Finite Group Databasebeta

← #13155996

Subgroups — #13155996

Order 1728.

758 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 102 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×#88811C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic
3#11723410C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5#11722050C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591444—31nomaximal
9#10590876—31nomaximal
10#10590553—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×#6290—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88811—22no—
21#88807—21no—
22Dic3×C3⋊Dic3—21no—
23C3×#3716—22no—
24#64149—31no—
25#64096—31no—
26C2×C3×#214—31no—
27C2×C3×#204—31no—
28#63914—31no—
29#63914—31no—
30#63914—61no—
31#63843—31no—
32#63843—31no—
33#63843—61no—
34C2×#3717—31no—
35C2×#3716—31no—
36C2×#3716D311yescharacteristic, complemented
37C2×#3715—31no—
38#63641—31no—
39C3×#832—31no—
40C3×#821—31no—
41C3×#809—31no—
42C3×#809—31no—
43#63572—31no—
44#63572—31no—
45#63572—61no—
46#63564D311yescharacteristic, complemented
47#63562D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C2×C3⋊(C3⋊Dic3)D411yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×#634—21no—
57C3×#634D412yescomplemented
58C2×C32×Dic3—21no—
59C32×D12—21no—
60C32×D12—21no—
61#6290—42no—
62#6290—42no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65#5163—91no—
66#4762—91no—
67#4762—91no—
68C2×C32×D4—31no—
69C2×#638—61no—
70C2×#638—61no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—21no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12C2×D311yescharacteristic
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×#625—61no—
83C2×#625—61no—
84C2×#625—61no—
85C2×#625—61no—
86C2×#625—62no—
87C2×#625—62no—
88C2×C32×C8—31no—
89C32×C4⋊C4—31no—
90#3750—61no—
91C4×C3⋊Dic3—31no—
92C4×C3⋊Dic3—31no—
93C4×C3⋊Dic3—31no—
94C4×C3⋊Dic3—61no—
95C4×C3⋊Dic3—61no—
96C4×C3⋊Dic3—61no—
97C4×C3⋊Dic3—61no—
98C4×C3⋊Dic3—121no—
99C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
100C3×#214—62no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4⋊Dic3C2×D311yescharacteristic
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C3×#204—62no—
111Dic3⋊Dic3—61no—
112Dic3⋊Dic3—61no—
113Dic3⋊Dic3—62no—
114Dic32—61no—
115Dic32—61no—
116Dic32—62no—
117#3717—62no—
118#3716—22no—
119#3716—62no—
120#3715—62no—
121C22×C33C2×D411yescharacteristic
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C33×C4C2×D411yescharacteristic
129C33×C4#4111yescharacteristic
130C3⋊(C3⋊Dic3)—21no—
131C3⋊(C3⋊Dic3)SD1612yescomplemented
132C3×C3⋊Dic3—22no—
133C32×Dic3—41no—
134C2×C3×SD16—91no—
135Q8×Dic3—91no—
136C2×#215—91no—
137#940—91no—
138#940—91no—
139#940—91no—
140#940—181no—
141C2×#214—31no—
142C2×#214—91no—
143C2×#204—31no—
144C2×#204—91no—
145#887—91no—
146#887—91no—
147#887—181no—
148C3×#102—91no—
149C3×#102—91no—
150#832—31no—
151#821—31no—
152Dic3⋊C8—91no—
153Dic3⋊C8—91no—
154Dic3⋊C8—91no—
155Dic3⋊C8—181no—
156#809—31no—
157#809—31no—
158C23×C32—61no—
159C22×C3⋊D3—21no—
160C22×C3×D3—22no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—62no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C32×D4—61no—
167C32×D4—61no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—41no—
172C2×C32×C4—61no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176#638—122no—
177#638—122no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—61no—
186C2×C3⋊Dic3—61no—
187C2×C3⋊Dic3—61no—
188C2×C3⋊Dic3—61no—
189C2×C3⋊Dic3—61no—
190C2×C3⋊Dic3—61no—
191C2×C3⋊Dic3—121no—
192C2×C3⋊Dic3—121no—
193C4⋊(C3⋊D3)—21no—
194C4⋊(C3⋊D3)—21no—
195#634—21no—
196#634#6712yescomplemented
197C2×C3×Dic3—22no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—41no—
208C3×D12—41no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—121no—
214C3×D12—121no—
215C3×Dic6—32no—
216C3×Dic6—32no—
217C3×Dic6—62no—
218C3×Dic6—62no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221#625—122no—
222#625—122no—
223#625—124no—
224#625—124no—
225#625—122no—
226#625—122no—
227C32×C8—61no—
228C3⋊(C3⋊C8)—21no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—21no—
231C3×C3⋊C8—41no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—121no—
235#460—271noSylow
236C2×C33#12111yescharacteristic
237C2×C33C2×SD1611yescharacteristic
238C2×C33AGL(1,ℤ/8)11yescharacteristic
239C3×C3⋊D3—81no—
240C32×D3—81no—
241C2×C3×Q8—91no—
242C2×C3×D4—31no—
243C2×C3×D4—31no—
244C2×C3×D4—61no—
245C2×C3×D4—91no—
246C2×#67—181no—
247C2×#67—181no—
248C2×#67—181no—
249C2×#67—181no—
250C2×#67—181no—
251C2×#67—181no—
252C2×#67—182no—
253C2×#67—182no—
254C2×D12—31no—
255C2×D12—31no—
256C2×D12—61no—
257C2×D12D3211yescharacteristic
258C2×Dic6—31no—
259C2×Dic6—31no—
260C2×Dic6—61no—
261C3×SD16—182no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—61no—
265C2×C3×C8—91no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—61no—
269#215—182no—
270#214—62no—
271#214—182no—
272C4⋊Dic3—181no—
273C4⋊Dic3D3211yescharacteristic
274Dic3⋊C4—181no—
275Dic3⋊C4—181no—
276Dic3⋊C4—182no—
277C4×Dic3—91no—
278C4×Dic3—91no—
279C4×Dic3—91no—
280C4×Dic3—181no—
281C4×Dic3—181no—
282C4×Dic3—181no—
283C4×Dic3—181no—
284C4×Dic3—181no—
285C4×Dic3—181no—
286C4×Dic3—182no—
287C4×Dic3—361no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8D3211yescharacteristic
292#204—62no—
293#204—182no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—41no—
298C22×C32—121no—
299C22×C32—121no—
300C22×C32—121no—
301C22×C32D3×D411yescharacteristic
302C22×C32D3×D411yescharacteristic
303C22×C32D3×D411yescharacteristic
304C2×C3⋊D3—41no—
305C2×C3⋊D3—41no—
306C2×C3⋊D3—41no—
307C2×C3×D3—42no—
308C2×C3×D3—42no—
309C2×C3×D3—42no—
310C2×C3×D3—41no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C2×C3×D3—41no—
316C2×C3×D3—122no—
317C2×C3×D3—122no—
318C2×C3×D3—122no—
319C2×C3×D3—121no—
320C2×C3×D3—121no—
321C2×C3×D3—121no—
322C2×C3×D3—121no—
323C2×C3×D3—121no—
324C2×C3×D3—121no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—121no—
334C32×C4#23511yescharacteristic
335C32×C4#23511yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4#23711yescharacteristic
339C32×C4C2×#6711yescharacteristic
340C3⋊Dic3—22no—
341C3⋊Dic3—32no—
342C3⋊Dic3—32no—
343C3⋊Dic3—32no—
344C3⋊Dic3—62no—
345C3⋊Dic3—62no—
346C3⋊Dic3—62no—
347C3⋊Dic3—61no—
348C3⋊Dic3—62no—
349C3⋊Dic3—61no—
350C3⋊Dic3—61no—
351C3⋊Dic3—122no—
352C3⋊Dic3—121no—
353C3⋊Dic3—121no—
354C3⋊Dic3—122no—
355C3×Dic3—42no—
356C3×Dic3—41no—
357C3×Dic3—41no—
358C3×Dic3—64no—
359C3×Dic3—62no—
360C3×Dic3—62no—
361C2×SD16—91no—
362C2×SD16—271no—
363#127—271no—
364C4×Q8—271no—
365C4⋊C8—271no—
366#102—91no—
367#102—91no—
368C33#46011yescharacteristic, complemented, Sylow
369C23×C3—62no—
370C23×C3—61no—
371C23×C3—61no—
372C23×C3—181no—
373C22×D3—21no—
374C22×D3—62no—
375C22×D3—61no—
376C22×D3—61no—
377C3×Q8—92no—
378C3×Q8—181no—
379C3×D4—61no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—121no—
384C3×D4—121no—
385C3×D4—181no—
386C3×D4—181no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—41no—
391C2×C3×C4—62no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—181no—
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398#67—362no—
399#67—362no—
400#67—362no—
401#67—362no—
402#67—364no—
403#67—364no—
404#67—362no—
405#67—362no—
406C2×Dic3—21no—
407C2×Dic3—62no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—91no—
411C2×Dic3—91no—
412C2×Dic3—91no—
413C2×Dic3—91no—
414C2×Dic3—91no—
415C2×Dic3—91no—
416C2×Dic3—181no—
417C2×Dic3—181no—
418C2×Dic3—181no—
419C2×Dic3—181no—
420C2×Dic3—181no—
421C2×Dic3—181no—
422C2×Dic3—361no—
423C2×Dic3—361no—
424D12—21no—
425D12—21no—
426D12—61no—
427D12—61no—
428D12—61no—
429D12—61no—
430D12—121no—
431D12—121no—
432Dic6—32no—
433Dic6—32no—
434Dic6—62no—
435Dic6—62no—
436Dic6—61no—
437Dic6—61no—
438C3×C8—61no—
439C3×C8—61no—
440C3×C8—121no—
441C3×C8—181no—
442C3⋊C8—21no—
443C3⋊C8—61no—
444C3⋊C8—61no—
445C3⋊C8—121no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32—41no—
458C2×C32—241no—
459C2×C32#88411yescharacteristic
460C2×C32#88411yescharacteristic
461C2×C32#91111yescharacteristic
462C2×C32#91111yescharacteristic
463C2×C32D3×SD1611yescharacteristic
464C2×C32D3×SD1611yescharacteristic
465C2×C32D3×SD1611yescharacteristic
466C2×C32#91411yescharacteristic
467C2×C32#93911yescharacteristic
468C3⋊D3—81no—
469C3×D3—82no—
470C3×D3—81no—
471C3×D3—81no—
472C3×D3—242no—
473C3×D3—241no—
474C3×D3—241no—
475C2×Q8—91no—
476C2×D4—31no—
477C2×D4—91no—
478C2×D4—541no—
479C2×D4—541no—
480SD16—182no—
481SD16—542no—
482C2×C8—31no—
483C2×C8—91no—
484C4⋊C4—31no—
485C4⋊C4—541no—
486C42—271no—
487C42—541no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—122no—
493C22×C3—122no—
494C22×C3—122no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—121no—
499C22×C3—121no—
500C22×C3—121no—
501C22×C3—361no—
502C22×C3—361no—
503C22×C3—361no—
504C22×C3#380211yescharacteristic
505C22×C3#381111yescharacteristic
506C22×C3#381111yescharacteristic
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—41no—
510C2×D3—122no—
511C2×D3—122no—
512C2×D3—122no—
513C2×D3—121no—
514C2×D3—121no—
515C2×D3—121no—
516C2×D3—121no—
517C2×D3—121no—
518C2×D3—121no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—122no—
528C3×C4—121no—
529C3×C4—121no—
530C3×C4—182no—
531C3×C4#379511yescharacteristic
532C3×C4#379811yescharacteristic
533C3×C4#380211yescharacteristic
534C3×C4#380411yescharacteristic
535C3×C4D3×#6711yescharacteristic
536C3×C4D3×#6711yescharacteristic
537Dic3—41no—
538Dic3—64no—
539Dic3—62no—
540Dic3—62no—
541Dic3—92no—
542Dic3—92no—
543Dic3—92no—
544Dic3—182no—
545Dic3—182no—
546Dic3—182no—
547Dic3—182no—
548Dic3—181no—
549Dic3—181no—
550Dic3—181no—
551Dic3—362no—
552Dic3—361no—
553Dic3—361no—
554Dic3—362no—
555C32—21no—
556C32—21no—
557C32—21no—
558C32—41no—
559C32#476211yescharacteristic, complemented
560C32#476211yescharacteristic, complemented
561C32#516311yescharacteristic, complemented
562C23—61no—
563C23—181no—
564Q8—92no—
565Q8—181no—
566D4—61no—
567D4—61no—
568D4—181no—
569D4—181no—
570D4—1082no—
571D4—1082no—
572C2×C4—61no—
573C2×C4—181no—
574C2×C4—271no—
575C2×C4—271no—
576C2×C4D3311yescharacteristic
577C8—61no—
578C8—181no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—41no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—242no—
592C2×C3—241no—
593C2×C3—241no—
594C2×C3—721no—
595C2×C3#6379611yescharacteristic
596C2×C3#6379811yescharacteristic
597C2×C3#6388511yescharacteristic
598C2×C3#6391211yescharacteristic
599C2×C3#6392711yescharacteristic
600C2×C3#6393011yescharacteristic
601C2×C3#6393711yescharacteristic
602C2×C3#6394011yescharacteristic
603C2×C3#6397111yescharacteristic
604D3—81no—
605D3—242no—
606D3—241no—
607D3—241no—
608C22—121no—
609C22—121no—
610C22—121no—
611C22—361no—
612C22—361no—
613C22—361no—
614C22#8895411yescharacteristic
615C4—121no—
616C4—182no—
617C4—272no—
618C4—541no—
619C4#8895311yescharacteristic
620C4#8896011yescharacteristic
621C3—21no—
622C3—21no—
623C3—21no—
624C3—41no—
625C3#1059055311yescharacteristic, complemented
626C3#1059087611yescharacteristic, complemented
627C3#1059144411yescharacteristic, complemented
628C2—241no—
629C2—721no—
630C2#1172284111yescharacteristic
631C2#1172287711yescharacteristic
632C2#1172296711yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723529C2
hypocenterC33#460
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88811C2semidirect
3. #11723410C2semidirect
4. C2×C3×#3716C2semidirect
5. #11722050C2semidirect
6. C3×#63564C2semidirect
7. C3×#63562C2semidirect
36. C2×#3716D3semidirect
46. #63564D3semidirect
47. #63562D3semidirect
57. C3×#634D4semidirect
131. C3⋊(C3⋊Dic3)SD16semidirect
196. #634#67semidirect
368. C33#460semidirect
559. C32#4762semidirect
560. C32#4762semidirect
561. C32#5163semidirect
625. C3#10590553semidirect
626. C3#10590876semidirect
627. C3#10591444semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117235292
#117234102
C2×C3×#37162
#117220502
C3×#635642
C3×#635622
#105914443
#105908763
#105905533