Finite Group Databasebeta

← #13178738

Subgroups — #13178738

Order 1728.

1418 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89041C211yesmaximal, characteristic
3C3×#64156C211yesmaximal, characteristic, complemented
4C3×#64153C211yesmaximal, characteristic, complemented
5#11723694C214yesmaximal, complemented
6#11723688C214yesmaximal, complemented
7#11723678C211yesmaximal, characteristic, complemented
8#11723677C211yesmaximal, characteristic, complemented
9#11723673C211yesmaximal, characteristic, complemented
10#10595399—31nomaximal
11#10595364—31nomaximal
12#10595340—31nomaximal
13C2×C33×Q8C2211yescharacteristic
14C3×Q8×C3⋊D3—24no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic
20C2×C3×#634C2212yes—
21C2×#6295C2212yes—
22C2×#6294C2212yes—
23#89041—24no—
24C3×#3752C2214yes—
25C3×#3750C2214yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28#88815C2212yes—
29#88815C2214yes—
30#88814C2212yes—
31#88814C2214yes—
32C2×Q8×C3⋊D3D311yescharacteristic, complemented
33C2×C3×D3×Q8—31no—
34C2×C3×D3×Q8—31no—
35C2×C3×D3×Q8—61no—
36C2×#3799—31no—
37C2×#3796—31no—
38C2×#3795—31no—
39#64156D311yescharacteristic, complemented
40#64153D311yescharacteristic, complemented
41C3×#947—31no—
42C3×#947—31no—
43C3×#947—61no—
44C3×#944—31no—
45C3×#944—31no—
46C3×#944—61no—
47#63905—34no—
48#63896—34no—
49#63888—34no—
50#63887—34no—
51#63874—31no—
52#63873—31no—
53#63866—31no—
54#63866—31no—
55#63864—31no—
56#63863—31no—
57#63860—31no—
58#63855—31no—
59#63855—31no—
60#63853—34no—
61#63850—34no—
62C22×C3×C3⋊D3C2312yes—
63C22×C3×C3⋊D3C2311yescharacteristic
64C33×Q8—22no—
65C2×C33×C4C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4⋊(C3⋊D3)—22no—
73C3×C4×C3⋊D3—24no—
74C3×C4×C3⋊D3D414yes—
75C3×#634—22no—
76C3×#634—22no—
77C3×#634—24no—
78#6295—24no—
79#6294—24no—
80#5816—91no—
81#5724—91no—
82#5675—91no—
83C2×C32×Q8—21no—
84C2×C32×Q8—21no—
85C2×C32×Q8—21no—
86C2×C32×Q8—41no—
87C2×C32×Q8C2×D311yescharacteristic
88C2×C32×Q8C2×D311yescharacteristic
89C2×C32×Q8C2×D311yescharacteristic
90Q8×C3⋊D3—24no—
91C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
92C2×C4×C3⋊D3C2×D312yes—
93C2×C4×C3⋊D3C2×D311yescharacteristic
94C2×#634C2×D311yescharacteristic
95C2×#634C2×D311yescharacteristic
96C2×#634C2×D312yes—
97C3×D3×Q8—64no—
98C3×D3×Q8—64no—
99C3×D3×Q8—68no—
100C2×C3×D12—31no—
101C2×C3×D12—31no—
102C2×C3×D12—61no—
103C2×C3×C4×D3—31no—
104C2×C3×C4×D3—31no—
105C2×C3×C4×D3—32no—
106C2×C3×C4×D3—32no—
107C2×C3×C4×D3—61no—
108C2×C3×C4×D3—62no—
109C2×C3×Dic6—31no—
110C2×C3×Dic6—31no—
111C2×C3×Dic6—31no—
112C2×C3×Dic6—31no—
113C2×C3×Dic6—32no—
114C2×C3×Dic6—32no—
115C2×C3×Dic6—61no—
116C2×C3×Dic6—62no—
117C2×C3×Dic6—61no—
118C2×#626—32no—
119C2×#626—32no—
120C2×#625—32no—
121C2×#624—32no—
122C2×D3×Dic3—32no—
123C2×D3×Dic3—32no—
124#3799—64no—
125#3796—64no—
126#3795—64no—
127#3752C2×D314yes—
128#3750C2×D314yes—
129C4×C3⋊Dic3C2×D311yescharacteristic
130C4×C3⋊Dic3C2×D311yescharacteristic
131C3×(C2×D3)⋊C4—34no—
132C3×(C2×D3)⋊C4—34no—
133C3×(C2×D3)⋊C4—64no—
134C3×Dic3⋊C4—34no—
135C3×Dic3⋊C4—34no—
136C3×Dic3⋊C4—64no—
137C3×C4×Dic3—31no—
138C3×C4×Dic3—31no—
139C3×C4×Dic3—31no—
140C3×C4×Dic3—31no—
141C3×C4×Dic3—61no—
142C3×C4×Dic3—61no—
143#3724—34no—
144#3724—32no—
145Dic3⋊Dic3—34no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—34no—
149#3722—34no—
150#3722—32no—
151(C2×D3)⋊Dic3—32no—
152(C2×D3)⋊Dic3—34no—
153(C2×D3)⋊Dic3—32no—
154(C2×D3)⋊Dic3—34no—
155C22×C33C2411yescharacteristic
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—24no—
159C2×C3×C3⋊D3C2×D412yes—
160C33×C4—22no—
161C33×C4C2×D412yes—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3C2×D412yes—
167C22×C3×Q8—91no—
168C2×#239—91no—
169C2×D3×Q8—31no—
170C2×D3×Q8—31no—
171C2×D3×Q8—61no—
172C2×#237—91no—
173C2×#235—91no—
174C3×C4⋊Q8—91no—
175C3×#124—91no—
176#947—31no—
177#947—31no—
178#947—61no—
179#947—91no—
180#946—94no—
181#944—31no—
182#944—31no—
183#944—61no—
184#944—91no—
185#944—91no—
186#936—91no—
187#929—94no—
188#926—91no—
189#896—94no—
190#894—94no—
191#883—94no—
192#878—94no—
193#875—91no—
194#872—91no—
195#872—91no—
196#869—91no—
197C22×C3⋊D3C22×D312yes—
198C22×C3⋊D3C22×D311yescharacteristic
199C22×C3×D3—31no—
200C22×C3×D3—31no—
201C22×C3×D3—32no—
202C22×C3×D3—32no—
203C22×C3×D3—61no—
204C22×C3×D3—62no—
205C32×Q8—22no—
206C32×Q8—24no—
207C32×Q8—22no—
208C32×Q8—22no—
209C32×Q8—42no—
210C32×Q8—42no—
211C32×Q8—44no—
212C2×C32×C4—21no—
213C2×C32×C4—21no—
214C2×C32×C4—22no—
215C2×C32×C4—21no—
216C2×C32×C4—22no—
217C2×C32×C4—22no—
218C2×C32×C4—41no—
219C2×C32×C4—42no—
220C2×C32×C4C22×D312yes—
221C2×C32×C4C22×D312yes—
222C2×C32×C4C22×D312yes—
223C2×C32×C4C22×D311yescharacteristic
224C2×C32×C4C22×D311yescharacteristic
225C2×C32×C4C22×D311yescharacteristic
226C2×C3⋊Dic3C22×D312yes—
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D312yes—
229C2×C3⋊Dic3C22×D312yes—
230C2×C3⋊Dic3C22×D311yescharacteristic
231C4⋊(C3⋊D3)—22no—
232C4×C3⋊D3—24no—
233C4×C3⋊D3#6714yes—
234#634—22no—
235#634—24no—
236#634—22no—
237C2×C3×Dic3—31no—
238C2×C3×Dic3—31no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—32no—
241C2×C3×Dic3—32no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—32no—
244C2×C3×Dic3—32no—
245C2×C3×Dic3—32no—
246C2×C3×Dic3—32no—
247C2×C3×Dic3—62no—
248C2×C3×Dic3—61no—
249C2×C3×Dic3—62no—
250C2×C3×Dic3—62no—
251C2×C3×Dic3—62no—
252C3×D12—62no—
253C3×D12—62no—
254C3×D12—122no—
255C3×C4×D3—34no—
256C3×C4×D3—34no—
257C3×C4×D3—64no—
258C3×C4×D3—64no—
259C3×C4×D3—64no—
260C3×C4×D3—68no—
261C3×Dic6—64no—
262C3×Dic6—64no—
263C3×Dic6—68no—
264C3×Dic6—64no—
265C3×Dic6—62no—
266C3×Dic6—62no—
267C3×Dic6—62no—
268C3×Dic6—62no—
269C3×Dic6—122no—
270#626—64no—
271#626—64no—
272#625—64no—
273#624—64no—
274D3×Dic3—64no—
275D3×Dic3—64no—
276#536—271noSylow
277C2×C33C22×D411yescharacteristic
278C2×C332−1+412yes—
279C3×C3⋊D3—22no—
280C3×C3⋊D3—42no—
281C2×C3×Q8—21no—
282C2×C3×Q8—21no—
283C2×C3×Q8—21no—
284C2×C3×Q8—41no—
285C2×C3×Q8—91no—
286C2×C3×Q8—91no—
287C2×C3×Q8—92no—
288C2×C3×Q8—184no—
289C2×C3×Q8D3211yescharacteristic
290C2×C3×Q8D3211yescharacteristic
291C2×C3×Q8D3211yescharacteristic
292C2×C3×D4—91no—
293C22×C3×C4—91no—
294C22×C3×C4—92no—
295C2×#67—92no—
296C2×#67—92no—
297C22×Dic3—92no—
298#239—184no—
299D3×Q8—64no—
300D3×Q8—64no—
301D3×Q8—68no—
302#237—184no—
303#235—184no—
304C2×D12—31no—
305C2×D12—31no—
306C2×D12—61no—
307C2×D12—92no—
308C2×C4×D3—31no—
309C2×C4×D3—31no—
310C2×C4×D3—32no—
311C2×C4×D3—32no—
312C2×C4×D3—61no—
313C2×C4×D3—62no—
314C2×C4×D3—92no—
315C2×C4×D3—92no—
316C2×Dic6—31no—
317C2×Dic6—31no—
318C2×Dic6—31no—
319C2×Dic6—32no—
320C2×Dic6—32no—
321C2×Dic6—31no—
322C2×Dic6—62no—
323C2×Dic6—61no—
324C2×Dic6—61no—
325C3×C4⋊C4—94no—
326C3×C22⋊C4—94no—
327C3×C42—91no—
328C3×C42—91no—
329#217—94no—
330#217—92no—
331(C2×D3)⋊C4—34no—
332(C2×D3)⋊C4—34no—
333(C2×D3)⋊C4—64no—
334(C2×D3)⋊C4—92no—
335(C2×D3)⋊C4—92no—
336(C2×D3)⋊C4—94no—
337(C2×D3)⋊C4—94no—
338C4⋊Dic3—94no—
339C4⋊Dic3—92no—
340Dic3⋊C4—34no—
341Dic3⋊C4—34no—
342Dic3⋊C4—64no—
343Dic3⋊C4—92no—
344Dic3⋊C4—92no—
345Dic3⋊C4—94no—
346Dic3⋊C4—94no—
347C4×Dic3—31no—
348C4×Dic3—31no—
349C4×Dic3—31no—
350C4×Dic3—31no—
351C4×Dic3—61no—
352C4×Dic3—61no—
353C22×C32—21no—
354C22×C32—21no—
355C22×C32—21no—
356C22×C32—41no—
357C22×C32C23×D311yescharacteristic
358C22×C32C23×D311yescharacteristic
359C22×C32C23×D311yescharacteristic
360C2×C3⋊D3—22no—
361C2×C3⋊D3—22no—
362C2×C3⋊D3—24no—
363C2×C3⋊D3C2×#6712yes—
364C2×C3×D3—32no—
365C2×C3×D3—32no—
366C2×C3×D3—62no—
367C2×C3×D3—64no—
368C2×C3×D3—62no—
369C2×C3×D3—62no—
370C2×C3×D3—62no—
371C2×C3×D3—64no—
372C2×C3×D3—64no—
373C2×C3×D3—62no—
374C2×C3×D3—122no—
375C2×C3×D3—124no—
376C32×C4—22no—
377C32×C4—24no—
378C32×C4—22no—
379C32×C4—22no—
380C32×C4—22no—
381C32×C4—22no—
382C32×C4—22no—
383C32×C4—42no—
384C32×C4—42no—
385C32×C4—42no—
386C32×C4—44no—
387C32×C4D3×D412yes—
388C32×C4D3×D412yes—
389C32×C4C2×#6712yes—
390C3⋊Dic3—22no—
391C3⋊Dic3—22no—
392C3⋊Dic3—22no—
393C3⋊Dic3—22no—
394C3⋊Dic3C2×#6712yes—
395C3×Dic3—32no—
396C3×Dic3—32no—
397C3×Dic3—62no—
398C3×Dic3—64no—
399C3×Dic3—64no—
400C3×Dic3—62no—
401C3×Dic3—62no—
402C3×Dic3—62no—
403C3×Dic3—62no—
404C3×Dic3—62no—
405C3×Dic3—62no—
406C3×Dic3—62no—
407C3×Dic3—62no—
408C3×Dic3—122no—
409C3×Dic3—122no—
410C2×#41—271no—
411C22×Q8—91no—
412C4⋊Q8—91no—
413C4⋊Q8—271no—
414#124—91no—
415#124—271no—
416#123—274no—
417C22⋊Q8—274no—
418#117—271no—
419C33#53611yescharacteristic, complemented, Sylow
420C23×C3—91no—
421C23×C3—92no—
422C22×D3—31no—
423C22×D3—31no—
424C22×D3—32no—
425C22×D3—32no—
426C22×D3—61no—
427C22×D3—62no—
428C3×Q8—24no—
429C3×Q8—22no—
430C3×Q8—22no—
431C3×Q8—22no—
432C3×Q8—42no—
433C3×Q8—42no—
434C3×Q8—44no—
435C3×Q8—184no—
436C3×Q8—182no—
437C3×Q8—182no—
438C3×D4—182no—
439C2×C3×C4—21no—
440C2×C3×C4—21no—
441C2×C3×C4—22no—
442C2×C3×C4—21no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—42no—
446C2×C3×C4—41no—
447C2×C3×C4—91no—
448C2×C3×C4—94no—
449C2×C3×C4—92no—
450C2×C3×C4—92no—
451C2×C3×C4—92no—
452C2×C3×C4—92no—
453C2×C3×C4—184no—
454C2×C3×C4C2×D3212yes—
455C2×C3×C4C2×D3212yes—
456C2×C3×C4C2×D3212yes—
457C2×C3×C4C2×D3211yescharacteristic
458C2×C3×C4C2×D3211yescharacteristic
459C2×C3×C4C2×D3211yescharacteristic
460#67—184no—
461#67—184no—
462C2×Dic3—31no—
463C2×Dic3—31no—
464C2×Dic3—32no—
465C2×Dic3—32no—
466C2×Dic3—32no—
467C2×Dic3—32no—
468C2×Dic3—32no—
469C2×Dic3—32no—
470C2×Dic3—32no—
471C2×Dic3—32no—
472C2×Dic3—62no—
473C2×Dic3—61no—
474C2×Dic3—62no—
475C2×Dic3—62no—
476C2×Dic3—62no—
477C2×Dic3—184no—
478D12—62no—
479D12—62no—
480D12—122no—
481D12—184no—
482C4×D3—34no—
483C4×D3—34no—
484C4×D3—64no—
485C4×D3—64no—
486C4×D3—64no—
487C4×D3—68no—
488C4×D3—184no—
489C4×D3—184no—
490Dic6—64no—
491Dic6—64no—
492Dic6—68no—
493Dic6—64no—
494Dic6—62no—
495Dic6—62no—
496Dic6—62no—
497Dic6—62no—
498Dic6—122no—
499C2×C32—22no—
500C2×C32—22no—
501C2×C32—22no—
502C2×C32—21no—
503C2×C32—21no—
504C2×C32—21no—
505C2×C32—41no—
506C2×C32—42no—
507C2×C32C2×D3×D411yescharacteristic
508C2×C32C2×D3×D411yescharacteristic
509C2×C32#100712yes—
510C2×C32#100712yes—
511C2×C32#101012yes—
512C2×C32C22×#6711yescharacteristic
513C3⋊D3—22no—
514C3⋊D3—42no—
515C3×D3—64no—
516C3×D3—62no—
517C3×D3—62no—
518C3×D3—122no—
519C3×D3—122no—
520C3×D3—242no—
521#41—544no—
522C2×Q8—92no—
523C2×Q8—91no—
524C2×Q8—91no—
525C2×Q8—184no—
526C2×Q8#111311yescharacteristic
527C2×D4—91no—
528C2×D4—272no—
529C22×C4—91no—
530C22×C4—92no—
531C22×C4—272no—
532C4⋊C4—94no—
533C4⋊C4—272no—
534C4⋊C4—274no—
535C22⋊C4—94no—
536C22⋊C4—272no—
537C22⋊C4—274no—
538C42—91no—
539C42—91no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—41no—
544C22×C3—92no—
545C22×C3—182no—
546C22×C3—184no—
547C22×C3—182no—
548C22×C3C22×D3211yescharacteristic
549C22×C3C22×D3211yescharacteristic
550C22×C3C22×D3211yescharacteristic
551C2×D3—32no—
552C2×D3—32no—
553C2×D3—62no—
554C2×D3—64no—
555C2×D3—62no—
556C2×D3—62no—
557C2×D3—64no—
558C2×D3—64no—
559C2×D3—62no—
560C2×D3—62no—
561C2×D3—122no—
562C2×D3—124no—
563C3×C4—24no—
564C3×C4—22no—
565C3×C4—22no—
566C3×C4—22no—
567C3×C4—22no—
568C3×C4—22no—
569C3×C4—22no—
570C3×C4—42no—
571C3×C4—42no—
572C3×C4—44no—
573C3×C4—42no—
574C3×C4—92no—
575C3×C4—182no—
576C3×C4—182no—
577C3×C4—182no—
578C3×C4—182no—
579C3×C4D3×#6712yes—
580C3×C4D3×#6712yes—
581C3×C4#381112yes—
582Dic3—32no—
583Dic3—32no—
584Dic3—62no—
585Dic3—64no—
586Dic3—64no—
587Dic3—62no—
588Dic3—62no—
589Dic3—62no—
590Dic3—62no—
591Dic3—62no—
592Dic3—62no—
593Dic3—62no—
594Dic3—62no—
595Dic3—122no—
596Dic3—122no—
597C32—21no—
598C32—21no—
599C32—21no—
600C32—41no—
601C32#567511yescharacteristic, complemented
602C32#572411yescharacteristic, complemented
603C32#581611yescharacteristic, complemented
604C23—91no—
605C23—92no—
606Q8—22no—
607Q8—184no—
608Q8—182no—
609Q8—182no—
610D4—182no—
611D4—544no—
612C2×C4—91no—
613C2×C4—94no—
614C2×C4—92no—
615C2×C4—92no—
616C2×C4—92no—
617C2×C4—92no—
618C2×C4—184no—
619C2×C4—544no—
620C2×C4C2×#111312yes—
621C2×C4C2×#111311yescharacteristic
622C2×C3—22no—
623C2×C3—22no—
624C2×C3—22no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—21no—
628C2×C3—42no—
629C2×C3—41no—
630C2×C3—182no—
631C2×C3—362no—
632C2×C3#6431512yes—
633C2×C3#6431512yes—
634C2×C3#6431612yes—
635C2×C3C2×D3×#6711yescharacteristic
636C2×C3C2×D3×#6711yescharacteristic
637C2×C3C2×#381111yescharacteristic
638D3—64no—
639D3—62no—
640D3—62no—
641D3—122no—
642D3—122no—
643D3—242no—
644C22—92no—
645C22—182no—
646C22—184no—
647C22—182no—
648C22C22×#111311yescharacteristic
649C4—22no—
650C4—92no—
651C4—182no—
652C4—182no—
653C4—182no—
654C4—182no—
655C4#8904912yes—
656C3—21no—
657C3—21no—
658C3—21no—
659C3—41no—
660C3#1059534011yescharacteristic, complemented
661C3#1059536411yescharacteristic, complemented
662C3#1059539911yescharacteristic, complemented
663C2—182no—
664C2—362no—
665C2#1172462412yes—
666C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×#89041C2
hypercenterC2×Q8#1113
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
3. C3×#64156C2semidirect
4. C3×#64153C2semidirect
5. #11723694C2semidirect
6. #11723688C2semidirect
7. #11723678C2semidirect
8. #11723677C2semidirect
9. #11723673C2semidirect
32. C2×Q8×C3⋊D3D3semidirect
39. #64156D3semidirect
40. #64153D3semidirect
419. C33#536semidirect
601. C32#5675semidirect
602. C32#5724semidirect
603. C32#5816semidirect
660. C3#10595340semidirect
661. C3#10595364semidirect
662. C3#10595399semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×#890412
C3×#641562
C3×#641532
#117236942
#117236882
#117236782
#117236772
#117236732
#105953993
#105953643
#105953403