Finite Group Databasebeta

← #13162033

Subgroups — #13162033

Order 1728.

848 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic
2C3×#64155C211yesmaximal, characteristic, complemented
3C2×C3×#3756C211yesmaximal, characteristic, complemented
4C2×C3×#3755C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6#11722075C211yesmaximal, characteristic, complemented
7#11722072C211yesmaximal, characteristic, complemented
8#10591657—31nomaximal
9#10591656—31nomaximal
10#10591655—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C2×#6295—21no—
16C2×#6294—21no—
17#89041—22no—
18#89041—22no—
19C3×#3756—22no—
20C3×#3755—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3750—21no—
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
26C2×#3799—31no—
27C2×#3796—31no—
28C2×#3795—31no—
29#64155D311yescharacteristic, complemented
30C2×#3756D311yescharacteristic, complemented
31C2×#3755D311yescharacteristic, complemented
32C3×#946—31no—
33C3×#946—31no—
34C3×#946—61no—
35C2×C3×#216—31no—
36C2×C3×#216—31no—
37C2×C3×#216—61no—
38C2×C3×#215—31no—
39C2×C3×#215—31no—
40C2×C3×#215—61no—
41#63567—31no—
42#63566—31no—
43#63565—31no—
44#63564—31no—
45#63563—31no—
46#63562—31no—
47#63557—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—31no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C33×Q8—21no—
53C33×Q8—21no—
54C2×C33×C4—21no—
55C2×C33×C4C2311yescharacteristic
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3D411yescharacteristic
59C3×C4⋊(C3⋊D3)—21no—
60C3×C4⋊(C3⋊D3)D412yes—
61C3×C4×C3⋊D3—21no—
62C3×C4×C3⋊D3—21no—
63C3×#634—21no—
64C3×#634D412yes—
65#6295—24no—
66#6294—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C3×C3⋊(C3⋊C8)—21no—
69#5187—91no—
70#5173—91no—
71#5047—91no—
72C2×C32×Q8—21no—
73C2×C32×Q8—21no—
74C2×C32×Q8—21no—
75C2×C32×Q8—41no—
76C2×C32×Q8C2×D311yescharacteristic
77C2×C32×Q8C2×D311yescharacteristic
78C2×C32×Q8C2×D311yescharacteristic
79C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×#634C2×D311yescharacteristic
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—61no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—61no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—61no—
91C2×#626—61no—
92C2×#626—61no—
93C2×#625—61no—
94C2×#624—61no—
95C2×D3×Dic3—61no—
96C2×D3×Dic3—61no—
97#3799—62no—
98#3799—62no—
99#3796—62no—
100#3796—62no—
101#3795—62no—
102#3795—62no—
103#3756—22no—
104#3755—22no—
105#3752—21no—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107#3750—21no—
108C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
109C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
110C3×#216—62no—
111C3×#216—62no—
112C3×#216—64no—
113C3×#215—62no—
114C3×#215—62no—
115C3×#215—64no—
116C3×(C2×D3)⋊C4—61no—
117C3×(C2×D3)⋊C4—61no—
118C3×(C2×D3)⋊C4—62no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—61no—
122C3×Dic3⋊C4—61no—
123C3×Dic3⋊C4—61no—
124C3×Dic3⋊C4—62no—
125C2×C3×C3⋊C8—31no—
126C2×C3×C3⋊C8—31no—
127C2×C3×C3⋊C8—31no—
128C2×C3×C3⋊C8—31no—
129C2×C3×C3⋊C8—61no—
130C2×C3×C3⋊C8—61no—
131C22×C33C2×D411yescharacteristic
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—41no—
137C2×C3×C3⋊D3—41no—
138C33×C4—41no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—41no—
144C2×#239—91no—
145C2×#237—91no—
146C2×#235—91no—
147C2×C3×Q16—91no—
148C2×C3×SD16—91no—
149C3×C22⋊Q8—91no—
150#946—31no—
151#946—31no—
152#946—61no—
153C2×#216—31no—
154C2×#216—31no—
155C2×#216—61no—
156C2×#215—31no—
157C2×#215—31no—
158C2×#215—61no—
159#835—91no—
160#832—91no—
161#830—91no—
162#821—91no—
163(C2×D3)⋊C8—91no—
164(C2×D3)⋊C8—91no—
165#816—91no—
166#810—91no—
167#809—91no—
168C22×C3⋊D3—21no—
169C22×C3⋊D3#6711yescharacteristic
170C22×C3×D3—31no—
171C22×C3×D3—31no—
172C22×C3×D3—61no—
173C22×C3×D3—61no—
174C22×C3×D3—61no—
175C22×C3×D3—121no—
176C32×Q8—21no—
177C32×Q8—21no—
178C32×Q8—22no—
179C32×Q8—22no—
180C32×Q8—21no—
181C32×Q8—21no—
182C32×Q8—21no—
183C32×Q8—21no—
184C32×Q8—41no—
185C32×Q8—41no—
186C32×Q8—41no—
187C32×Q8—41no—
188C32×Q8—42no—
189C32×Q8—42no—
190C2×C32×C4—21no—
191C2×C32×C4—22no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—41no—
198C2×C32×C4—41no—
199C2×C32×C4—41no—
200C2×C32×C4—42no—
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204C2×C3⋊Dic3—21no—
205C2×C3⋊Dic3—21no—
206C2×C3⋊Dic3#6711yescharacteristic
207C4⋊(C3⋊D3)—21no—
208C4⋊(C3⋊D3)#6712yes—
209C4×C3⋊D3—21no—
210C4×C3⋊D3—21no—
211#634—21no—
212#634#6712yes—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—62no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—121no—
222C3×D12—32no—
223C3×D12—32no—
224C3×D12—62no—
225C3×D12—61no—
226C3×D12—61no—
227C3×D12—121no—
228C3×C4×D3—61no—
229C3×C4×D3—61no—
230C3×C4×D3—61no—
231C3×C4×D3—61no—
232C3×C4×D3—121no—
233C3×C4×D3—121no—
234C3×Dic6—32no—
235C3×Dic6—32no—
236C3×Dic6—62no—
237C3×Dic6—61no—
238C3×Dic6—61no—
239C3×Dic6—121no—
240#626—64no—
241#626—64no—
242#625—64no—
243#624—64no—
244D3×Dic3—64no—
245D3×Dic3—64no—
246C3⋊(C3⋊C8)—21no—
247C3⋊(C3⋊C8)—21no—
248C3×C3⋊C8—62no—
249C3×C3⋊C8—62no—
250C3×C3⋊C8—61no—
251C3×C3⋊C8—61no—
252C3×C3⋊C8—61no—
253C3×C3⋊C8—61no—
254#452—271noSylow
255C2×C33#12011yescharacteristic
256C2×C33#13511yescharacteristic
257C2×C33#13711yescharacteristic
258C3×C3⋊D3—41no—
259C3×C3⋊D3—42no—
260C2×C3×Q8—21no—
261C2×C3×Q8—21no—
262C2×C3×Q8—21no—
263C2×C3×Q8—41no—
264C2×C3×Q8—91no—
265C2×C3×Q8D3211yescharacteristic
266C2×C3×Q8D3211yescharacteristic
267C2×C3×Q8D3211yescharacteristic
268C2×C3×D4—91no—
269C22×C3×C4—91no—
270C2×#67—181no—
271C2×#67—181no—
272C22×Dic3—181no—
273#239—182no—
274#239—182no—
275#237—182no—
276#237—182no—
277#235—182no—
278#235—182no—
279C2×D12—31no—
280C2×D12—31no—
281C2×D12—61no—
282C2×D12—181no—
283C2×C4×D3—31no—
284C2×C4×D3—31no—
285C2×C4×D3—61no—
286C2×C4×D3—181no—
287C2×C4×D3—181no—
288C2×Dic6—31no—
289C2×Dic6—31no—
290C2×Dic6—61no—
291C3×Q16—182no—
292C3×SD16—182no—
293C2×C3×C8—91no—
294C2×C3×C8—91no—
295C3×C4⋊C4—91no—
296C3×C4⋊C4—181no—
297C3×C22⋊C4—181no—
298#216—62no—
299#216—62no—
300#216—64no—
301#215—62no—
302#215—62no—
303#215—64no—
304(C2×D3)⋊C4—61no—
305(C2×D3)⋊C4—61no—
306(C2×D3)⋊C4—62no—
307C4⋊Dic3—31no—
308C4⋊Dic3—31no—
309C4⋊Dic3—61no—
310Dic3⋊C4—61no—
311Dic3⋊C4—61no—
312Dic3⋊C4—62no—
313C2×C3⋊C8—31no—
314C2×C3⋊C8—31no—
315C2×C3⋊C8—31no—
316C2×C3⋊C8—31no—
317C2×C3⋊C8—61no—
318C2×C3⋊C8—61no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—41no—
323C22×C32D3×D411yescharacteristic
324C22×C32D3×D411yescharacteristic
325C22×C32C2×#6711yescharacteristic
326C2×C3⋊D3—21no—
327C2×C3⋊D3—22no—
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3⋊D3—41no—
331C2×C3⋊D3—41no—
332C2×C3×D3—62no—
333C2×C3×D3—61no—
334C2×C3×D3—61no—
335C2×C3×D3—61no—
336C2×C3×D3—61no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—61no—
340C2×C3×D3—61no—
341C2×C3×D3—121no—
342C2×C3×D3—121no—
343C2×C3×D3—122no—
344C2×C3×D3—121no—
345C2×C3×D3—121no—
346C2×C3×D3—121no—
347C2×C3×D3—121no—
348C2×C3×D3—241no—
349C2×C3×D3—241no—
350C32×C4—21no—
351C32×C4—21no—
352C32×C4—21no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—21no—
356C32×C4—41no—
357C32×C4—41no—
358C32×C4—42no—
359C32×C4—42no—
360C32×C4—42no—
361C32×C4—44no—
362C32×C4—41no—
363C32×C4—41no—
364C32×C4—41no—
365C32×C4D3×D411yescharacteristic
366C32×C4D3×D411yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4C2×#6711yescharacteristic
370C32×C4C2×#6711yescharacteristic
371C3⋊Dic3—21no—
372C3⋊Dic3—22no—
373C3⋊Dic3—41no—
374C3×Dic3—61no—
375C3×Dic3—62no—
376C3×Dic3—61no—
377C3×Dic3—62no—
378C3×Dic3—122no—
379C3×Dic3—121no—
380C3×Dic3—122no—
381C3×Dic3—121no—
382C3×Dic3—121no—
383C2×#41—271no—
384C2×Q16—91no—
385C2×SD16—91no—
386C22⋊Q8—91no—
387#103—271no—
388#102—271no—
389C22⋊C8—271no—
390C33#45211yescharacteristic, complemented, Sylow
391C23×C3—91no—
392C23×C3—181no—
393C22×D3—31no—
394C22×D3—31no—
395C22×D3—61no—
396C22×D3—61no—
397C22×D3—61no—
398C22×D3—121no—
399C3×Q8—22no—
400C3×Q8—22no—
401C3×Q8—21no—
402C3×Q8—21no—
403C3×Q8—21no—
404C3×Q8—21no—
405C3×Q8—21no—
406C3×Q8—21no—
407C3×Q8—41no—
408C3×Q8—41no—
409C3×Q8—41no—
410C3×Q8—41no—
411C3×Q8—42no—
412C3×Q8—42no—
413C3×Q8—92no—
414C3×Q8—181no—
415C3×D4—92no—
416C3×D4—181no—
417C2×C3×C4—22no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—42no—
427C2×C3×C4—41no—
428C2×C3×C4—91no—
429C2×C3×C4—181no—
430C2×C3×C4—181no—
431C2×C3×C4—181no—
432C2×C3×C4—181no—
433C2×C3×C4C2×D3211yescharacteristic
434C2×C3×C4C2×D3211yescharacteristic
435C2×C3×C4C2×D3211yescharacteristic
436#67—184no—
437#67—184no—
438C2×Dic3—31no—
439C2×Dic3—31no—
440C2×Dic3—61no—
441C2×Dic3—61no—
442C2×Dic3—61no—
443C2×Dic3—62no—
444C2×Dic3—61no—
445C2×Dic3—61no—
446C2×Dic3—121no—
447C2×Dic3—184no—
448D12—32no—
449D12—32no—
450D12—62no—
451D12—61no—
452D12—61no—
453D12—121no—
454D12—184no—
455C4×D3—61no—
456C4×D3—61no—
457C4×D3—61no—
458C4×D3—61no—
459C4×D3—121no—
460C4×D3—121no—
461C4×D3—184no—
462C4×D3—184no—
463Dic6—32no—
464Dic6—32no—
465Dic6—62no—
466Dic6—61no—
467Dic6—61no—
468Dic6—121no—
469C3×C8—181no—
470C3×C8—181no—
471C3⋊C8—62no—
472C3⋊C8—62no—
473C3⋊C8—61no—
474C3⋊C8—61no—
475C3⋊C8—61no—
476C3⋊C8—61no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—21no—
482C2×C32—21no—
483C2×C32—21no—
484C2×C32—21no—
485C2×C32—21no—
486C2×C32—41no—
487C2×C32—41no—
488C2×C32—41no—
489C2×C32#91511yescharacteristic
490C2×C32#91611yescharacteristic
491C2×C32#91811yescharacteristic
492C2×C32#91911yescharacteristic
493C2×C32#93711yescharacteristic
494C2×C32#93711yescharacteristic
495C2×C32#93711yescharacteristic
496C2×C32#94211yescharacteristic
497C2×C32#95011yescharacteristic
498C3⋊D3—41no—
499C3⋊D3—42no—
500C3×D3—122no—
501C3×D3—121no—
502C3×D3—121no—
503C3×D3—122no—
504C3×D3—122no—
505C3×D3—242no—
506#41—542no—
507#41—542no—
508C2×Q8—91no—
509C2×Q8#111311yescharacteristic
510C2×D4—91no—
511C2×D4—541no—
512C22×C4—91no—
513C22×C4—541no—
514Q16—182no—
515SD16—182no—
516C2×C8—91no—
517C2×C8—91no—
518C4⋊C4—91no—
519C4⋊C4—181no—
520C22⋊C4—181no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—41no—
525C22×C3—181no—
526C22×C3—181no—
527C22×C3—182no—
528C22×C3—181no—
529C22×C3—361no—
530C22×C3—361no—
531C22×C3D3×#6711yescharacteristic
532C22×C3D3×#6711yescharacteristic
533C22×C3#381111yescharacteristic
534C2×D3—62no—
535C2×D3—61no—
536C2×D3—61no—
537C2×D3—61no—
538C2×D3—61no—
539C2×D3—61no—
540C2×D3—61no—
541C2×D3—62no—
542C2×D3—62no—
543C2×D3—121no—
544C2×D3—121no—
545C2×D3—122no—
546C2×D3—121no—
547C2×D3—121no—
548C2×D3—121no—
549C2×D3—121no—
550C2×D3—241no—
551C2×D3—241no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—42no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4—44no—
562C3×C4—41no—
563C3×C4—41no—
564C3×C4—41no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—181no—
568C3×C4—182no—
569C3×C4—361no—
570C3×C4D3×#6711yescharacteristic
571C3×C4D3×#6711yescharacteristic
572C3×C4D3×#6711yescharacteristic
573C3×C4D3×#6711yescharacteristic
574C3×C4#381111yescharacteristic
575C3×C4#381111yescharacteristic
576Dic3—62no—
577Dic3—62no—
578Dic3—61no—
579Dic3—61no—
580Dic3—122no—
581Dic3—121no—
582Dic3—122no—
583Dic3—121no—
584Dic3—121no—
585C32—21no—
586C32—21no—
587C32—21no—
588C32—41no—
589C32#504711yescharacteristic, complemented
590C32#517311yescharacteristic, complemented
591C32#518711yescharacteristic, complemented
592C23—91no—
593C23—181no—
594Q8—21no—
595Q8—21no—
596Q8—92no—
597Q8—181no—
598D4—92no—
599D4—181no—
600D4—544no—
601C2×C4—21no—
602C2×C4—91no—
603C2×C4—181no—
604C2×C4—181no—
605C2×C4—181no—
606C2×C4—181no—
607C2×C4—544no—
608C2×C4C2×#111311yescharacteristic
609C8—181no—
610C8—181no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—41no—
621C2×C3—41no—
622C2×C3—41no—
623C2×C3—361no—
624C2×C3—362no—
625C2×C3#6394611yescharacteristic
626C2×C3#6394711yescharacteristic
627C2×C3#6394811yescharacteristic
628C2×C3#6394911yescharacteristic
629C2×C3#6395011yescharacteristic
630C2×C3#6395111yescharacteristic
631C2×C3#6398311yescharacteristic
632C2×C3#6398311yescharacteristic
633C2×C3#6398311yescharacteristic
634D3—122no—
635D3—121no—
636D3—122no—
637D3—122no—
638D3—121no—
639D3—242no—
640C22—181no—
641C22—181no—
642C22—181no—
643C22—182no—
644C22—361no—
645C22—361no—
646C22#8904911yescharacteristic
647C4—41no—
648C4—181no—
649C4—182no—
650C4—361no—
651C4#8904911yescharacteristic
652C4#8904911yescharacteristic
653C3—21no—
654C3—21no—
655C3—21no—
656C3—41no—
657C3#1059165511yescharacteristic, complemented
658C3#1059165611yescharacteristic, complemented
659C3#1059165711yescharacteristic, complemented
660C2—361no—
661C2—362no—
662C2#1172371211yescharacteristic
663C2#1172371311yescharacteristic
664C2#1172372911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89041C2
hypercenterC2×Q8#1113
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64155C2semidirect
3. C2×C3×#3756C2semidirect
4. C2×C3×#3755C2semidirect
5. #11722076C2semidirect
6. #11722075C2semidirect
7. #11722072C2semidirect
29. #64155D3semidirect
30. C2×#3756D3semidirect
31. C2×#3755D3semidirect
390. C33#452semidirect
589. C32#5047semidirect
590. C32#5173semidirect
591. C32#5187semidirect
657. C3#10591655semidirect
658. C3#10591656semidirect
659. C3#10591657semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C3×#641552
C2×C3×#37562
C2×C3×#37552
#117220762
#117220752
#117220722
#105916573
#105916563
#105916553