Finite Group Databasebeta

← #13161753

Subgroups — #13161753

Order 1728.

790 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic
2C2×C3×#3744C211yesmaximal, characteristic, complemented
3#11723674C211yesmaximal, characteristic, complemented
4C2×#88812C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic, complemented
8#10590970—31nomaximal
9#10590961—31nomaximal
10#10590960—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C2×#6295—21no—
16C2×#6294—21no—
17#89041—22no—
18#89041—22no—
19C2×C33×C8C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#3744—22no—
23#88815—21no—
24#88814—21no—
25#88812—22no—
26C2×#3799—31no—
27C2×#3796—31no—
28C2×#3795—31no—
29C2×#3744D311yescharacteristic, complemented
30C2×C3×#204—31no—
31C2×C3×#204—31no—
32C2×C3×#204—61no—
33#63871—31no—
34#63857—31no—
35#63856—31no—
36C2×#3719—31no—
37C2×#3719—31no—
38C2×#3718—31no—
39#63647D311yescharacteristic, complemented
40#63643D311yescharacteristic, complemented
41C3×(C2×D3)⋊C8—31no—
42C3×(C2×D3)⋊C8—31no—
43C3×(C2×D3)⋊C8—61no—
44C3×#816—31no—
45C3×#816—31no—
46C3×#816—61no—
47#63564—31no—
48#63563—31no—
49#63562—31no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D412yes—
59C3×C4×C3⋊D3—21no—
60C3×C4×C3⋊D3—21no—
61C3×#634—21no—
62C3×#634—21no—
63C3×#634—21no—
64C3×#634D412yes—
65#6295—24no—
66#6294—24no—
67C33×C8—21no—
68C3×C3⋊(C3⋊C8)—21no—
69#5173—91no—
70#4808—91no—
71#4733—91no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×#634C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—61no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—61no—
87C2×C3×Dic6—61no—
88C2×#626—61no—
89C2×#626—61no—
90C2×#625—61no—
91C2×#624—61no—
92C2×D3×Dic3—61no—
93C2×D3×Dic3—61no—
94#3799—62no—
95#3799—62no—
96#3796—62no—
97#3796—62no—
98#3795—62no—
99#3795—62no—
100C2×C32×C8—21no—
101C2×C32×C8—21no—
102C2×C32×C8—21no—
103C2×C32×C8—41no—
104C2×C32×C8C2×D311yescharacteristic
105C2×C32×C8C2×D311yescharacteristic
106C2×C32×C8C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
109#3744—22no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—61no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116C3×#204—62no—
117C3×#204—62no—
118C3×#204—64no—
119#3724—61no—
120Dic3⋊Dic3—61no—
121Dic3⋊Dic3—61no—
122#3722—61no—
123(C2×D3)⋊Dic3—61no—
124(C2×D3)⋊Dic3—61no—
125#3719—62no—
126#3719—62no—
127#3718—62no—
128C22×C33C2×D411yescharacteristic
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—21no—
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—41no—
134C2×C3×C3⋊D3—41no—
135C33×C4C2×D411yescharacteristic
136C33×C4C2×D411yescharacteristic
137C3×C3⋊Dic3—21no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—41no—
140C3×C3⋊Dic3—41no—
141C2×#239—91no—
142C2×#237—91no—
143C2×#235—91no—
144C2×C3×SD16—91no—
145#946—91no—
146C2×#216—91no—
147C2×#216—91no—
148#924—91no—
149C2×Dic12—91no—
150C2×#204—31no—
151C2×#204—31no—
152C2×#204—61no—
153#897—91no—
154C3×#103—91no—
155C3×C22⋊C8—91no—
156#832—91no—
157#821—91no—
158(C2×D3)⋊C8—31no—
159(C2×D3)⋊C8—31no—
160(C2×D3)⋊C8—61no—
161#816—31no—
162#816—31no—
163#816—61no—
164#809—91no—
165C22×C3⋊D3—21no—
166C22×C3⋊D3D1211yescharacteristic
167C22×C3×D3—31no—
168C22×C3×D3—31no—
169C22×C3×D3—61no—
170C22×C3×D3—61no—
171C22×C3×D3—61no—
172C22×C3×D3—121no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—41no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3D1211yescharacteristic
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)#6712yes—
186C4×C3⋊D3—21no—
187C4×C3⋊D3—21no—
188#634—21no—
189#634—21no—
190#634—21no—
191#634D1212yes—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—121no—
202C2×C3×Dic3—121no—
203C2×C3×Dic3—121no—
204C3×D12—32no—
205C3×D12—32no—
206C3×D12—62no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×C4×D3—61no—
211C3×C4×D3—61no—
212C3×C4×D3—61no—
213C3×C4×D3—61no—
214C3×C4×D3—121no—
215C3×C4×D3—121no—
216C3×Dic6—32no—
217C3×Dic6—32no—
218C3×Dic6—62no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—62no—
222C3×Dic6—61no—
223C3×Dic6—61no—
224C3×Dic6—61no—
225C3×Dic6—61no—
226C3×Dic6—121no—
227C3×Dic6—121no—
228#626—64no—
229#626—64no—
230#625—64no—
231#624—64no—
232D3×Dic3—64no—
233D3×Dic3—64no—
234C32×C8—21no—
235C32×C8—22no—
236C32×C8—21no—
237C32×C8—21no—
238C32×C8—41no—
239C32×C8—41no—
240C32×C8—42no—
241C3⋊(C3⋊C8)—21no—
242C3×C3⋊C8—61no—
243C3×C3⋊C8—61no—
244C3×C3⋊C8—121no—
245#452—271noSylow
246C2×C33#12011yescharacteristic
247C2×C33#13511yescharacteristic
248C2×C33#13711yescharacteristic
249C3×C3⋊D3—41no—
250C3×C3⋊D3—42no—
251C2×C3×Q8—91no—
252C2×C3×Q8—91no—
253C2×C3×D4—91no—
254C22×C3×C4—91no—
255C2×#67—181no—
256C2×#67—181no—
257C22×Dic3—181no—
258#239—182no—
259#239—182no—
260#237—182no—
261#237—182no—
262#235—182no—
263#235—182no—
264C2×D12—31no—
265C2×D12—31no—
266C2×D12—61no—
267C2×D12—181no—
268C2×C4×D3—31no—
269C2×C4×D3—31no—
270C2×C4×D3—61no—
271C2×C4×D3—181no—
272C2×C4×D3—181no—
273C2×Dic6—31no—
274C2×Dic6—31no—
275C2×Dic6—31no—
276C2×Dic6—31no—
277C2×Dic6—61no—
278C2×Dic6—61no—
279C3×SD16—182no—
280C2×C3×C8—21no—
281C2×C3×C8—21no—
282C2×C3×C8—21no—
283C2×C3×C8—41no—
284C2×C3×C8—91no—
285C2×C3×C8D3211yescharacteristic
286C2×C3×C8D3211yescharacteristic
287C2×C3×C8D3211yescharacteristic
288C3×C4⋊C4—91no—
289#217—181no—
290#216—182no—
291#216—182no—
292(C2×D3)⋊C4—181no—
293(C2×D3)⋊C4—181no—
294C4⋊Dic3—31no—
295C4⋊Dic3—31no—
296C4⋊Dic3—61no—
297C4⋊Dic3—181no—
298Dic3⋊C4—181no—
299Dic3⋊C4—181no—
300C2×C3⋊C8—31no—
301C2×C3⋊C8—31no—
302C2×C3⋊C8—61no—
303Dic12—182no—
304#204—62no—
305#204—62no—
306#204—64no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—41no—
311C22×C32C2×D1211yescharacteristic
312C22×C32D3×D411yescharacteristic
313C22×C32D3×D411yescharacteristic
314C2×C3⋊D3—21no—
315C2×C3⋊D3—21no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—41no—
319C2×C3⋊D3—41no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3—61no—
324C2×C3×D3—61no—
325C2×C3×D3—61no—
326C2×C3×D3—61no—
327C2×C3×D3—61no—
328C2×C3×D3—61no—
329C2×C3×D3—121no—
330C2×C3×D3—121no—
331C2×C3×D3—122no—
332C2×C3×D3—121no—
333C2×C3×D3—121no—
334C2×C3×D3—121no—
335C2×C3×D3—121no—
336C2×C3×D3—241no—
337C2×C3×D3—241no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—41no—
345C32×C4—41no—
346C32×C4C2×D1211yescharacteristic
347C32×C4D3×D411yescharacteristic
348C32×C4D3×D411yescharacteristic
349C32×C4D3×D411yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4C2×#6711yescharacteristic
352C3⋊Dic3—21no—
353C3⋊Dic3—22no—
354C3⋊Dic3—41no—
355C3⋊Dic3—41no—
356C3×Dic3—61no—
357C3×Dic3—62no—
358C3×Dic3—61no—
359C3×Dic3—62no—
360C3×Dic3—122no—
361C3×Dic3—121no—
362C3×Dic3—122no—
363C3×Dic3—121no—
364C3×Dic3—121no—
365C3×Dic3—121no—
366C3×Dic3—121no—
367C3×Dic3—241no—
368C2×#41—271no—
369C2×Q16—271no—
370C2×SD16—91no—
371C22⋊Q8—271no—
372#103—91no—
373#102—271no—
374C22⋊C8—91no—
375C33#45211yescharacteristic, complemented, Sylow
376C23×C3—91no—
377C23×C3—181no—
378C22×D3—31no—
379C22×D3—31no—
380C22×D3—61no—
381C22×D3—61no—
382C22×D3—61no—
383C22×D3—121no—
384C3×Q8—92no—
385C3×Q8—181no—
386C3×Q8—181no—
387C3×Q8—181no—
388C3×D4—92no—
389C3×D4—181no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—91no—
395C2×C3×C4—181no—
396C2×C3×C4—181no—
397C2×C3×C4—181no—
398C2×C3×C4—181no—
399C2×C3×C4—181no—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403#67—184no—
404#67—184no—
405C2×Dic3—31no—
406C2×Dic3—31no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—121no—
415C2×Dic3—121no—
416C2×Dic3—121no—
417C2×Dic3—184no—
418D12—32no—
419D12—32no—
420D12—62no—
421D12—61no—
422D12—61no—
423D12—121no—
424D12—184no—
425C4×D3—61no—
426C4×D3—61no—
427C4×D3—61no—
428C4×D3—61no—
429C4×D3—121no—
430C4×D3—121no—
431C4×D3—184no—
432C4×D3—184no—
433Dic6—32no—
434Dic6—32no—
435Dic6—62no—
436Dic6—61no—
437Dic6—61no—
438Dic6—62no—
439Dic6—61no—
440Dic6—61no—
441Dic6—61no—
442Dic6—61no—
443Dic6—121no—
444Dic6—121no—
445C3×C8—22no—
446C3×C8—21no—
447C3×C8—21no—
448C3×C8—21no—
449C3×C8—41no—
450C3×C8—41no—
451C3×C8—42no—
452C3×C8—181no—
453C3⋊C8—61no—
454C3⋊C8—61no—
455C3⋊C8—121no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—41no—
466C2×C32—41no—
467C2×C32—41no—
468C2×C32#88211yescharacteristic
469C2×C32#88211yescharacteristic
470C2×C32#90411yescharacteristic
471C2×C32#90911yescharacteristic
472C2×C32#91511yescharacteristic
473C2×C32#91611yescharacteristic
474C2×C32#91611yescharacteristic
475C2×C32#91811yescharacteristic
476C2×C32#93711yescharacteristic
477C3⋊D3—41no—
478C3⋊D3—42no—
479C3×D3—122no—
480C3×D3—121no—
481C3×D3—122no—
482C3×D3—121no—
483C3×D3—122no—
484C3×D3—242no—
485#41—542no—
486#41—542no—
487C2×Q8—91no—
488C2×Q8—91no—
489C2×D4—91no—
490C2×D4—541no—
491C22×C4—91no—
492C22×C4—541no—
493Q16—542no—
494SD16—182no—
495C2×C8—91no—
496C2×C8#111311yescharacteristic
497C4⋊C4—91no—
498C4⋊C4—541no—
499C22⋊C4—541no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—41no—
504C22×C3—181no—
505C22×C3—181no—
506C22×C3—181no—
507C22×C3—182no—
508C22×C3—361no—
509C22×C3—361no—
510C22×C3D3×D1211yescharacteristic
511C22×C3D3×D1211yescharacteristic
512C22×C3#380211yescharacteristic
513C2×D3—62no—
514C2×D3—61no—
515C2×D3—61no—
516C2×D3—61no—
517C2×D3—61no—
518C2×D3—61no—
519C2×D3—62no—
520C2×D3—62no—
521C2×D3—61no—
522C2×D3—121no—
523C2×D3—121no—
524C2×D3—122no—
525C2×D3—121no—
526C2×D3—121no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—241no—
530C2×D3—241no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—181no—
540C3×C4—182no—
541C3×C4—361no—
542C3×C4—361no—
543C3×C4D3×D1211yescharacteristic
544C3×C4D3×D1211yescharacteristic
545C3×C4#380211yescharacteristic
546C3×C4D3×#6711yescharacteristic
547C3×C4D3×#6711yescharacteristic
548C3×C4#381111yescharacteristic
549Dic3—62no—
550Dic3—61no—
551Dic3—62no—
552Dic3—61no—
553Dic3—122no—
554Dic3—121no—
555Dic3—122no—
556Dic3—121no—
557Dic3—121no—
558Dic3—121no—
559Dic3—121no—
560Dic3—241no—
561C32—21no—
562C32—21no—
563C32—21no—
564C32—41no—
565C32#473311yescharacteristic, complemented
566C32#480811yescharacteristic, complemented
567C32#517311yescharacteristic, complemented
568C23—91no—
569C23—181no—
570Q8—92no—
571Q8—181no—
572Q8—181no—
573Q8—181no—
574D4—92no—
575D4—181no—
576D4—544no—
577C2×C4—91no—
578C2×C4—181no—
579C2×C4—181no—
580C2×C4—181no—
581C2×C4—181no—
582C2×C4—181no—
583C2×C4—544no—
584C2×C4C2×#111311yescharacteristic
585C8—21no—
586C8—181no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—361no—
600C2×C3—362no—
601C2×C3#6380011yescharacteristic
602C2×C3#6380111yescharacteristic
603C2×C3#6380611yescharacteristic
604C2×C3#6380611yescharacteristic
605C2×C3#6380911yescharacteristic
606C2×C3#6381111yescharacteristic
607C2×C3#6392211yescharacteristic
608C2×C3#6392211yescharacteristic
609C2×C3#6392311yescharacteristic
610D3—122no—
611D3—121no—
612D3—122no—
613D3—122no—
614D3—121no—
615D3—242no—
616C22—181no—
617C22—181no—
618C22—181no—
619C22—182no—
620C22—361no—
621C22—361no—
622C22C4⋊#111311yescharacteristic
623C4—181no—
624C4—182no—
625C4—361no—
626C4—361no—
627C4C4⋊#111311yescharacteristic
628C4#8904911yescharacteristic
629C3—21no—
630C3—21no—
631C3—21no—
632C3—41no—
633C3#1059096011yescharacteristic, complemented
634C3#1059096111yescharacteristic, complemented
635C3#1059097011yescharacteristic, complemented
636C2—361no—
637C2—362no—
638C2#1172365011yescharacteristic
639C2#1172365311yescharacteristic
640C2#1172370111yescharacteristic
subgroupG/H
centerC22C4⋊#1113
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89041C2
hypercenterC2×C8#1113
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3744C2semidirect
3. #11723674C2semidirect
4. C2×#88812C2semidirect
5. C3×#63647C2semidirect
6. C3×#63643C2semidirect
7. #11722075C2semidirect
29. C2×#3744D3semidirect
39. #63647D3semidirect
40. #63643D3semidirect
375. C33#452semidirect
565. C32#4733semidirect
566. C32#4808semidirect
567. C32#5173semidirect
633. C3#10590960semidirect
634. C3#10590961semidirect
635. C3#10590970semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C2×C3×#37442
#117236742
C2×#888122
C3×#636472
C3×#636432
#117220752
#105909703
#105909613
#105909603