Finite Group Databasebeta

← #13155995

Subgroups — #13155995

Order 1728.

792 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic, complemented
2#11723524C211yesmaximal, characteristic
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591444—31nomaximal
9#10590879—31nomaximal
10#10590799—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—21no—
15C2×#6291—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88815—21no—
21#88813—21no—
22#88797—22no—
23C3×#3716—22no—
24#64149—31no—
25C2×#3744—31no—
26C2×C3×#214—31no—
27C2×C3×#204—31no—
28#63911—31no—
29#63911—31no—
30#63911—61no—
31#63844—31no—
32#63843—31no—
33Dic3×Dic6—31no—
34C2×#3717—31no—
35C2×#3717—31no—
36C2×#3717—61no—
37C2×#3716D311yescharacteristic, complemented
38#63648—31no—
39C3×#832—31no—
40C3×#821—31no—
41C3×Dic3⋊C8—31no—
42C3×Dic3⋊C8—31no—
43#63572D311yescharacteristic, complemented
44#63564—31no—
45#63564—31no—
46#63564—61no—
47#63563D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C4⋊(C3⋊(C3⋊D3))—21no—
52C4⋊(C3⋊(C3⋊D3))—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×#634—21no—
58C3×#634D412yescomplemented
59C32×D12—21no—
60C32×D12—21no—
61#6291—42no—
62#6291—42no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8—21no—
65#5163—91no—
66#4848—91no—
67#4762—91no—
68C2×C32×D4—31no—
69C2×#638—61no—
70C2×#638—61no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—61no—
77C2×C4⋊(C3⋊D3)—121no—
78C2×#634C2×D311yescharacteristic
79C2×C3×D12—21no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—61no—
85C2×#626—61no—
86C2×#626—61no—
87C2×#626—61no—
88C2×#626—61no—
89C2×#626—62no—
90C2×#626—62no—
91C2×C32×C8—31no—
92C2×C32×C8—31no—
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94C4×C3⋊Dic3C2×D311yescharacteristic
95#3744—62no—
96C3×#214—62no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—61no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—61no—
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C3×#204—62no—
110#3724—61no—
111Dic3⋊Dic3—61no—
112Dic3⋊Dic3—61no—
113Dic32—61no—
114Dic32—61no—
115Dic32—61no—
116#3717—62no—
117#3717—62no—
118#3717—64no—
119#3716—22no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—41no—
122C2×C3⋊(C3⋊D3)—41no—
123C2×C3⋊(C3⋊D3)—41no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3—21no—
131C3×C3⋊Dic3—41no—
132C3×C3⋊Dic3SD1612yescomplemented
133C2×C3×SD16—91no—
134Q8×Dic3—91no—
135C2×#215—91no—
136#940—91no—
137#940—91no—
138#940—181no—
139C2×#214—31no—
140C2×#204—31no—
141C2×#204—91no—
142C2×#204—91no—
143C2×#204—181no—
144#887—91no—
145#874—91no—
146C4×Dic6—91no—
147C3×C4⋊C8—91no—
148C3×#102—91no—
149#832—31no—
150#821—31no—
151#821—91no—
152#821—91no—
153#821—181no—
154Dic3⋊C8—31no—
155Dic3⋊C8—31no—
156#809—91no—
157C23×C32—61no—
158C22×C3⋊D3—61no—
159C22×C3⋊D3—62no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—61no—
162C22×C3⋊D3—121no—
163C22×C3⋊D3—121no—
164C22×C3⋊D3—122no—
165C22×C3×D3—22no—
166C22×C3×D3—21no—
167C22×C3×D3—21no—
168C32×D4—61no—
169C32×D4—61no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177#638—122no—
178#638—122no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3D1211yescharacteristic
182C2×C3⋊Dic3D1211yescharacteristic
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—62no—
186C4⋊(C3⋊D3)—62no—
187C4⋊(C3⋊D3)—61no—
188C4⋊(C3⋊D3)—61no—
189C4⋊(C3⋊D3)—61no—
190C4⋊(C3⋊D3)—61no—
191C4⋊(C3⋊D3)—121no—
192C4⋊(C3⋊D3)—121no—
193C4⋊(C3⋊D3)—121no—
194C4⋊(C3⋊D3)—121no—
195C4⋊(C3⋊D3)—122no—
196C4⋊(C3⋊D3)—122no—
197#634—21no—
198#634#6712yescomplemented
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—121no—
210C2×C3×Dic3—121no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—21no—
215C3×D12—41no—
216C3×D12—41no—
217C3×Dic6—32no—
218C3×Dic6—32no—
219C3×Dic6—62no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—121no—
223#626—122no—
224#626—122no—
225#626—124no—
226#626—124no—
227#626—122no—
228#626—122no—
229C32×C8—61no—
230C32×C8—61no—
231C3×C3⋊C8—22no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—21no—
234C3×C3⋊C8—21no—
235C3×C3⋊C8—21no—
236C3×C3⋊C8—41no—
237#460—271noSylow
238C2×C33#12111yescharacteristic
239C2×C33C2×SD1611yescharacteristic
240C2×C33AGL(1,ℤ/8)11yescharacteristic
241C3⋊(C3⋊D3)—81no—
242C32×D3—81no—
243C2×C3×Q8—91no—
244C2×C3×D4—31no—
245C2×C3×D4—31no—
246C2×C3×D4—61no—
247C2×#67—181no—
248C2×#67—181no—
249C2×#67—181no—
250C2×#67—181no—
251C2×#67—182no—
252C2×#67—182no—
253C2×D12—91no—
254C2×D12—91no—
255C2×D12—91no—
256C2×D12—181no—
257C2×D12—181no—
258C2×D12—181no—
259C2×D12—181no—
260C2×D12—181no—
261C2×D12—361no—
262C2×D12D3211yescharacteristic
263C2×Dic6—31no—
264C2×Dic6—31no—
265C2×Dic6—61no—
266C3×SD16—182no—
267C2×C3×C8—31no—
268C2×C3×C8—31no—
269C2×C3×C8—31no—
270C2×C3×C8—31no—
271C2×C3×C8—61no—
272C2×C3×C8—61no—
273C3×C4⋊C4—91no—
274C3×C42—91no—
275#215—182no—
276#214—62no—
277C4⋊Dic3—31no—
278C4⋊Dic3—31no—
279C4⋊Dic3—61no—
280C4⋊Dic3—181no—
281Dic3⋊C4—181no—
282Dic3⋊C4—181no—
283C4×Dic3—31no—
284C4×Dic3—31no—
285C4×Dic3—61no—
286C4×Dic3—181no—
287C4×Dic3—181no—
288C4×Dic3—181no—
289C2×C3⋊C8D3211yescharacteristic
290C2×C3⋊C8D3211yescharacteristic
291#204—62no—
292#204—182no—
293#204—182no—
294#204—184no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—121no—
300C22×C32—121no—
301C22×C32—121no—
302C22×C32C2×D1211yescharacteristic
303C22×C32D3×D411yescharacteristic
304C22×C32D3×D411yescharacteristic
305C2×C3⋊D3—122no—
306C2×C3⋊D3—122no—
307C2×C3⋊D3—122no—
308C2×C3⋊D3—124no—
309C2×C3⋊D3—121no—
310C2×C3⋊D3—121no—
311C2×C3⋊D3—121no—
312C2×C3⋊D3—121no—
313C2×C3⋊D3—121no—
314C2×C3⋊D3—122no—
315C2×C3⋊D3—122no—
316C2×C3⋊D3—121no—
317C2×C3⋊D3—121no—
318C2×C3⋊D3—121no—
319C2×C3⋊D3—121no—
320C2×C3⋊D3—241no—
321C2×C3⋊D3—241no—
322C2×C3⋊D3—241no—
323C2×C3⋊D3—241no—
324C2×C3⋊D3—242no—
325C2×C3⋊D3—242no—
326C2×C3×D3—42no—
327C2×C3×D3—42no—
328C2×C3×D3—42no—
329C2×C3×D3—41no—
330C2×C3×D3—41no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—41no—
342C32×C4—41no—
343C32×C4#23511yescharacteristic
344C32×C4D3×D411yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4#23711yescharacteristic
347C32×C4#23911yescharacteristic
348C32×C4C2×#6711yescharacteristic
349C3⋊Dic3—21no—
350C3⋊Dic3—22no—
351C3⋊Dic3—41no—
352C3⋊Dic3#20412yescomplemented
353C3×Dic3—32no—
354C3×Dic3—32no—
355C3×Dic3—62no—
356C3×Dic3—62no—
357C3×Dic3—62no—
358C3×Dic3—62no—
359C3×Dic3—61no—
360C3×Dic3—61no—
361C3×Dic3—122no—
362C3×Dic3—121no—
363C3×Dic3—121no—
364C3×Dic3—241no—
365C2×SD16—91no—
366C2×SD16—271no—
367#127—271no—
368C4×Q8—271no—
369C4⋊C8—91no—
370#102—91no—
371#102—271no—
372C33#46011yescharacteristic, complemented, Sylow
373C23×C3—62no—
374C23×C3—61no—
375C23×C3—61no—
376C22×D3—21no—
377C22×D3—182no—
378C22×D3—181no—
379C22×D3—181no—
380C22×D3—181no—
381C22×D3—361no—
382C22×D3—361no—
383C22×D3—362no—
384C3×Q8—92no—
385C3×Q8—181no—
386C3×D4—61no—
387C3×D4—61no—
388C3×D4—61no—
389C3×D4—61no—
390C3×D4—121no—
391C3×D4—121no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—91no—
397C2×C3×C4—91no—
398C2×C3×C4—181no—
399C2×C3×C4—181no—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403#67—362no—
404#67—362no—
405#67—364no—
406#67—364no—
407#67—362no—
408#67—362no—
409C2×Dic3—31no—
410C2×Dic3—31no—
411C2×Dic3—31no—
412C2×Dic3—31no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—61no—
416C2×Dic3—61no—
417C2×Dic3—61no—
418C2×Dic3—61no—
419C2×Dic3—121no—
420C2×Dic3—121no—
421D12—21no—
422D12—21no—
423D12—182no—
424D12—182no—
425D12—181no—
426D12—181no—
427D12—181no—
428D12—181no—
429D12—181no—
430D12—181no—
431D12—361no—
432D12—361no—
433D12—361no—
434D12—361no—
435D12—362no—
436D12—362no—
437D12—362no—
438D12—362no—
439Dic6—32no—
440Dic6—32no—
441Dic6—62no—
442Dic6—61no—
443Dic6—61no—
444Dic6—121no—
445C3×C8—62no—
446C3×C8—61no—
447C3×C8—61no—
448C3×C8—61no—
449C3×C8—61no—
450C3×C8—121no—
451C3⋊C8—21no—
452C3⋊C8—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—41no—
463C2×C32—41no—
464C2×C32—41no—
465C2×C32—241no—
466C2×C32#88411yescharacteristic
467C2×C32#89511yescharacteristic
468C2×C32C2×#20411yescharacteristic
469C2×C32#90811yescharacteristic
470C2×C32#91111yescharacteristic
471C2×C32D3×SD1611yescharacteristic
472C2×C32D3×SD1611yescharacteristic
473C2×C32#91411yescharacteristic
474C2×C32#93911yescharacteristic
475C3⋊D3—242no—
476C3⋊D3—242no—
477C3⋊D3—242no—
478C3⋊D3—244no—
479C3⋊D3—241no—
480C3⋊D3—241no—
481C3⋊D3—241no—
482C3×D3—82no—
483C3×D3—81no—
484C3×D3—81no—
485C2×Q8—91no—
486C2×D4—31no—
487C2×D4—271no—
488C2×D4—541no—
489C2×D4—541no—
490SD16—182no—
491SD16—542no—
492C2×C8—31no—
493C2×C8—31no—
494C4⋊C4—91no—
495C4⋊C4—541no—
496C42—91no—
497C42—541no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—41no—
502C22×C3—122no—
503C22×C3—122no—
504C22×C3—122no—
505C22×C3—121no—
506C22×C3—121no—
507C22×C3—121no—
508C22×C3—121no—
509C22×C3—121no—
510C22×C3—121no—
511C22×C3D3×D1211yescharacteristic
512C22×C3D3×D1211yescharacteristic
513C22×C3#381111yescharacteristic
514C2×D3—41no—
515C2×D3—41no—
516C2×D3—41no—
517C2×D3—362no—
518C2×D3—362no—
519C2×D3—362no—
520C2×D3—362no—
521C2×D3—362no—
522C2×D3—364no—
523C2×D3—361no—
524C2×D3—361no—
525C2×D3—361no—
526C2×D3—361no—
527C2×D3—361no—
528C2×D3—361no—
529C2×D3—361no—
530C2×D3—361no—
531C2×D3—361no—
532C2×D3—721no—
533C2×D3—721no—
534C2×D3—721no—
535C2×D3—721no—
536C2×D3—722no—
537C2×D3—722no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—92no—
547C3×C4—182no—
548C3×C4—181no—
549C3×C4—361no—
550C3×C4#379611yescharacteristic
551C3×C4#379911yescharacteristic
552C3×C4#380411yescharacteristic
553C3×C4D3×#6711yescharacteristic
554C3×C4D3×#6711yescharacteristic
555C3×C4#381111yescharacteristic
556Dic3—32no—
557Dic3—32no—
558Dic3—62no—
559Dic3—62no—
560Dic3—61no—
561Dic3—61no—
562Dic3—62no—
563Dic3—62no—
564Dic3—122no—
565Dic3—121no—
566Dic3—121no—
567Dic3—241no—
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#476211yescharacteristic, complemented
573C32#484811yescharacteristic, complemented
574C32#516311yescharacteristic, complemented
575C23—61no—
576C23—541no—
577Q8—92no—
578Q8—181no—
579D4—61no—
580D4—61no—
581D4—541no—
582D4—541no—
583D4—1082no—
584D4—1082no—
585C2×C4—91no—
586C2×C4—91no—
587C2×C4—181no—
588C2×C4—181no—
589C2×C4D3311yescharacteristic
590C8—61no—
591C8—61no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—41no—
604C2×C3—242no—
605C2×C3—241no—
606C2×C3—241no—
607C2×C3D3×#20411yescharacteristic
608C2×C3D3×#20411yescharacteristic
609C2×C3#6379411yescharacteristic
610C2×C3#6379511yescharacteristic
611C2×C3#6388611yescharacteristic
612C2×C3#6391011yescharacteristic
613C2×C3#6393011yescharacteristic
614C2×C3#6393711yescharacteristic
615C2×C3#6397111yescharacteristic
616D3—81no—
617D3—722no—
618D3—722no—
619D3—722no—
620D3—724no—
621D3—721no—
622D3—721no—
623D3—721no—
624C22—121no—
625C22—121no—
626C22—121no—
627C22—1081no—
628C22—1081no—
629C22—1081no—
630C22#8895511yescharacteristic
631C4—92no—
632C4—181no—
633C4—182no—
634C4—361no—
635C4#8895211yescharacteristic
636C4#8896211yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059079911yescharacteristic, complemented
642C3#1059087911yescharacteristic, complemented
643C3#1059144411yescharacteristic, complemented
644C2—241no—
645C2—2161no—
646C2#1172284211yescharacteristic
647C2#1172287911yescharacteristic
648C2#1172296611yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723524C2
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. #11723668C2semidirect
3. C2×#88797C2semidirect
4. C2×C3×#3716C2semidirect
5. #11722036C2semidirect
6. C3×#63572C2semidirect
7. C3×#63563C2semidirect
37. C2×#3716D3semidirect
43. #63572D3semidirect
47. #63563D3semidirect
58. C3×#634D4semidirect
132. C3×C3⋊Dic3SD16semidirect
198. #634#67semidirect
352. C3⋊Dic3#204semidirect
372. C33#460semidirect
572. C32#4762semidirect
573. C32#4848semidirect
574. C32#5163semidirect
641. C3#10590799semidirect
642. C3#10590879semidirect
643. C3#10591444semidirect

Maximal subgroups

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

subgroupindex
#117236682
#117235242
C2×#887972
C2×C3×#37162
#117220362
C3×#635722
C3×#635632
#105914443
#105908793
#105907993