no—
101#896—32no—
102#896—32no—
103#896—31no—
104#895—31no—
105#894—32no—
106#894—31no—
107#893—32no—
108#892—31no—
109#888—32no—
110#887—31no—
111#885—32no—
112#884—31no—
113#883—31no—
114#882—31no—
115#67⋊C4—32no—
116#878—31no—
117#875—31no—
118C4×D12—31no—
119#872—31no—
120C4×Dic6—31no—
121C4×SL(2,𝔽3)—21no—
122C4×SL(2,𝔽3)—31no—
123C4×SL(2,𝔽3)C2×D311yescharacteristic, complemented
124C23×C32—41no—
125C23×C32—41no—
126C22×C3×D3—41no—
127C22×C3×D3—41no—
128C22×C3×D3—82no—
129C22×C3×D3—82no—
130C22×C3×D3—82no—
131C32×D4—82no—
132C2×C32×C4—41no—
133C3×#67—82no—
134C3×#67—82no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—41no—
137C3×SL(2,𝔽3)C2×D411yescharacteristic
138C2×2−1+4—31no—
139C23×Q8—31no—
140D4×Q8—31no—
141D4×Q8—91no—
142D4×Q8—91no—
143D4×Q8C3×D311yescharacteristic, complemented
144#548—31no—
145#548—31no—
146#548—91no—
147#548—91no—
148#536—91no—
149#536—91no—
150#525—91no—
151#525—91no—
152#519—31no—
153C24×C3—121no—
154C23×D3C2×A411yescharacteristic
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—31no—
158C2×C3×Q8—31no—
159C2×C3×Q8—61no—
160C2×C3×Q8—61no—
161C2×C3×Q8C23×C311yescharacteristic
162C2×C3×D4—41no—
163C2×C3×D4—81no—
164C2×C3×D4—121no—
165C2×C3×D4—121no—
166C2×C3×D4C2×A411yescharacteristic
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C2×#67—31no—
170C2×#67—31no—
171C2×#67C2×A411yescharacteristic
172C2×#67C2×A411yescharacteristic
173C22×Dic3—31no—
174#239—21no—
175#239—62no—
176#239—62no—
177#239—61no—
178D3×Q8—21no—
179D3×Q8—32no—
180D3×Q8—62no—
181D3×Q8—62no—
182D3×Q8—62no—
183D3×Q8—62no—
184D3×Q8—62no—
185D3×Q8—62no—
186D3×Q8—61no—
187D3×Q8C3×D412yescomplemented
188#235—62no—
189#235—62no—
190#235—62no—
191#235—62no—
192C2×D12—31no—
193C2×D12—31no—
194C2×D12—31no—
195C2×C4×D3—31no—
196C2×C4×D3—32no—
197C2×C4×D3—31no—
198C2×C4×D3—31no—
199C2×C4×D3—31no—
200C2×C4×D3—31no—
201C2×C4×D3—62no—
202C2×C4×D3—62no—
203C2×Dic6—31no—
204C2×Dic6—32no—
205C2×Dic6—31no—
206C2×Dic6—31no—
207C2×Dic6—62no—
208C2×Dic6—62no—
209#41⋊C3—61no—
210C2×SL(2,𝔽3)—21no—
211C2×SL(2,𝔽3)—21no—
212C2×SL(2,𝔽3)—21no—
213C2×SL(2,𝔽3)—32no—
214C2×SL(2,𝔽3)—41no—
215C2×SL(2,𝔽3)—41no—
216C2×SL(2,𝔽3)—61no—
217C2×SL(2,𝔽3)C22×D311yescharacteristic
218C3×C4⋊C4—31no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—31no—
221C3×C22⋊C4—31no—
222C3×C22⋊C4—31no—
223C3×C22⋊C4—121no—
224C3×C22⋊C4—121no—
225C3×C22⋊C4—121no—
226C3×C42—31no—
227#217—31no—
228#217C2×A411yescharacteristic
229(C2×D3)⋊C4—32no—
230(C2×D3)⋊C4—32no—
231(C2×D3)⋊C4—31no—
232(C2×D3)⋊C4—31no—
233(C2×D3)⋊C4—31no—
234(C2×D3)⋊C4—31no—
235(C2×D3)⋊C4C2×A411yescharacteristic
236(C2×D3)⋊C4C2×A411yescharacteristic
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239Dic3⋊C4—32no—
240Dic3⋊C4—31no—
241Dic3⋊C4—31no—
242Dic3⋊C4—31no—
243Dic3⋊C4—31no—
244C4×Dic3—32no—
245C4×Dic3—31no—
246C22×C32—42no—
247C22×C32—41no—
248C22×C32—81no—
249C22×C32—81no—
250C22×C32—81no—
251C22×C32—81no—
252C22×C32—81no—
253C2×C3×D3—42no—
254C2×C3×D3—42no—
255C2×C3×D3—81no—
256C2×C3×D3—81no—
257C2×C3×D3—81no—
258C2×C3×D3—81no—
259C2×C3×D3—88no—
260C32×C4—81no—
261C3×Dic3—81no—
262C3×Dic3—81no—
2632−1+4—62no—
2642−1+4—182no—
265C2×#41—31no—
266C2×#41—91no—
267C2×#41—91no—
268C2×#41—91no—
269C22×Q8—31no—
270C22×Q8—31no—
271C22×Q8—62no—
272C22×Q8—91no—
273C22×Q8—91no—
274C22×Q8—91no—
275C22×Q8—91no—
276C22×Q8—182no—
277C22×Q8C2×C3×D311yescharacteristic
278C22×Q8C2×C3×D311yescharacteristic, complemented
279C23×C4—91no—
280C4⋊Q8—31no—
281C4⋊Q8—92no—
282C4⋊Q8—91no—
283#124—91no—
284#124—92no—
285#124—91no—
286#123—91no—
287#123—91no—
288#123—92no—
289C22⋊Q8—31no—
290C22⋊Q8—31no—
291C22⋊Q8—91no—
292C22⋊Q8—91no—
293C22⋊Q8—92no—
294C22⋊Q8—92no—
295C22⋊Q8—91no—
296C22⋊Q8—91no—
297C22⋊Q8—91no—
298C22⋊Q8—91no—
299#121—91no—
300#121—91no—
301#121—91no—
302#121—91no—
303#120—31no—
304#120—91no—
305C4×Q8—31no—
306C4×Q8—91no—
307C4×Q8—91no—
308C4×Q8C2×C3×D311yescharacteristic, complemented
309C4×D4—31no—
310C4×D4—91no—
311C4×D4—91no—
312C4×D4—92no—
313C4×D4—92no—
314C4×D4—91no—
315#117—91no—
316#117—91no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—81no—
320C23×C3—81no—
321C23×C3—121no—
322C23×C3—121no—
323C23×C3—242no—
324C23×C3—242no—
325C23×C3—242no—
326C23×C3C22×A411yescharacteristic
327C23×C3C22×A411yescharacteristic
328C22×D3—22no—
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—31no—
332C22×D3C22×A411yescharacteristic
333C22×D3C22×A411yescharacteristic
334C3×Q8—31no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8C2×C3×D411yescharacteristic
340C3×D4—22no—
341C3×D4—82no—
342C3×D4—84no—
343C3×D4—242no—
344C3×D4—242no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—41no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—81no—
355C2×C3×C4—121no—
356C2×C3×C4—121no—
357C2×C3×C4C22×A411yescharacteristic
358#67—22no—
359#67—22no—
360#67—62no—
361#67—62no—
362C2×Dic3—31no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—62no—
366C2×Dic3C22×A411yescharacteristic
367C2×Dic3C22×A411yescharacteristic
368D12—61no—
369D12—61no—
370D12—62no—
371D12—62no—
372C4×D3—32no—
373C4×D3—32no—
374C4×D3—61no—
375C4×D3—61no—
376C4×D3—62no—
377C4×D3—64no—
378C4×D3—61no—
379C4×D3—61no—
380C4×D3—62no—
381Dic6—32no—
382Dic6—32no—
383Dic6—62no—
384Dic6—64no—
385Dic6—61no—
386Dic6—61no—
387SL(2,𝔽3)—21no—
388SL(2,𝔽3)D3×D411yescharacteristic
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—82no—
393C2×C32—161no—
394C3×D3—82no—
395C3×D3—82no—
396C24—31no—
397#41—61no—
398#41—182no—
399#41—182no—
400#41—182no—
401#41—182no—
402#41—182no—
403#41—182no—
404#41—181no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—31no—
408C2×Q8—31no—
409C2×Q8—32no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—91no—
414C2×Q8—92no—
415C2×Q8—91no—
416C2×Q8—91no—
417C2×Q8—92no—
418C2×Q8—182no—
419C2×Q8—182no—
420C2×Q8—182no—
421C2×Q8—182no—
422C2×Q8—182no—
423C2×Q8—182no—
424C2×Q8—182no—
425C2×Q8—181no—
426C2×Q8—182no—
427C2×Q8C22×C3×D311yescharacteristic
428C2×D4—31no—
429C2×D4—31no—
430C2×D4—91no—
431C2×D4—91no—
432C2×D4—91no—
433C2×D4—91no—
434C2×D4—91no—
435C2×D4D3×A411yescharacteristic
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—91no—
442C22×C4—91no—
443C22×C4—92no—
444C22×C4—91no—
445C22×C4—182no—
446C22×C4—182no—
447C4⋊C4—31no—
448C4⋊C4—32no—
449C4⋊C4—31no—
450C4⋊C4—91no—
451C4⋊C4—91no—
452C4⋊C4—91no—
453C4⋊C4—91no—
454C4⋊C4—91no—
455C4⋊C4—92no—
456C4⋊C4—91no—
457C22⋊C4—31no—
458C22⋊C4—31no—
459C22⋊C4—31no—
460C22⋊C4—31no—
461C22⋊C4—31no—
462C22⋊C4—91no—
463C22⋊C4—91no—
464C22⋊C4—91no—
465C22⋊C4—92no—
466C22⋊C4—92no—
467C22⋊C4—91no—
468C22⋊C4—91no—
469C42—31no—
470C42—91no—
471C42—92no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—42no—
478C22×C3—41no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—81no—
487C22×C3—122no—
488C22×C3—122no—
489C22×C3—161no—
490C22×C3—161no—
491C22×C3—161no—
492C22×C3—161no—
493C22×C3—241no—
494C22×C3—241no—
495C22×C3—241no—
496C22×C3—241no—
497C22×C3—248no—
498C22×C32−1+4⋊C312yescomplemented
499C22×C3C23×A411yescharacteristic
500C2×D3—28no—
501C2×D3—21no—
502C2×D3—21no—
503C2×D3—21no—
504C2×D3—21no—
505C2×D3—61no—
506C2×D3—61no—
507C2×D3—61no—
508C2×D3D4×A412yes—
509C2×D32−1+4⋊C312yescomplemented
510C3×C4—21no—
511C3×C4—31no—
512C3×C4—31no—
513C3×C4—61no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—81no—
517C3×C4—82no—
518C3×C4—241no—
519C3×C4—241no—
520Dic3—21no—
521Dic3—21no—
522Dic3—32no—
523Dic3—61no—
524Dic3—61no—
525C32—41noSylow
526C23—31no—
527C23—31no—
528C23—62no—
529C23—62no—
530C23—62no—
531C23—91no—
532C23C2×D3×A411yescharacteristic
533C23C2×D3×A411yescharacteristic
534Q8—31no—
535Q8—61no—
536Q8—61no—
537Q8—61no—
538Q8—61no—
539Q8—92no—
540Q8—92no—
541Q8—181no—
542Q8—181no—
543Q8—182no—
544Q8—184no—
545Q8C3×D3×D411yescharacteristic
546D4—22no—
547D4—62no—
548D4—62no—
549D4—181no—
550D4—181no—
551D4—182no—
552D4—182no—
553D4—182no—
554D4—182no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—31no—
558C2×C4—31no—
559C2×C4—31no—
560C2×C4—31no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—92no—
566C2×C4—92no—
567C2×C4—91no—
568C2×C4—91no—
569C2×C4—91no—
570C2×C4—181no—
571C2×C4—181no—
572C2×C4—181no—
573C2×C4—181no—
574C2×C4—182no—
575C2×C4—184no—
576C2×C4—182no—
577C2×C4—182no—
578C2×C4C2×D3×A411yescharacteristic
579C2×C3—22no—
580C2×C3—41no—
581C2×C3—41no—
582C2×C3—41no—
583C2×C3—41no—
584C2×C3—82no—
585C2×C3—81no—
586C2×C3—81no—
587C2×C3—81no—
588C2×C3—162no—
589C2×C3—161no—
590C2×C3—162no—
591C2×C3—242no—
592C2×C3—242no—
593C2×C3C2×D4×A411yescharacteristic
594C2×C3C2×2−1+4⋊C311yescharacteristic
595C2×C3C2×2−1+4⋊C311yescharacteristic
596D3—22no—
597D3—22no—
598D3—121no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—32no—
605C22—32no—
606C22—68no—
607C22—61no—
608C22—61no—
609C22—61no—
610C22—61no—
611C22—181no—
612C22—181no—
613C22—181no—
614C22#1007⋊C312yescomplemented
615C22C22×D3×A411yescharacteristic
616C4—21no—
617C4—31no—
618C4—31no—
619C4—61no—
620C4—61no—
621C4—61no—
622C4—61no—
623C4—61no—
624C4—92no—
625C4—181no—
626C4—181no—
627C3—41no—
628C3—81no—
629C3#3486⋊C311yescharacteristic, complemented
630C2—22no—
631C2—41no—
632C2—62no—
633C2—62no—
634C2—361no—
635C2D3×D4×A411yescharacteristic
636C2C2×#1007⋊C311yescharacteristic
637C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×D4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#1007⋊C3C2semidirect
2. C22×D3×SL(2,𝔽3)C2semidirect
3. C3×D4×SL(2,𝔽3)C2semidirect
4. SL(2,𝔽3)×#67C2semidirect
5. #5817⋊C3C2semidirect
6. #5576⋊C3C2semidirect
7. #5572⋊C3C2semidirect
8. #86759C3semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
16. C2×C3⋊(#41⋊C3)C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. Dic3×SL(2,𝔽3)C22semidirect
20. C2×#1007C2×C3semidirect
21. C22×D3×Q8C2×C3semidirect
24. C3×D4×Q8C2×C3semidirect
25. #5817C2×C3semidirect
26. Q8×#67C2×C3semidirect
33. #5576C2×C3semidirect
34. #5572C2×C3semidirect
39. D4×SL(2,𝔽3)D3semidirect
51. D3×SL(2,𝔽3)D4semidirect
58. C22×C3×Q8C22×C3semidirect
62. C2×#239C22×C3semidirect
82. C22×SL(2,𝔽3)C2×D3semidirect
87. C3×C4×Q8C22×C3semidirect
92. Q8×Dic3C22×C3semidirect
123. C4×SL(2,𝔽3)C2×D3semidirect
143. D4×Q8C3×D3semidirect
187. D3×Q8C3×D4semidirect
278. C22×Q8C2×C3×D3semidirect
308. C4×Q8C2×C3×D3semidirect
498. C22×C32−1+4⋊C3semidirect
509. C2×D32−1+4⋊C3semidirect
614. C22#1007⋊C3semidirect
629. C3#3486⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
C22×D3×SL(2,𝔽3)2
C3×D4×SL(2,𝔽3)2
SL(2,𝔽3)×#672
#5817⋊C32
#5576⋊C32
#5572⋊C32
#867593
#3486⋊C33
C3×#9374