Finite Group Databasebeta

← #86735⋊C3 #11923247

Subgroups — #86735⋊C3 #11923247

Order 1152.

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

normal subgroups only

This group has 829 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×#1010⋊C3C211yesmaximal, characteristic, complemented
2C22×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C211yesmaximal, characteristic, complemented
4#5831⋊C3C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
6#5572⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
8#86735C311yesmaximal, characteristic, complemented
9#3486⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12#1010⋊C3—22no—
13C2×D3×SL(2,𝔽3)—22no—
14C2×D3×SL(2,𝔽3)C2211yescharacteristic
15C2×D3×SL(2,𝔽3)C2211yescharacteristic
16C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
17C3×#882—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
20C2×#1010C2×C311yescharacteristic, complemented
21C22×D3×Q8C2×C311yescharacteristic, complemented
22C2×2−1+4⋊C3—31no—
23C23×SL(2,𝔽3)—31no—
24C3×#548C2×C311yescharacteristic, complemented
25#5831C2×C311yescharacteristic, complemented
26Q8×#67C2×C311yescharacteristic, complemented
27#5677—31no—
28#5675—31no—
29#5632—31no—
30#5628—31no—
31#5606—31no—
32#5598—31no—
33Q8×D12C2×C311yescharacteristic, complemented
34#5572C2×C311yescharacteristic, complemented
35#5547—31no—
36D4×Dic6—31no—
37D4×SL(2,𝔽3)—31no—
38D4×SL(2,𝔽3)—31no—
39#548⋊C3—21no—
40#548⋊C3—31no—
41#548⋊C3D311yescharacteristic, complemented
42#519⋊C3—31no—
43C23×C3×D3—41no—
44C2×C3×#67—41no—
45C2×C3×D12—41no—
46C2×C3×D12—41no—
47C3×#41⋊C3—21no—
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)C2311yescharacteristic
50D3×SL(2,𝔽3)—21no—
51D3×SL(2,𝔽3)—21no—
52D3×SL(2,𝔽3)D412yescomplemented
53C32×C22⋊C4—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×(C2×D3)⋊C4—41no—
56#3486—31noSylow
57C2×C3×#41C22×C311yescharacteristic, complemented
58C22×C3×Q8C22×C311yescharacteristic
59#1010—22no—
60#1010—62no—
61C2×D3×Q8—22no—
62C2×D3×Q8—31no—
63C2×D3×Q8—31no—
64C2×D3×Q8—62no—
65C2×D3×Q8C22×C311yescharacteristic
66C2×D3×Q8C22×C311yescharacteristic
67C2×D3×Q8C22×C311yescharacteristic, complemented
68C2×#237—31no—
69C2×#237—31no—
70C2×#235—31no—
71C22×C4×D3—31no—
72C22×Dic6—31no—
73C22×Dic6—31no—
742−1+4⋊C3—62no—
75C2×#41⋊C3—21no—
76C2×#41⋊C3C2×D311yescharacteristic, complemented
77C22×SL(2,𝔽3)—21no—
78C22×SL(2,𝔽3)—31no—
79C22×SL(2,𝔽3)—31no—
80C22×SL(2,𝔽3)—31no—
81C22×SL(2,𝔽3)—62no—
82C22×SL(2,𝔽3)C2×D311yescharacteristic
83C3×#124—31no—
84C3×C22⋊Q8—31no—
85C3×#121—31no—
86C3×#120—121no—
87C3×C4×Q8C22×C311yescharacteristic, complemented
88C3×C4×D4—31no—
89#946—31no—
90Q8×Dic3C22×C311yescharacteristic, complemented
91#944—31no—
92#939—31no—
93#938—31no—
94#937—31no—
95#936—31no—
96D4×Dic3—32no—
97C4×#67—31no—
98C4×#67—31no—
99#924—31no—
100#924—31no—
101#897—31no—
102#897—31no—
103#896—31no—
104#894—31no—
105#892—31no—
106#888—32no—
107#887—31no—
108#885—32no—
109#884—31no—
110#883—32no—
111#883—31no—
112#882A411yescharacteristic
113#67⋊C4—32no—
114#878—32no—
115#878—32no—
116#878—31no—
117C4×D12—31no—
118#872—31no—
119#869—31no—
120C4×Dic6—31no—
121C4×SL(2,𝔽3)—21no—
122C4×SL(2,𝔽3)—31no—
123C4×SL(2,𝔽3)C2×D311yescharacteristic, complemented
124C23×C32—41no—
125C22×C3×D3—41no—
126C22×C3×D3—41no—
127C22×C3×D3—41no—
128C22×C3×D3—82no—
129C22×C3×D3—82no—
130C22×C3×D3—82no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C3×#67—82no—
134C2×C3×Dic3—41no—
135C3×D12—82no—
136C3×D12—82no—
137C3×SL(2,𝔽3)C2×D411yescharacteristic
138C2×2−1+4—31no—
139C23×Q8—31no—
140D4×Q8—31no—
141D4×Q8—31no—
142D4×Q8—91no—
143D4×Q8—91no—
144#548—31no—
145#548—91no—
146#548—91no—
147#548C3×D311yescharacteristic, complemented
148#536—91no—
149#536—91no—
150#525—91no—
151#525—91no—
152#519—31no—
153C24×C3—121no—
154C23×D3C2×A411yescharacteristic
155C3×#41—21no—
156C3×#41—61no—
157C2×C3×Q8—21no—
158C2×C3×Q8—31no—
159C2×C3×Q8—61no—
160C2×C3×Q8C23×C311yescharacteristic
161C2×C3×D4—31no—
162C2×C3×D4—31no—
163C2×C3×D4—121no—
164C2×C3×D4—121no—
165C2×C3×D4—121no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C2×#67—31no—
169C2×#67—31no—
170C2×#67—31no—
171C2×#67C2×A411yescharacteristic
172C22×Dic3—32no—
173C22×Dic3—31no—
174D3×Q8—21no—
175D3×Q8—21no—
176D3×Q8—32no—
177D3×Q8—62no—
178D3×Q8—62no—
179D3×Q8—62no—
180D3×Q8—62no—
181D3×Q8—61no—
182D3×Q8—61no—
183D3×Q8C3×D412yescomplemented
184#237—62no—
185#237—62no—
186#237—62no—
187#237—62no—
188#235—62no—
189#235—62no—
190C2×D12C2×A411yescharacteristic
191C2×D12C2×A411yescharacteristic
192C2×C4×D3—31no—
193C2×C4×D3—31no—
194C2×C4×D3—31no—
195C2×C4×D3—31no—
196C2×C4×D3—31no—
197C2×C4×D3—62no—
198C2×C4×D3—62no—
199C2×Dic6—31no—
200C2×Dic6—32no—
201C2×Dic6—31no—
202C2×Dic6—31no—
203C2×Dic6—31no—
204C2×Dic6—62no—
205C2×Dic6—62no—
206C2×Dic6—62no—
207C2×Dic6—62no—
208#41⋊C3—21no—
209#41⋊C3—41no—
210C2×SL(2,𝔽3)—21no—
211C2×SL(2,𝔽3)—21no—
212C2×SL(2,𝔽3)—32no—
213C2×SL(2,𝔽3)—41no—
214C2×SL(2,𝔽3)—61no—
215C2×SL(2,𝔽3)—61no—
216C2×SL(2,𝔽3)C22×D311yescharacteristic
217C3×C4⋊C4—31no—
218C3×C4⋊C4—31no—
219C3×C22⋊C4—32no—
220C3×C22⋊C4—31no—
221C3×C22⋊C4—41no—
222C3×C22⋊C4—81no—
223C3×C22⋊C4—121no—
224C3×C22⋊C4—121no—
225C3×C22⋊C4C2×A411yescharacteristic
226C3×C42—31no—
227#217—32no—
228#217—31no—
229#217—31no—
230(C2×D3)⋊C4—31no—
231(C2×D3)⋊C4—31no—
232(C2×D3)⋊C4—31no—
233(C2×D3)⋊C4—31no—
234(C2×D3)⋊C4C2×A411yescharacteristic
235(C2×D3)⋊C4C2×A411yescharacteristic
236C4⋊Dic3—32no—
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—
250C2×C3×D3—42no—
251C2×C3×D3—42no—
252C2×C3×D3—81no—
253C2×C3×D3—81no—
254C2×C3×D3—81no—
255C2×C3×D3—81no—
256C2×C3×D3—88no—
257C2×C3×D3—81no—
258C2×C3×D3—81no—
259C2×C3×D3—81no—
260C32×C4—81no—
261C32×C4—81no—
262C3×Dic3—81no—
2632−1+4—62no—
2642−1+4—182no—
265C2×#41—91no—
266C2×#41—91no—
267C2×#41—91no—
268C2×#41C2×C3×D311yescharacteristic, complemented
269C22×Q8—31no—
270C22×Q8—31no—
271C22×Q8—31no—
272C22×Q8—62no—
273C22×Q8—91no—
274C22×Q8—91no—
275C22×Q8—91no—
276C22×Q8—91no—
277C22×Q8—182no—
278C22×Q8C2×C3×D311yescharacteristic
279C23×C4—91no—
280C4⋊Q8—91no—
281C4⋊Q8—92no—
282C4⋊Q8—91no—
283#124—31no—
284#124—92no—
285#124—91no—
286#123—91no—
287#123—92no—
288#123—91no—
289C22⋊Q8—31no—
290C22⋊Q8—91no—
291C22⋊Q8—92no—
292C22⋊Q8—92no—
293C22⋊Q8—91no—
294C22⋊Q8—91no—
295C22⋊Q8—91no—
296C22⋊Q8—91no—
297C22⋊Q8—91no—
298C22⋊Q8—91no—
299#121—31no—
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—91no—
314C4×D4—92no—
315#117—91no—
316#117—91no—
317C23×C3—31no—
318C23×C3—41no—
319C23×C3—81no—
320C23×C3—121no—
321C23×C3—121no—
322C23×C3—121no—
323C23×C3—242no—
324C23×C3—242no—
325C23×C3—242no—
326C23×C3C22×A411yescharacteristic
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
330C22×D3C22×A411yescharacteristic
331C22×D3C22×A411yescharacteristic
332C22×D3C22×A411yescharacteristic
333C3×Q8—31no—
334C3×Q8—61no—
335C3×Q8—61no—
336C3×Q8C2×C3×D411yescharacteristic
337C3×D4—61no—
338C3×D4—61no—
339C3×D4—62no—
340C3×D4—242no—
341C3×D4—242no—
342C3×D4—242no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—81no—
353C2×C3×C4—81no—
354C2×C3×C4—121no—
355C2×C3×C4C22×A411yescharacteristic
356C2×C3×C4C22×A411yescharacteristic
357#67—22no—
358#67—62no—
359#67—62no—
360#67—62no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—62no—
366C2×Dic3—64no—
367C2×Dic3C22×A411yescharacteristic
368D12—22no—
369D12—22no—
370C4×D3—32no—
371C4×D3—32no—
372C4×D3—62no—
373C4×D3—61no—
374C4×D3—61no—
375C4×D3—61no—
376C4×D3—61no—
377C4×D3—62no—
378Dic6—32no—
379Dic6—32no—
380Dic6—62no—
381Dic6—64no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—61no—
386SL(2,𝔽3)—21no—
387SL(2,𝔽3)D3×D411yescharacteristic
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—82no—
392C3×D3—82no—
393C3×D3—82no—
394C3×D3—161no—
395C24—31no—
396#41—21no—
397#41—61no—
398#41—182no—
399#41—182no—
400#41—182no—
401#41—182no—
402#41—182no—
403#41—182no—
404C2×Q8—21no—
405C2×Q8—31no—
406C2×Q8—32no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—92no—
411C2×Q8—91no—
412C2×Q8—91no—
413C2×Q8—91no—
414C2×Q8—91no—
415C2×Q8—92no—
416C2×Q8—182no—
417C2×Q8—182no—
418C2×Q8—182no—
419C2×Q8—182no—
420C2×Q8—182no—
421C2×Q8—182no—
422C2×Q8—182no—
423C2×Q8—181no—
424C2×Q8—181no—
425C2×Q8—182no—
426C2×Q8C22×C3×D311yescharacteristic
427C2×D4—31no—
428C2×D4—31no—
429C2×D4—31no—
430C2×D4—31no—
431C2×D4—31no—
432C2×D4—91no—
433C2×D4—91no—
434C2×D4—91no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—91no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—91no—
442C22×C4—92no—
443C22×C4—91no—
444C22×C4—182no—
445C22×C4—182no—
446C4⋊C4—31no—
447C4⋊C4—31no—
448C4⋊C4—91no—
449C4⋊C4—91no—
450C4⋊C4—91no—
451C4⋊C4—92no—
452C4⋊C4—92no—
453C4⋊C4—91no—
454C4⋊C4—91no—
455C4⋊C4—91no—
456C22⋊C4—32no—
457C22⋊C4—31no—
458C22⋊C4—31no—
459C22⋊C4—31no—
460C22⋊C4—91no—
461C22⋊C4—91no—
462C22⋊C4—91no—
463C22⋊C4—91no—
464C22⋊C4—91no—
465C22⋊C4—92no—
466C22⋊C4—91no—
467C22⋊C4D3×A411yescharacteristic
468C42—31no—
469C42—91no—
470C42—92no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—42no—
474C22×C3—41no—
475C22×C3—61no—
476C22×C3—61no—
477C22×C3—61no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—82no—
481C22×C3—81no—
482C22×C3—122no—
483C22×C3—122no—
484C22×C3—161no—
485C22×C3—161no—
486C22×C3—241no—
487C22×C3—241no—
488C22×C3—241no—
489C22×C3—241no—
490C22×C3—241no—
491C22×C3—241no—
492C22×C3—241no—
493C22×C3—248no—
494C22×C32−1+4⋊C312yescomplemented
495C22×C3C23×A411yescharacteristic
496C2×D3—28no—
497C2×D3—21no—
498C2×D3—21no—
499C2×D3—21no—
500C2×D3—21no—
501C2×D3—21no—
502C2×D3—21no—
503C2×D3—21no—
504C2×D3D4×A412yes—
505C2×D32−1+4⋊C312yescomplemented
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—31no—
509C3×C4—31no—
510C3×C4—61no—
511C3×C4—61no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4—161no—
515C3×C4—161no—
516C3×C4—241no—
517Dic3—21no—
518Dic3—32no—
519Dic3—61no—
520Dic3—61no—
521Dic3—61no—
522C32—41noSylow
523C23—31no—
524C23—31no—
525C23—31no—
526C23—31no—
527C23—62no—
528C23—62no—
529C23—62no—
530C23C2×D3×A411yescharacteristic
531Q8—31no—
532Q8—61no—
533Q8—61no—
534Q8—92no—
535Q8—92no—
536Q8—182no—
537Q8—181no—
538Q8—181no—
539Q8—181no—
540Q8—181no—
541Q8—184no—
542Q8C3×D3×D411yescharacteristic
543D4—62no—
544D4—62no—
545D4—61no—
546D4—61no—
547D4—62no—
548D4—62no—
549D4—182no—
550D4—182no—
551D4—182no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—92no—
561C2×C4—92no—
562C2×C4—91no—
563C2×C4—91no—
564C2×C4—91no—
565C2×C4—91no—
566C2×C4—181no—
567C2×C4—181no—
568C2×C4—181no—
569C2×C4—181no—
570C2×C4—182no—
571C2×C4—182no—
572C2×C4—182no—
573C2×C4—184no—
574C2×C4C2×D3×A411yescharacteristic
575C2×C4C2×D3×A411yescharacteristic
576C2×C3—22no—
577C2×C3—41no—
578C2×C3—41no—
579C2×C3—41no—
580C2×C3—82no—
581C2×C3—81no—
582C2×C3—81no—
583C2×C3—81no—
584C2×C3—121no—
585C2×C3—162no—
586C2×C3—242no—
587C2×C3—242no—
588C2×C3—481no—
589C2×C3C2×D4×A411yescharacteristic
590C2×C3C2×2−1+4⋊C311yescharacteristic
591C2×C3C2×2−1+4⋊C311yescharacteristic
592D3—22no—
593D3—22no—
594D3—41no—
595C22—21no—
596C22—21no—
597C22—32no—
598C22—32no—
599C22—68no—
600C22—61no—
601C22—61no—
602C22—61no—
603C22—61no—
604C22—61no—
605C22—61no—
606C22—61no—
607C22—61no—
608C22—61no—
609C22—61no—
610C22#1010⋊C312yescomplemented
611C22C22×D3×A411yescharacteristic
612C4—21no—
613C4—21no—
614C4—31no—
615C4—31no—
616C4—61no—
617C4—61no—
618C4—61no—
619C4—92no—
620C4—181no—
621C4—181no—
622C4—181no—
623C3—41no—
624C3—81no—
625C3#3486⋊C311yescharacteristic, complemented
626C2—22no—
627C2—62no—
628C2—62no—
629C2—121no—
630C2—121no—
631C2D3×D4×A411yescharacteristic
632C2C2×#1010⋊C311yescharacteristic
633C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#548C2×C3
socleC22×C3C23×A4
hypercenterC22⋊C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#1010⋊C3C2semidirect
2. C22×D3×SL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. #5831⋊C3C2semidirect
5. SL(2,𝔽3)×#67C2semidirect
6. #5572⋊C3C2semidirect
7. SL(2,𝔽3)×D12C2semidirect
8. #86735C3semidirect
10. C2×C3×#41⋊C3C22semidirect
16. C2×D3×SL(2,𝔽3)C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. Dic3×SL(2,𝔽3)C22semidirect
20. C2×#1010C2×C3semidirect
21. C22×D3×Q8C2×C3semidirect
24. C3×#548C2×C3semidirect
25. #5831C2×C3semidirect
26. Q8×#67C2×C3semidirect
33. Q8×D12C2×C3semidirect
34. #5572C2×C3semidirect
41. #548⋊C3D3semidirect
52. D3×SL(2,𝔽3)D4semidirect
57. C2×C3×#41C22×C3semidirect
67. C2×D3×Q8C22×C3semidirect
76. C2×#41⋊C3C2×D3semidirect
87. C3×C4×Q8C22×C3semidirect
90. Q8×Dic3C22×C3semidirect
123. C4×SL(2,𝔽3)C2×D3semidirect
147. #548C3×D3semidirect
183. D3×Q8C3×D4semidirect
268. C2×#41C2×C3×D3semidirect
308. C4×Q8C2×C3×D3semidirect
494. C22×C32−1+4⋊C3semidirect
505. C2×D32−1+4⋊C3semidirect
610. C22#1010⋊C3semidirect
625. C3#3486⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1010⋊C32
C22×D3×SL(2,𝔽3)2
C3×#548⋊C32
#5831⋊C32
SL(2,𝔽3)×#672
#5572⋊C32
SL(2,𝔽3)×D122
#867353
#3486⋊C33
C3×#8824