Finite Group Databasebeta

← #67×#41⋊C3 #11923385

Subgroups — #67×#41⋊C3 #11923385

Order 1152.

839 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×D3×#41⋊C3C211yesmaximal, characteristic, complemented
3#5831⋊C3C211yesmaximal, characteristic, complemented
4#5834⋊C3C211yesmaximal, characteristic, complemented
5Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
6SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
7#5817⋊C3C211yesmaximal, characteristic, complemented
8#41×#67C311yesmaximal, characteristic, complemented
9D4×#41⋊C3—31nomaximal
10C2×C3×#41⋊C3—22no—
11C2×C3×#41⋊C3C2211yescharacteristic
12C2×C3×#41⋊C3C2211yescharacteristic, complemented
13C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
14D3×#41⋊C3—22no—
15C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
16C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
17C3×C4×#67—41nomaximal
18Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
19Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
20C22×C3×#41C2×C311yescharacteristic, complemented
21C2×D3×#41C2×C311yescharacteristic, complemented
22C22×#41⋊C3—21no—
23C22×#41⋊C3—31no—
24C22×#41⋊C3D311yescharacteristic, complemented
25#5834C2×C311yescharacteristic, complemented
26#5832—31no—
27#5831C2×C311yescharacteristic, complemented
28#5829—31no—
29#5828—31no—
30Dic3×#41C2×C311yescharacteristic, complemented
31#5824—31no—
32#5817C2×C311yescharacteristic, complemented
33Q8×#67C2×C311yescharacteristic
34#5806—31no—
35D4×#67—31no—
36#5791—31no—
37C2×C4×#67—31no—
38C4⋊(#41⋊C3)—31no—
39D4×SL(2,𝔽3)—31no—
40#548⋊C3—31no—
41#548⋊C3—31no—
42C4×#41⋊C3—31no—
43C22×C32×C4—41no—
44C2×C3×#67—41no—
45C2×C3×C4×D3—41no—
46C3×#41⋊C3—21no—
47C3×#41⋊C3D412yescomplemented
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)C2311yescharacteristic
50D3×SL(2,𝔽3)—21no—
51C3⋊(#41⋊C3)—21no—
52C3×#217—41no—
53C3×(C2×D3)⋊C4—41no—
54C3×Dic3⋊C4—41no—
55C3×C4×Dic3—41no—
56D4×#41—31noSylow
57C2×C3×#41—22no—
58C2×C3×#41—31no—
59C2×C3×#41—31no—
60C2×C3×#41—62no—
61C2×C3×#41C22×C311yescharacteristic
62C2×C3×#41C22×C311yescharacteristic
63C22×C3×Q8C22×C311yescharacteristic
64C22×C3×D4—31no—
65C23×C3×C4—31no—
66C22×#67—31no—
67D3×#41—22no—
68D3×#41—62no—
69C2×#239C22×C311yescharacteristic
70C2×D3×Q8C22×C311yescharacteristic
71C2×#237—31no—
72C2×D3×D4—31no—
73C2×#235—31no—
74C22×C4×D3—31no—
75C2×#41⋊C3—24no—
76C2×#41⋊C3—22no—
77C2×#41⋊C3—21no—
78C2×#41⋊C3—21no—
79C2×#41⋊C3—31no—
80C2×#41⋊C3—62no—
81C2×#41⋊C3C2×D311yescharacteristic
82C2×#41⋊C3C2×D311yescharacteristic, complemented
83C22×SL(2,𝔽3)—21no—
84C22×SL(2,𝔽3)—31no—
85C22×SL(2,𝔽3)C2×D311yescharacteristic, complemented
86C3×C4×D4—121no—
87#953—32no—
88C2×#217—31no—
89#947—31no—
90#946—31no—
91#946—31no—
92Q8×Dic3C22×C311yescharacteristic
93Q8×Dic3C22×C311yescharacteristic
94#944—31no—
95#940—31no—
96#939—32no—
97#939—32no—
98#938—31no—
99#938—31no—
100#937—32no—
101#936—31no—
102#935—32no—
103D4×Dic3—31no—
104D4×Dic3—31no—
105#930—31no—
106#930—31no—
107#929—32no—
108C4×#67—32no—
109C4×#67—31no—
110C4×#67—31no—
111C4×#67—31no—
112C4×#67—31no—
113C4×#67—31no—
114C4×#67—31no—
115C4×#67—31no—
116C4×#67A411yescharacteristic
117C2×(C2×D3)⋊C4—31no—
118#926—31no—
119#924—31no—
120#924—31no—
121C2×Dic3⋊C4—31no—
122C2×C4×Dic3—31no—
123C4×SL(2,𝔽3)—31no—
124C4×SL(2,𝔽3)—31no—
125C23×C32—41no—
126C22×C3×D3—41no—
127C2×C32×C4—41no—
128C2×C32×C4—41no—
129C2×C32×C4—82no—
130C3×#67—44no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—41no—
134C3×C4×D3—82no—
135C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
136C22×#41—31no—
137C22×#41C3×D311yescharacteristic, complemented
138#550—31no—
139D4×Q8—31no—
140#548—31no—
141#548—31no—
142#547—91no—
143#546—91no—
144#546—91no—
145D42—91no—
146#532—91no—
147#525—91no—
148#525—91no—
149C4×#41—31no—
150C2×C4×D4—91no—
151C24×C3—31no—
152C23×D3—31no—
153C3×#41—21no—
154C3×#41—32no—
155C3×#41—62no—
156C3×#41—62no—
157C3×#41—62no—
158C3×#41—62no—
159C3×#41—61no—
160C3×#41C3×D412yescomplemented
161C2×C3×Q8—21no—
162C2×C3×Q8—31no—
163C2×C3×Q8—61no—
164C2×C3×Q8C23×C311yescharacteristic
165C2×C3×D4—31no—
166C2×C3×D4—31no—
167C2×C3×D4—32no—
168C2×C3×D4—62no—
169C2×C3×D4—62no—
170C2×C3×D4—121no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—41no—
177C22×C3×C4—62no—
178C22×C3×C4—62no—
179C22×C3×C4—81no—
180C22×C3×C4—121no—
181C22×C3×C4C2×A411yescharacteristic
182C2×#67—32no—
183C2×#67—31no—
184C2×#67—31no—
185C2×#67—32no—
186C2×#67—31no—
187C2×#67—64no—
188C2×#67C2×A411yescharacteristic
189C22×Dic3—31no—
190C22×Dic3—31no—
191C22×Dic3—31no—
192#239—21no—
193#239—61no—
194D3×Q8—21no—
195D3×Q8—61no—
196#237—62no—
197#237—62no—
198D3×D4—62no—
199D3×D4—62no—
200#235—62no—
201#235—62no—
202C2×D12—31no—
203C2×C4×D3—31no—
204C2×C4×D3—31no—
205C2×C4×D3—31no—
206C2×C4×D3—62no—
207C2×C4×D3—62no—
208C2×C4×D3C2×A411yescharacteristic
209C2×Dic6—31no—
210#41⋊C3—22no—
211#41⋊C3—21no—
212#41⋊C3—22no—
213#41⋊C3—61no—
214#41⋊C3#6712yesdirect product factor
215C2×SL(2,𝔽3)—22no—
216C2×SL(2,𝔽3)—21no—
217C2×SL(2,𝔽3)—21no—
218C2×SL(2,𝔽3)—61no—
219C2×SL(2,𝔽3)C22×D311yescharacteristic
220C3×C4⋊C4—121no—
221C3×C22⋊C4—121no—
222C3×C22⋊C4—121no—
223C3×C42—121no—
224#217—32no—
225#217—32no—
226#217—31no—
227#217—31no—
228#217—31no—
229#217C2×A411yescharacteristic
230(C2×D3)⋊C4—32no—
231(C2×D3)⋊C4—31no—
232(C2×D3)⋊C4—31no—
233(C2×D3)⋊C4—31no—
234(C2×D3)⋊C4C2×A411yescharacteristic
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237Dic3⋊C4—32no—
238Dic3⋊C4—31no—
239Dic3⋊C4—31no—
240Dic3⋊C4—31no—
241Dic3⋊C4C2×A411yescharacteristic
242C4×Dic3—31no—
243C4×Dic3—31no—
244C4×Dic3—31no—
245C4×Dic3C2×A411yescharacteristic
246C22×C32—42no—
247C22×C32—41no—
248C22×C32—81no—
249C22×C32—81no—
250C2×C3×D3—42no—
251C2×C3×D3—81no—
252C2×C3×D3—81no—
253C32×C4—42no—
254C32×C4—81no—
255C3×Dic3—42no—
256C3×Dic3—81no—
257C3×Dic3—81no—
258C2×#41—22no—
259C2×#41—31no—
260C2×#41—31no—
261C2×#41—31no—
262C2×#41—62no—
263C2×#41—62no—
264C2×#41—91no—
265C2×#41—91no—
266C2×#41—182no—
267C2×#41C2×C3×D311yescharacteristic
268C2×#41C2×C3×D311yescharacteristic
269C22×Q8—31no—
270C22×Q8C2×C3×D311yescharacteristic
271C22×D4—31no—
272C22×D4—91no—
273C22×D4—91no—
274C23×C4—31no—
275C23×C4—91no—
276C4⋊Q8—91no—
277#127—91no—
278#124—91no—
279#124—91no—
280#123—92no—
281#123—92no—
282C22⋊Q8—91no—
283C22⋊Q8—91no—
284C22⋊Q8—91no—
285C22⋊Q8—91no—
286#121—91no—
287#121—91no—
288#121—92no—
289#121—92no—
290#121—91no—
291#121—91no—
292#120—92no—
293#120—92no—
294C4×Q8—31no—
295C4×Q8—31no—
296C4×D4—31no—
297C4×D4—91no—
298C4×D4—91no—
299C4×D4—91no—
300C4×D4—91no—
301C4×D4—91no—
302C4×D4—91no—
303C4×D4—92no—
304C4×D4—91no—
305C4×D4—91no—
306C4×D4—91no—
307#117—91no—
308C2×C4⋊C4—91no—
309C2×C22⋊C4—91no—
310C2×C22⋊C4—91no—
311C2×C42—91no—
312C23×C3—31no—
313C23×C3—31no—
314C23×C3—41no—
315C23×C3—62no—
316C23×C3—62no—
317C23×C3—81no—
318C23×C3—121no—
319C23×C3—121no—
320C23×C3C22×A411yescharacteristic
321C22×D3—31no—
322C22×D3—62no—
323C22×D3—62no—
324C22×D3—121no—
325C22×D3C22×A411yescharacteristic
326C3×Q8—31no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8C2×C3×D411yescharacteristic
330C3×D4—32no—
331C3×D4—32no—
332C3×D4—61no—
333C3×D4—61no—
334C3×D4—64no—
335C3×D4—124no—
336C2×C3×C4—22no—
337C2×C3×C4—31no—
338C2×C3×C4—32no—
339C2×C3×C4—32no—
340C2×C3×C4—31no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—62no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—62no—
349C2×C3×C4—82no—
350C2×C3×C4—84no—
351C2×C3×C4—81no—
352C2×C3×C4—81no—
353C2×C3×C4—121no—
354C2×C3×C4—121no—
355C2×C3×C4—121no—
356C2×C3×C4—242no—
357C2×C3×C4C22×A411yescharacteristic
358C2×C3×C4C22×A411yescharacteristic
359#67—64no—
360#67—62no—
361#67—62no—
362#67—62no—
363#67—122no—
364#67#41⋊C314yesdirect product factor
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—62no—
369C2×Dic3—62no—
370C2×Dic3—121no—
371C2×Dic3C22×A411yescharacteristic
372C2×Dic3C22×A411yescharacteristic
373C2×Dic3C22×A411yescharacteristic
374D12—61no—
375D12—61no—
376C4×D3—22no—
377C4×D3—61no—
378C4×D3—61no—
379C4×D3—61no—
380C4×D3—61no—
381C4×D3—62no—
382Dic6—61no—
383Dic6—61no—
384SL(2,𝔽3)—21no—
385SL(2,𝔽3)C2×#6711yescharacteristic, complemented
386C2×C32—41no—
387C2×C32—41no—
388C2×C32—41no—
389C2×C32—82no—
390C3×D3—82no—
391C24—31no—
392C24—91no—
393#41—21no—
394#41—32no—
395#41—62no—
396#41—62no—
397#41—62no—
398#41—62no—
399#41—61no—
400#41—61no—
401#41—182no—
402#41—182no—
403#41—182no—
404#41—182no—
405#41—181no—
406#41C3×#6712yescomplemented
407C2×Q8—21no—
408C2×Q8—31no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—91no—
412C2×Q8—181no—
413C2×Q8C22×C3×D311yescharacteristic
414C2×D4—31no—
415C2×D4—31no—
416C2×D4—31no—
417C2×D4—32no—
418C2×D4—62no—
419C2×D4—62no—
420C2×D4—91no—
421C2×D4—91no—
422C2×D4—91no—
423C2×D4—92no—
424C2×D4—91no—
425C2×D4—92no—
426C2×D4—184no—
427C2×D4—182no—
428C2×D4—182no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—91no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—91no—
442C22×C4—91no—
443C22×C4—182no—
444C22×C4—182no—
445C22×C4D3×A411yescharacteristic
446C4⋊C4—31no—
447C4⋊C4—91no—
448C4⋊C4—91no—
449C4⋊C4—91no—
450C4⋊C4—91no—
451C4⋊C4—92no—
452C4⋊C4—91no—
453C22⋊C4—31no—
454C22⋊C4—31no—
455C22⋊C4—91no—
456C22⋊C4—91no—
457C22⋊C4—91no—
458C22⋊C4—91no—
459C22⋊C4—92no—
460C22⋊C4—92no—
461C22⋊C4—92no—
462C22⋊C4—91no—
463C22⋊C4—91no—
464C42—31no—
465C42—91no—
466C42—91no—
467C42—91no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—32no—
471C22×C3—42no—
472C22×C3—41no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—61no—
476C22×C3—61no—
477C22×C3—61no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—82no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3—81no—
484C22×C3—122no—
485C22×C3—124no—
486C22×C3—241no—
487C22×C3—241no—
488C22×C3C2×#41⋊C312yescomplemented
489C22×C3C23×A411yescharacteristic
490C2×D3—21no—
491C2×D3—21no—
492C2×D3—61no—
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—124no—
496C2×D3C2×#41⋊C312yescomplemented
497C3×C4—21no—
498C3×C4—31no—
499C3×C4—31no—
500C3×C4—42no—
501C3×C4—61no—
502C3×C4—81no—
503C3×C4—82no—
504C3×C4—82no—
505C3×C4—122no—
506C3×C4—241no—
507C3×C4—241no—
508C3×C4D4×A412yes—
509Dic3—21no—
510Dic3—21no—
511Dic3—61no—
512Dic3—61no—
513Dic3—61no—
514Dic3C2×#41⋊C312yescomplemented
515C32—41noSylow
516C23—31no—
517C23—31no—
518C23—31no—
519C23—62no—
520C23—62no—
521C23—91no—
522C23—121no—
523C23—182no—
524C23—182no—
525C23—361no—
526C23C2×D3×A411yescharacteristic
527Q8—31no—
528Q8—61no—
529Q8—61no—
530Q8—181no—
531Q8—181no—
532Q8C2×C3×#6711yescharacteristic
533D4—32no—
534D4—32no—
535D4—34no—
536D4—61no—
537D4—61no—
538D4—64no—
539D4—182no—
540D4—181no—
541D4—181no—
542D4—182no—
543D4—184no—
544D4—182no—
545D4—362no—
546C2×C4—22no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—32no—
550C2×C4—32no—
551C2×C4—31no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—91no—
562C2×C4—91no—
563C2×C4—91no—
564C2×C4—181no—
565C2×C4—181no—
566C2×C4—181no—
567C2×C4—181no—
568C2×C4—182no—
569C2×C4—182no—
570C2×C4—182no—
571C2×C4—361no—
572C2×C4C2×D3×A411yescharacteristic
573C2×C4C2×D3×A411yescharacteristic
574C2×C3—22no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—41no—
578C2×C3—62no—
579C2×C3—82no—
580C2×C3—84no—
581C2×C3—81no—
582C2×C3—81no—
583C2×C3—81no—
584C2×C3—121no—
585C2×C3—242no—
586C2×C3C2×D4×A411yescharacteristic
587C2×C3C22×#41⋊C311yescharacteristic
588C2×C3#594511yescharacteristic
589D3—22no—
590D3—121no—
591C22—21no—
592C22—21no—
593C22—32no—
594C22—32no—
595C22—61no—
596C22—61no—
597C22—61no—
598C22—61no—
599C22—61no—
600C22—61no—
601C22—61no—
602C22—124no—
603C22—181no—
604C22—181no—
605C22—181no—
606C22—364no—
607C22D3×#41⋊C312yescomplemented
608C22C22×D3×A411yescharacteristic
609C4—21no—
610C4—32no—
611C4—31no—
612C4—31no—
613C4—61no—
614C4—61no—
615C4—61no—
616C4—181no—
617C4—181no—
618C4—181no—
619C4A4×#6712yes—
620C3—41no—
621C3—81no—
622C3D4×#41⋊C311yescharacteristic, complemented
623C2—22no—
624C2—62no—
625C2—62no—
626C2—121no—
627C2—361no—
628C2C2×D3×#41⋊C311yescharacteristic
629C2#5964⋊C311yescharacteristic
630C2C2×A4×#6711yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC22×C3×#41C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×D3×#41⋊C3C2semidirect
3. #5831⋊C3C2semidirect
4. #5834⋊C3C2semidirect
5. Dic3×#41⋊C3C2semidirect
6. SL(2,𝔽3)×#67C2semidirect
7. #5817⋊C3C2semidirect
8. #41×#67C3semidirect
12. C2×C3×#41⋊C3C22semidirect
13. C22×C3×SL(2,𝔽3)C22semidirect
15. C2×D3×SL(2,𝔽3)C22semidirect
16. C2×C3⋊(#41⋊C3)C22semidirect
18. Dic3×SL(2,𝔽3)C22semidirect
19. Dic3×SL(2,𝔽3)C22semidirect
20. C22×C3×#41C2×C3semidirect
21. C2×D3×#41C2×C3semidirect
24. C22×#41⋊C3D3semidirect
25. #5834C2×C3semidirect
27. #5831C2×C3semidirect
30. Dic3×#41C2×C3semidirect
32. #5817C2×C3semidirect
47. C3×#41⋊C3D4semidirect
82. C2×#41⋊C3C2×D3semidirect
85. C22×SL(2,𝔽3)C2×D3semidirect
135. C3×SL(2,𝔽3)C2×D4semidirect
137. C22×#41C3×D3semidirect
160. C3×#41C3×D4semidirect
214. #41⋊C3#67direct (N × H)
364. #67#41⋊C3direct (N × H)
385. SL(2,𝔽3)C2×#67semidirect
406. #41C3×#67semidirect
488. C22×C3C2×#41⋊C3semidirect
496. C2×D3C2×#41⋊C3semidirect
514. Dic3C2×#41⋊C3semidirect
607. C22D3×#41⋊C3semidirect
622. C3D4×#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×D3×#41⋊C32
#5831⋊C32
#5834⋊C32
Dic3×#41⋊C32
SL(2,𝔽3)×#672
#5817⋊C32
#41×#673
D4×#41⋊C33
C3×C4×#674