Finite Group Databasebeta

← #13155835

Subgroups — #13155835

Order 1728.

960 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89014C211yesmaximal, characteristic, complemented
2#11723696C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic, complemented
5#11722042C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic, complemented
8#10591457—31nomaximal
9#10590968—31nomaximal
10#10590962—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6295—21no—
16C2×#6290—21no—
17C2×D3×C3⋊Dic3—21no—
18#89014—22no—
19#89014—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88814—21no—
24#88796—22no—
25C3×#3714—22no—
26C2×#3830—31no—
27C2×#3795—31no—
28C2×#3795—31no—
29C2×#3795—61no—
30C2×#3744—31no—
31C2×C3×#213—31no—
32C2×C3×D24—31no—
33#63916—31no—
34#63909—31no—
35#63908—31no—
36C2×#3716—31no—
37C2×#3716—31no—
38C2×#3716—61no—
39C2×#3714D311yescharacteristic, complemented
40#63643—31no—
41C3×#832—31no—
42C3×#821—31no—
43C3×(C2×D3)⋊C8—31no—
44C3×(C2×D3)⋊C8—31no—
45#63567—31no—
46#63567—31no—
47#63567—61no—
48#63563D311yescharacteristic, complemented
49#63557D311yescharacteristic, complemented
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C22×C32×D3—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)—21no—
55#6308—21no—
56#6308D412yescomplemented
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3D411yescharacteristic
59C3×C4⋊(C3⋊D3)—21no—
60C3×C4⋊(C3⋊D3)—21no—
61C3×C4×C3⋊D3—21no—
62C3×C4×C3⋊D3—21no—
63C32×D12—21no—
64C32×D12D412yescomplemented
65#6295—42no—
66#6290—24no—
67D3×C3⋊Dic3—24no—
68C32×C3⋊C8—21no—
69C32×C3⋊C8—21no—
70#5158—91no—
71#4781—91no—
72#4734—91no—
73C2×C32×D4—31no—
74C2×#638—61no—
75C22×C3⋊Dic3—61no—
76#3830—62no—
77#3830—62no—
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
79C2×C4×C3⋊D3C2×D311yescharacteristic
80C2×#634—31no—
81C2×#634—31no—
82C2×#634—31no—
83C2×#634—61no—
84C2×#634—61no—
85C2×#634—61no—
86C2×#634—121no—
87C2×C3×D12—21no—
88C2×C3×D12—31no—
89C2×C3×D12—31no—
90C2×C3×D12—61no—
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×D12C2×D311yescharacteristic
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—61no—
96C2×#626—61no—
97C2×#626—61no—
98C2×#625—61no—
99C2×#625—61no—
100C2×#625—62no—
101C2×#625—61no—
102C2×D3×Dic3—61no—
103C2×D3×Dic3—61no—
104C2×D3×Dic3—62no—
105#3795—62no—
106#3795—62no—
107#3795—64no—
108#3795—64no—
109#3795—62no—
110#3795—62no—
111C2×C32×C8—31no—
112C2×C32×C8—31no—
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic
114#3744—62no—
115C3×#213—62no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—61no—
119C2×C3×C3⋊C8—21no—
120C2×C3×C3⋊C8—21no—
121C2×C3×C3⋊C8C2×D311yescharacteristic
122C2×C3×C3⋊C8C2×D311yescharacteristic
123C2×C3×C3⋊C8C2×D311yescharacteristic
124C2×C3×C3⋊C8C2×D311yescharacteristic
125C3×D24—62no—
126#3722—61no—
127(C2×D3)⋊Dic3—61no—
128(C2×D3)⋊Dic3—61no—
129#3716—62no—
130#3716—62no—
131#3716—64no—
132#3714—22no—
133C22×C33C2×D411yescharacteristic
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—41no—
138C2×C3×C3⋊D3—41no—
139C2×C3×C3⋊D3—41no—
140C2×C32×D3—22no—
141C2×C32×D3—41no—
142C2×C32×D3—41no—
143C33×C4C2×D411yescharacteristic
144C33×C4C2×D411yescharacteristic
145C3⋊(C3⋊Dic3)—22no—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—41no—
148C2×#237—91no—
149C2×#237—91no—
150C2×#237—181no—
151C2×#235—91no—
152C2×C3×D8—91no—
153#938—91no—
154C2×#214—91no—
155C2×#213—31no—
156#930—91no—
157C2×D24—31no—
158C2×#204—91no—
159C2×#204—91no—
160C2×#204—181no—
161#895—91no—
162C3×#102—91no—
163C3×C22⋊C8—91no—
164#832—31no—
165#821—31no—
166(C2×D3)⋊C8—31no—
167(C2×D3)⋊C8—31no—
168#816—91no—
169#816—91no—
170#816—181no—
171#810—91no—
172C23×C32—61no—
173C22×C3⋊D3—21no—
174C22×C3⋊D3D1211yescharacteristic
175C22×C3×D3—22no—
176C22×C3×D3—21no—
177C22×C3×D3—21no—
178C22×C3×D3—31no—
179C22×C3×D3—31no—
180C22×C3×D3—61no—
181C22×C3×D3—61no—
182C22×C3×D3—61no—
183C22×C3×D3—121no—
184C32×D4—32no—
185C32×D4—61no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—41no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193#638—64no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—64no—
196C2×C3⋊Dic3—61no—
197C2×C3⋊Dic3—62no—
198C2×C3⋊Dic3—61no—
199C2×C3⋊Dic3—61no—
200C2×C3⋊Dic3—121no—
201C2×C3⋊Dic3—121no—
202C2×C3⋊Dic3—122no—
203C2×C3⋊Dic3D1211yescharacteristic
204C4⋊(C3⋊D3)—21no—
205C4⋊(C3⋊D3)—21no—
206C4×C3⋊D3—21no—
207C4×C3⋊D3—21no—
208#634—32no—
209#634—32no—
210#634—32no—
211#634—62no—
212#634—62no—
213#634—62no—
214#634—61no—
215#634—62no—
216#634—61no—
217#634—61no—
218#634—122no—
219#634—121no—
220#634—121no—
221#634—122no—
222C2×C3×Dic3—31no—
223C2×C3×Dic3—31no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—61no—
226C2×C3×Dic3—61no—
227C2×C3×Dic3—121no—
228C3×D12—22no—
229C3×D12—21no—
230C3×D12—21no—
231C3×D12—22no—
232C3×D12—62no—
233C3×D12—61no—
234C3×D12—61no—
235C3×D12—61no—
236C3×D12—61no—
237C3×D12—121no—
238C3×D12D1212yescomplemented
239C3×D12#6712yescomplemented
240C3×C4×D3—62no—
241C3×C4×D3—62no—
242C3×C4×D3—61no—
243C3×C4×D3—61no—
244C3×C4×D3—61no—
245C3×C4×D3—61no—
246#626—122no—
247#626—122no—
248#625—68no—
249#625—64no—
250#625—64no—
251#625—122no—
252D3×Dic3—68no—
253D3×Dic3—64no—
254D3×Dic3—64no—
255C32×C8—61no—
256C32×C8—61no—
257C3×C3⋊C8—22no—
258C3×C3⋊C8—21no—
259C3×C3⋊C8—21no—
260C3×C3⋊C8—21no—
261C3×C3⋊C8—21no—
262C3×C3⋊C8—41no—
263#449—271noSylow
264C2×C33#12011yescharacteristic
265C2×C33#13511yescharacteristic
266C2×C33AGL(1,ℤ/8)11yescharacteristic
267C3×C3⋊D3—41no—
268C3×C3⋊D3—81no—
269C32×D3—42no—
270C2×C3×D4—31no—
271C2×C3×D4—31no—
272C2×C3×D4—61no—
273C2×C3×D4—91no—
274C22×C3×C4—91no—
275C2×#67—181no—
276C2×#67—181no—
277C2×#67—181no—
278C2×#67—181no—
279C2×#67—181no—
280C2×#67—182no—
281C22×Dic3—181no—
282C22×Dic3—181no—
283C22×Dic3—182no—
284#237—182no—
285#237—182no—
286#237—182no—
287#237—182no—
288#237—184no—
289#237—184no—
290#235—182no—
291#235—182no—
292C2×D12—31no—
293C2×D12—31no—
294C2×D12—61no—
295C2×D12—181no—
296C2×D12D3211yescharacteristic
297C2×C4×D3—31no—
298C2×C4×D3—31no—
299C2×C4×D3—61no—
300C2×C4×D3—181no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—181no—
305C2×Dic6—181no—
306C2×Dic6—181no—
307C2×Dic6—361no—
308C3×D8—182no—
309C2×C3×C8—31no—
310C2×C3×C8—31no—
311C2×C3×C8—31no—
312C2×C3×C8—31no—
313C2×C3×C8—61no—
314C2×C3×C8—61no—
315C3×C4⋊C4—91no—
316#217—181no—
317#214—182no—
318#213—62no—
319(C2×D3)⋊C4—181no—
320(C2×D3)⋊C4—181no—
321C4⋊Dic3—31no—
322C4⋊Dic3—31no—
323C4⋊Dic3—61no—
324C2×C3⋊C8D3211yescharacteristic
325C2×C3⋊C8D3211yescharacteristic
326D24—62no—
327#204—182no—
328#204—182no—
329#204—184no—
330C22×C32—21no—
331C22×C32—21no—
332C22×C32—21no—
333C22×C32—41no—
334C22×C32—62no—
335C22×C32—121no—
336C22×C32—121no—
337C22×C32C2×D1211yescharacteristic
338C22×C32D3×D411yescharacteristic
339C22×C32D3×D411yescharacteristic
340C2×C3⋊D3—21no—
341C2×C3⋊D3—21no—
342C2×C3⋊D3—21no—
343C2×C3⋊D3—41no—
344C2×C3⋊D3—41no—
345C2×C3⋊D3—41no—
346C2×C3×D3—24no—
347C2×C3×D3—22no—
348C2×C3×D3—22no—
349C2×C3×D3—42no—
350C2×C3×D3—42no—
351C2×C3×D3—41no—
352C2×C3×D3—41no—
353C2×C3×D3—41no—
354C2×C3×D3—41no—
355C2×C3×D3—62no—
356C2×C3×D3—61no—
357C2×C3×D3—61no—
358C2×C3×D3—61no—
359C2×C3×D3—61no—
360C2×C3×D3—61no—
361C2×C3×D3—61no—
362C2×C3×D3—121no—
363C2×C3×D3—121no—
364C2×C3×D3—122no—
365C2×C3×D3—121no—
366C2×C3×D3—121no—
367C2×C3×D3—121no—
368C2×C3×D3—121no—
369C2×C3×D3—121no—
370C2×C3×D3—121no—
371C2×C3×D3—241no—
372C2×C3×D3—241no—
373C32×C4—21no—
374C32×C4—21no—
375C32×C4—21no—
376C32×C4—21no—
377C32×C4—21no—
378C32×C4—21no—
379C32×C4—41no—
380C32×C4—41no—
381C32×C4C2×D1211yescharacteristic
382C32×C4D3×D411yescharacteristic
383C32×C4D3×D411yescharacteristic
384C32×C4D3×D411yescharacteristic
385C32×C4D3×D411yescharacteristic
386C32×C4C2×#6711yescharacteristic
387C3⋊Dic3—21no—
388C3⋊Dic3—41no—
389C3⋊Dic3—62no—
390C3⋊Dic3—64no—
391C3⋊Dic3—62no—
392C3⋊Dic3—62no—
393C3⋊Dic3—122no—
394C3⋊Dic3—122no—
395C3⋊Dic3—124no—
396C3×Dic3—62no—
397C3×Dic3—61no—
398C3×Dic3—61no—
399C3×Dic3—121no—
400C3×Dic3—121no—
401C3×Dic3—241no—
402C2×#41—271no—
403C2×SD16—271no—
404C2×D8—91no—
405#121—271no—
406#103—271no—
407#102—91no—
408C22⋊C8—91no—
409C33#44911yescharacteristic, complemented, Sylow
410C23×C3—62no—
411C23×C3—61no—
412C23×C3—61no—
413C23×C3—91no—
414C23×C3—181no—
415C22×D3—21no—
416C22×D3—31no—
417C22×D3—31no—
418C22×D3—61no—
419C22×D3—61no—
420C22×D3—61no—
421C22×D3—121no—
422C3×D4—32no—
423C3×D4—32no—
424C3×D4—62no—
425C3×D4—62no—
426C3×D4—61no—
427C3×D4—61no—
428C3×D4—181no—
429C3×D4—181no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—41no—
434C2×C3×C4—91no—
435C2×C3×C4—181no—
436C2×C3×C4—181no—
437C2×C3×C4—181no—
438C2×C3×C4C2×D3211yescharacteristic
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C2×D3211yescharacteristic
441#67—184no—
442#67—188no—
443#67—184no—
444#67—184no—
445#67—362no—
446#67—362no—
447C2×Dic3—31no—
448C2×Dic3—31no—
449C2×Dic3—61no—
450C2×Dic3—61no—
451C2×Dic3—61no—
452C2×Dic3—121no—
453C2×Dic3—188no—
454C2×Dic3—184no—
455C2×Dic3—184no—
456C2×Dic3—182no—
457C2×Dic3—181no—
458C2×Dic3—181no—
459C2×Dic3—181no—
460C2×Dic3—361no—
461C2×Dic3—361no—
462C2×Dic3—362no—
463D12—21no—
464D12—62no—
465D12—61no—
466D12—61no—
467D12—61no—
468D12—61no—
469D12—121no—
470D12—362no—
471D12#62612yescomplemented
472C4×D3—62no—
473C4×D3—62no—
474C4×D3—61no—
475C4×D3—61no—
476C4×D3—61no—
477C4×D3—61no—
478C4×D3—184no—
479Dic6—92no—
480Dic6—92no—
481Dic6—92no—
482Dic6—182no—
483Dic6—182no—
484Dic6—182no—
485Dic6—182no—
486Dic6—181no—
487Dic6—181no—
488Dic6—181no—
489Dic6—362no—
490Dic6—361no—
491Dic6—361no—
492Dic6—362no—
493C3×C8—62no—
494C3×C8—61no—
495C3×C8—61no—
496C3×C8—61no—
497C3×C8—61no—
498C3×C8—121no—
499C3⋊C8—21no—
500C3⋊C8—21no—
501C2×C32—21no—
502C2×C32—21no—
503C2×C32—21no—
504C2×C32—21no—
505C2×C32—21no—
506C2×C32—21no—
507C2×C32—21no—
508C2×C32—21no—
509C2×C32—21no—
510C2×C32—41no—
511C2×C32—41no—
512C2×C32—41no—
513C2×C32—122no—
514C2×C32#88211yescharacteristic
515C2×C32#88211yescharacteristic
516C2×C32#90411yescharacteristic
517C2×C32#90811yescharacteristic
518C2×C32#91111yescharacteristic
519C2×C32#91111yescharacteristic
520C2×C32#91211yescharacteristic
521C2×C32#91611yescharacteristic
522C2×C32#93711yescharacteristic
523C3⋊D3—41no—
524C3⋊D3—81no—
525C3×D3—44no—
526C3×D3—42no—
527C3×D3—42no—
528C3×D3—122no—
529C3×D3—121no—
530C3×D3—121no—
531C3×D3—242no—
532C3×D3—241no—
533C3×D3—241no—
534#41—542no—
535#41—542no—
536C2×Q8—271no—
537C2×D4—31no—
538C2×D4—91no—
539C2×D4—541no—
540C2×D4—541no—
541C22×C4—91no—
542C22×C4—541no—
543SD16—542no—
544D8—182no—
545C2×C8—31no—
546C2×C8—31no—
547C4⋊C4—91no—
548C22⋊C4—541no—
549C22×C3—21no—
550C22×C3—21no—
551C22×C3—21no—
552C22×C3—41no—
553C22×C3—64no—
554C22×C3—62no—
555C22×C3—62no—
556C22×C3—122no—
557C22×C3—122no—
558C22×C3—121no—
559C22×C3—121no—
560C22×C3—121no—
561C22×C3—121no—
562C22×C3—181no—
563C22×C3—181no—
564C22×C3—181no—
565C22×C3—361no—
566C22×C3—361no—
567C22×C3—361no—
568C22×C3D3×D1211yescharacteristic
569C22×C3D3×D1211yescharacteristic
570C22×C3#381111yescharacteristic
571C2×D3—22no—
572C2×D3—41no—
573C2×D3—41no—
574C2×D3—62no—
575C2×D3—61no—
576C2×D3—61no—
577C2×D3—61no—
578C2×D3—61no—
579C2×D3—61no—
580C2×D3—61no—
581C2×D3—121no—
582C2×D3—121no—
583C2×D3—122no—
584C2×D3—121no—
585C2×D3—121no—
586C2×D3—121no—
587C2×D3—121no—
588C2×D3—121no—
589C2×D3—121no—
590C2×D3—241no—
591C2×D3—241no—
592C3×C4—21no—
593C3×C4—21no—
594C3×C4—21no—
595C3×C4—21no—
596C3×C4—21no—
597C3×C4—21no—
598C3×C4—41no—
599C3×C4—41no—
600C3×C4—181no—
601C3×C4—361no—
602C3×C4D3×D1211yescharacteristic
603C3×C4#380211yescharacteristic
604C3×C4C2×#62611yescharacteristic
605C3×C4D3×#6711yescharacteristic
606C3×C4#381111yescharacteristic
607C3×C4#381111yescharacteristic
608Dic3—62no—
609Dic3—61no—
610Dic3—61no—
611Dic3—121no—
612Dic3—121no—
613Dic3—184no—
614Dic3—182no—
615Dic3—182no—
616Dic3—182no—
617Dic3—241no—
618Dic3—362no—
619Dic3—362no—
620Dic3—364no—
621C32—21no—
622C32—21no—
623C32—21no—
624C32—41no—
625C32#473411yescharacteristic, complemented
626C32#478111yescharacteristic, complemented
627C32#515811yescharacteristic, complemented
628C23—61no—
629C23—91no—
630C23—181no—
631Q8—272no—
632Q8—541no—
633D4—32no—
634D4—61no—
635D4—181no—
636D4—181no—
637D4—544no—
638D4—1082no—
639C2×C4—91no—
640C2×C4—181no—
641C2×C4—181no—
642C2×C4—181no—
643C2×C4—544no—
644C2×C4—541no—
645C2×C4D3311yescharacteristic
646C8—61no—
647C8—61no—
648C2×C3—21no—
649C2×C3—21no—
650C2×C3—21no—
651C2×C3—21no—
652C2×C3—21no—
653C2×C3—21no—
654C2×C3—21no—
655C2×C3—21no—
656C2×C3—21no—
657C2×C3—41no—
658C2×C3—41no—
659C2×C3—41no—
660C2×C3—124no—
661C2×C3—122no—
662C2×C3—122no—
663C2×C3—361no—
664C2×C3—721no—
665C2×C3#6379511yescharacteristic
666C2×C3#6379511yescharacteristic
667C2×C3#6380611yescharacteristic
668C2×C3#6380711yescharacteristic
669C2×C3#6392211yescharacteristic
670C2×C3#6392311yescharacteristic
671C2×C3#6392711yescharacteristic
672C2×C3#6393611yescharacteristic
673C2×C3#6398111yescharacteristic
674D3—42no—
675D3—122no—
676D3—121no—
677D3—121no—
678D3—242no—
679D3—241no—
680D3—241no—
681C22—62no—
682C22—121no—
683C22—121no—
684C22—181no—
685C22—181no—
686C22—181no—
687C22—361no—
688C22—361no—
689C22—361no—
690C22#8895511yescharacteristic
691C4—181no—
692C4—361no—
693C4—542no—
694C4D3×#62611yescharacteristic
695C4#8895411yescharacteristic
696C3—21no—
697C3—21no—
698C3—21no—
699C3—41no—
700C3#1059096211yescharacteristic, complemented
701C3#1059096811yescharacteristic, complemented
702C3#1059145711yescharacteristic, complemented
703C2—122no—
704C2—361no—
705C2—721no—
706C2#1172286411yescharacteristic
707C2#1172288011yescharacteristic
708C2#1172295011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#449
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×#89014C2semidirect
2. #11723696C2semidirect
3. C2×#88796C2semidirect
4. C2×C3×#3714C2semidirect
5. #11722042C2semidirect
6. C3×#63563C2semidirect
7. C3×#63557C2semidirect
39. C2×#3714D3semidirect
48. #63563D3semidirect
49. #63557D3semidirect
56. #6308D4semidirect
64. C32×D12D4semidirect
238. C3×D12D12semidirect
239. C3×D12#67semidirect
409. C33#449semidirect
471. D12#626semidirect
625. C32#4734semidirect
626. C32#4781semidirect
627. C32#5158semidirect
700. C3#10590962semidirect
701. C3#10590968semidirect
702. C3#10591457semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
#117236962
C2×#887962
C2×C3×#37142
#117220422
C3×#635632
C3×#635572
#105914573
#105909683
#105909623