Finite Group Databasebeta

← #13178739

Subgroups — #13178739

Order 1728.

1586 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×#89041C211yesmaximal, characteristic, complemented
3C3×#64156C211yesmaximal, characteristic, complemented
4C3×#64153C211yesmaximal, characteristic, complemented
5#11723697C211yesmaximal, characteristic, complemented
6#11723687C214yesmaximal, complemented
7#11723684C214yesmaximal, complemented
8C4×#6294C211yesmaximal, characteristic, complemented
9#11723677C211yesmaximal, characteristic, complemented
10#10595363—31nomaximal
11#10595360—31nomaximal
12#10595359—31nomaximal
13C2×C33×Q8C2211yescharacteristic
14C3×#3832—24no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2212yes—
18C2×C3×C4×C3⋊D3C2212yes—
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C3×#634C2211yescharacteristic
21C2×#6295C2212yes—
22C2×#6295C2214yes—
23C2×#6294C2212yes—
24C2×#6294C2212yes—
25#89041—24no—
26C3×#3752C2214yes—
27C3×#3750C2214yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C3×C4×C3⋊Dic3C2211yescharacteristic
30#88814C2214yes—
31#88813C2212yes—
32C2×#3832D311yescharacteristic, complemented
33C2×C3×#239—31no—
34C2×C3×#239—31no—
35C2×C3×#239—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#63914—31no—
48#63911—31no—
49#63911—31no—
50#63910—34no—
51#63904—34no—
52#63886—34no—
53#63885—34no—
54C4×#624—31no—
55#63881—34no—
56#63879—34no—
57C4×D3×Dic3—31no—
58C4×D3×Dic3—31no—
59#63864—31no—
60#63863—31no—
61#63860—31no—
62C22×C3×C3⋊D3C2312yes—
63C22×C3×C3⋊D3C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C33×Q8—22no—
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C3×C4⋊(C3⋊D3)—24no—
73C3×C4⋊(C3⋊D3)—22no—
74C3×C4⋊(C3⋊D3)—22no—
75C3×C4×C3⋊D3—24no—
76C3×C4×C3⋊D3D414yescomplemented
77C3×#634—22no—
78#6295—24no—
79#6295—28no—
80#6294—24no—
81#6294—24no—
82#5832—91no—
83#5725—91no—
84#5668—91no—
85C2×C32×Q8—21no—
86C2×C32×Q8—21no—
87C2×C32×Q8—21no—
88C2×C32×Q8—41no—
89C2×C32×Q8C2×D311yescharacteristic
90C2×C32×Q8C2×D311yescharacteristic
91C2×C32×Q8C2×D311yescharacteristic
92#3832—24no—
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
94C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
95C2×C4⋊(C3⋊D3)C2×D312yes—
96C2×C4×C3⋊D3C2×D312yes—
97C2×C4×C3⋊D3C2×D311yescharacteristic
98C2×#634C2×D311yescharacteristic
99C3×#239—64no—
100C3×#239—64no—
101C3×#239—68no—
102C2×C3×D12—31no—
103C2×C3×D12—31no—
104C2×C3×D12—31no—
105C2×C3×D12—31no—
106C2×C3×D12—32no—
107C2×C3×D12—32no—
108C2×C3×D12—61no—
109C2×C3×D12—62no—
110C2×C3×D12—61no—
111C2×C3×C4×D3—31no—
112C2×C3×C4×D3—31no—
113C2×C3×C4×D3—32no—
114C2×C3×C4×D3—32no—
115C2×C3×C4×D3—61no—
116C2×C3×C4×D3—62no—
117C2×C3×Dic6—31no—
118C2×C3×Dic6—31no—
119C2×C3×Dic6—61no—
120C2×#626—32no—
121C2×#626—32no—
122C2×#626—34no—
123C2×#626—34no—
124C2×#625—32no—
125C2×#625—34no—
126C2×#624—32no—
127C2×#624—32no—
128C2×D3×Dic3—32no—
129C2×D3×Dic3—32no—
130C2×D3×Dic3—32no—
131C2×D3×Dic3—32no—
132#3799—64no—
133#3796—64no—
134#3795—64no—
135#3752C2×D314yes—
136#3750C2×D314yes—
137C4×C3⋊Dic3C2×D311yescharacteristic
138C4×C3⋊Dic3C2×D311yescharacteristic
139C3×(C2×D3)⋊C4—34no—
140C3×(C2×D3)⋊C4—34no—
141C3×(C2×D3)⋊C4—64no—
142C3×Dic3⋊C4—34no—
143C3×Dic3⋊C4—34no—
144C3×Dic3⋊C4—64no—
145C3×C4×Dic3—31no—
146C3×C4×Dic3—31no—
147C3×C4×Dic3—31no—
148C3×C4×Dic3—31no—
149C3×C4×Dic3—61no—
150C3×C4×Dic3—61no—
151#3722—34no—
152(C2×D3)⋊Dic3—34no—
153(C2×D3)⋊Dic3—34no—
154Dic32—32no—
155Dic32—32no—
156Dic32—32no—
157C22×C33C2411yescharacteristic
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—22no—
160C2×C3×C3⋊D3—22no—
161C2×C3×C3⋊D3—24no—
162C2×C3×C3⋊D3—24no—
163C2×C3×C3⋊D3C2×D412yes—
164C33×C4—22no—
165C33×C4C2×D412yes—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3C2×D412yes—
169C3×C3⋊Dic3#4114yescomplemented
170C2×C3×#41—91no—
171C2×#239—31no—
172C2×#239—31no—
173C2×#239—61no—
174C2×#239—91no—
175C2×#237—91no—
176C2×#235—91no—
177C3×C4⋊Q8—91no—
178C3×#124—91no—
179#947—31no—
180#947—31no—
181#947—61no—
182#947—91no—
183#944—31no—
184#944—31no—
185#944—61no—
186#940—91no—
187#940—91no—
188#939—94no—
189#936—91no—
190C4×#67—94no—
191C2×C4×Dic3—91no—
192#895—94no—
193#893—94no—
194#884—94no—
195#67⋊C4—94no—
196#875—91no—
197#874—91no—
198C42×D3—91no—
199C42×D3—91no—
200C22×C3⋊D3C22×D312yes—
201C22×C3⋊D3C22×D312yes—
202C22×C3⋊D3C22×D311yescharacteristic
203C22×C3×D3—31no—
204C22×C3×D3—31no—
205C22×C3×D3—32no—
206C22×C3×D3—32no—
207C22×C3×D3—32no—
208C22×C3×D3—32no—
209C22×C3×D3—61no—
210C22×C3×D3—62no—
211C22×C3×D3—62no—
212C32×Q8—22no—
213C32×Q8—24no—
214C32×Q8—22no—
215C32×Q8—22no—
216C32×Q8—42no—
217C32×Q8—42no—
218C32×Q8—44no—
219C2×C32×C4—21no—
220C2×C32×C4—21no—
221C2×C32×C4—22no—
222C2×C32×C4—21no—
223C2×C32×C4—22no—
224C2×C32×C4—22no—
225C2×C32×C4—41no—
226C2×C32×C4—42no—
227C2×C32×C4C22×D312yes—
228C2×C32×C4C22×D312yes—
229C2×C32×C4C22×D312yes—
230C2×C32×C4C22×D311yescharacteristic
231C2×C32×C4C22×D311yescharacteristic
232C2×C32×C4C22×D311yescharacteristic
233C2×C3⋊Dic3C22×D312yes—
234C2×C3⋊Dic3C22×D312yes—
235C2×C3⋊Dic3C22×D312yes—
236C2×C3⋊Dic3C22×D311yescharacteristic
237C4⋊(C3⋊D3)—24no—
238C4⋊(C3⋊D3)—22no—
239C4⋊(C3⋊D3)—22no—
240C4×C3⋊D3—24no—
241C4×C3⋊D3#6714yescomplemented
242#634—22no—
243C2×C3×Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—32no—
246C2×C3×Dic3—32no—
247C2×C3×Dic3—32no—
248C2×C3×Dic3—32no—
249C2×C3×Dic3—32no—
250C2×C3×Dic3—32no—
251C2×C3×Dic3—62no—
252C2×C3×Dic3—61no—
253C2×C3×Dic3—62no—
254C2×C3×Dic3—62no—
255C3×D12—64no—
256C3×D12—64no—
257C3×D12—68no—
258C3×D12—64no—
259C3×D12—62no—
260C3×D12—62no—
261C3×D12—62no—
262C3×D12—62no—
263C3×D12—122no—
264C3×C4×D3—34no—
265C3×C4×D3—34no—
266C3×C4×D3—64no—
267C3×C4×D3—64no—
268C3×C4×D3—64no—
269C3×C4×D3—68no—
270C3×Dic6—62no—
271C3×Dic6—62no—
272C3×Dic6—122no—
273#626—68no—
274#626—68no—
275#626—64no—
276#626—64no—
277#625—64no—
278#625—68no—
279#624—64no—
280#624—64no—
281D3×Dic3—64no—
282D3×Dic3—64no—
283D3×Dic3—64no—
284D3×Dic3—64no—
285#532—271noSylow
286C2×C33C22×D411yescharacteristic
287C2×C33C2×#4112yes—
288C3×C3⋊D3—22no—
289C3×C3⋊D3—42no—
290C3×C3⋊D3—42no—
291C3×#41—184no—
292C2×C3×Q8—21no—
293C2×C3×Q8—21no—
294C2×C3×Q8—21no—
295C2×C3×Q8—41no—
296C2×C3×Q8—91no—
297C2×C3×Q8D3211yescharacteristic
298C2×C3×Q8D3211yescharacteristic
299C2×C3×Q8D3211yescharacteristic
300C2×C3×D4—91no—
301C2×C3×D4—92no—
302C2×C3×D4—91no—
303C22×C3×C4—91no—
304C22×C3×C4—92no—
305C2×#67—94no—
306C2×#67—92no—
307C2×#67—94no—
308C2×#67—92no—
309C22×Dic3—92no—
310C22×Dic3—92no—
311#239—64no—
312#239—64no—
313#239—68no—
314#239—184no—
315#237—184no—
316#235—184no—
317C2×D12—31no—
318C2×D12—31no—
319C2×D12—31no—
320C2×D12—32no—
321C2×D12—32no—
322C2×D12—31no—
323C2×D12—62no—
324C2×D12—61no—
325C2×D12—61no—
326C2×D12—94no—
327C2×D12—92no—
328C2×C4×D3—31no—
329C2×C4×D3—31no—
330C2×C4×D3—32no—
331C2×C4×D3—32no—
332C2×C4×D3—61no—
333C2×C4×D3—62no—
334C2×C4×D3—92no—
335C2×C4×D3—92no—
336C2×C4×D3—92no—
337C2×C4×D3—92no—
338C2×Dic6—31no—
339C2×Dic6—31no—
340C2×Dic6—61no—
341C3×C4⋊C4—94no—
342C3×C22⋊C4—94no—
343C3×C42—91no—
344C3×C42—91no—
345#217—94no—
346(C2×D3)⋊C4—34no—
347(C2×D3)⋊C4—34no—
348(C2×D3)⋊C4—64no—
349(C2×D3)⋊C4—94no—
350(C2×D3)⋊C4—94no—
351Dic3⋊C4—34no—
352Dic3⋊C4—34no—
353Dic3⋊C4—64no—
354C4×Dic3—31no—
355C4×Dic3—31no—
356C4×Dic3—31no—
357C4×Dic3—31no—
358C4×Dic3—61no—
359C4×Dic3—61no—
360C4×Dic3—92no—
361C4×Dic3—92no—
362C4×Dic3—92no—
363C22×C32—21no—
364C22×C32—21no—
365C22×C32—21no—
366C22×C32—41no—
367C22×C32C23×D311yescharacteristic
368C22×C32C23×D311yescharacteristic
369C22×C32C23×D311yescharacteristic
370C2×C3⋊D3—22no—
371C2×C3⋊D3—22no—
372C2×C3⋊D3—22no—
373C2×C3⋊D3—24no—
374C2×C3⋊D3—24no—
375C2×C3⋊D3C2×#6712yes—
376C2×C3×D3—32no—
377C2×C3×D3—32no—
378C2×C3×D3—62no—
379C2×C3×D3—62no—
380C2×C3×D3—62no—
381C2×C3×D3—64no—
382C2×C3×D3—62no—
383C2×C3×D3—62no—
384C2×C3×D3—62no—
385C2×C3×D3—64no—
386C2×C3×D3—64no—
387C2×C3×D3—64no—
388C2×C3×D3—64no—
389C2×C3×D3—62no—
390C2×C3×D3—122no—
391C2×C3×D3—124no—
392C2×C3×D3—122no—
393C2×C3×D3—124no—
394C32×C4—22no—
395C32×C4—24no—
396C32×C4—22no—
397C32×C4—22no—
398C32×C4—22no—
399C32×C4—22no—
400C32×C4—22no—
401C32×C4—42no—
402C32×C4—42no—
403C32×C4—42no—
404C32×C4—44no—
405C32×C4D3×D412yes—
406C32×C4D3×D412yes—
407C32×C4C2×#6712yes—
408C3⋊Dic3—22no—
409C3⋊Dic3—22no—
410C3⋊Dic3#23914yescomplemented
411C3⋊Dic3C2×#6712yes—
412C3×Dic3—32no—
413C3×Dic3—32no—
414C3×Dic3—34no—
415C3×Dic3—34no—
416C3×Dic3—62no—
417C3×Dic3—64no—
418C3×Dic3—62no—
419C3×Dic3—62no—
420C3×Dic3—62no—
421C3×Dic3—62no—
422C3×Dic3—122no—
423C3×Dic3—122no—
424C2×#41—91no—
425C2×#41—271no—
426C4⋊Q8—91no—
427#127—271no—
428#124—91no—
429#124—271no—
430#121—274no—
431C4×D4—274no—
432C2×C42—271no—
433C33#53211yescharacteristic, complemented, Sylow
434C23×C3—91no—
435C23×C3—92no—
436C23×C3—92no—
437C22×D3—31no—
438C22×D3—31no—
439C22×D3—32no—
440C22×D3—32no—
441C22×D3—32no—
442C22×D3—32no—
443C22×D3—61no—
444C22×D3—62no—
445C22×D3—62no—
446C3×Q8—24no—
447C3×Q8—22no—
448C3×Q8—22no—
449C3×Q8—22no—
450C3×Q8—42no—
451C3×Q8—42no—
452C3×Q8—44no—
453C3×Q8—182no—
454C3×D4—184no—
455C3×D4—182no—
456C3×D4—182no—
457C2×C3×C4—21no—
458C2×C3×C4—21no—
459C2×C3×C4—22no—
460C2×C3×C4—21no—
461C2×C3×C4—22no—
462C2×C3×C4—22no—
463C2×C3×C4—42no—
464C2×C3×C4—41no—
465C2×C3×C4—91no—
466C2×C3×C4—92no—
467C2×C3×C4—94no—
468C2×C3×C4—92no—
469C2×C3×C4—92no—
470C2×C3×C4—184no—
471C2×C3×C4C2×D3212yes—
472C2×C3×C4C2×D3212yes—
473C2×C3×C4C2×D3212yes—
474C2×C3×C4C2×D3211yescharacteristic
475C2×C3×C4C2×D3211yescharacteristic
476C2×C3×C4C2×D3211yescharacteristic
477#67—188no—
478#67—188no—
479#67—184no—
480#67—184no—
481C2×Dic3—32no—
482C2×Dic3—32no—
483C2×Dic3—31no—
484C2×Dic3—31no—
485C2×Dic3—32no—
486C2×Dic3—32no—
487C2×Dic3—32no—
488C2×Dic3—32no—
489C2×Dic3—62no—
490C2×Dic3—61no—
491C2×Dic3—62no—
492C2×Dic3—62no—
493C2×Dic3—184no—
494C2×Dic3—184no—
495D12—64no—
496D12—64no—
497D12—68no—
498D12—64no—
499D12—62no—
500D12—62no—
501D12—62no—
502D12—62no—
503D12—122no—
504D12—188no—
505D12—184no—
506C4×D3—34no—
507C4×D3—34no—
508C4×D3—64no—
509C4×D3—64no—
510C4×D3—64no—
511C4×D3—68no—
512C4×D3—184no—
513C4×D3—184no—
514C4×D3—184no—
515C4×D3—184no—
516Dic6—62no—
517Dic6—62no—
518Dic6—122no—
519C2×C32—22no—
520C2×C32—22no—
521C2×C32—22no—
522C2×C32—21no—
523C2×C32—21no—
524C2×C32—21no—
525C2×C32—41no—
526C2×C32—42no—
527C2×C32C2×D3×D411yescharacteristic
528C2×C32C2×D3×D411yescharacteristic
529C2×C32C2×#23912yes—
530C2×C32D3×#4112yes—
531C2×C32D3×#4112yes—
532C2×C32C22×#6711yescharacteristic
533C3⋊D3—22no—
534C3⋊D3—42no—
535C3⋊D3—42no—
536C3×D3—62no—
537C3×D3—62no—
538C3×D3—122no—
539C3×D3—124no—
540C3×D3—122no—
541C3×D3—122no—
542C3×D3—122no—
543C3×D3—122no—
544C3×D3—242no—
545#41—184no—
546#41—544no—
547C2×Q8—91no—
548C2×Q8#111311yescharacteristic
549C2×D4—92no—
550C2×D4—91no—
551C2×D4—91no—
552C2×D4—274no—
553C2×D4—272no—
554C22×C4—91no—
555C22×C4—92no—
556C22×C4—272no—
557C22×C4—272no—
558C4⋊C4—94no—
559C22⋊C4—94no—
560C22⋊C4—274no—
561C42—91no—
562C42—91no—
563C42—272no—
564C22×C3—21no—
565C22×C3—21no—
566C22×C3—21no—
567C22×C3—41no—
568C22×C3—92no—
569C22×C3—182no—
570C22×C3—184no—
571C22×C3—184no—
572C22×C3—182no—
573C22×C3—182no—
574C22×C3C22×D3211yescharacteristic
575C22×C3C22×D3211yescharacteristic
576C22×C3C22×D3211yescharacteristic
577C2×D3—32no—
578C2×D3—32no—
579C2×D3—62no—
580C2×D3—62no—
581C2×D3—62no—
582C2×D3—64no—
583C2×D3—64no—
584C2×D3—64no—
585C2×D3—64no—
586C2×D3—64no—
587C2×D3—62no—
588C2×D3—62no—
589C2×D3—62no—
590C2×D3—62no—
591C2×D3—122no—
592C2×D3—124no—
593C2×D3—122no—
594C2×D3—124no—
595C3×C4—24no—
596C3×C4—22no—
597C3×C4—22no—
598C3×C4—22no—
599C3×C4—22no—
600C3×C4—22no—
601C3×C4—22no—
602C3×C4—42no—
603C3×C4—42no—
604C3×C4—44no—
605C3×C4—42no—
606C3×C4—94no—
607C3×C4—92no—
608C3×C4—182no—
609C3×C4—182no—
610C3×C4D3×#6712yes—
611C3×C4D3×#6712yes—
612C3×C4#381112yes—
613Dic3—34no—
614Dic3—34no—
615Dic3—32no—
616Dic3—32no—
617Dic3—62no—
618Dic3—64no—
619Dic3—62no—
620Dic3—62no—
621Dic3—62no—
622Dic3—62no—
623Dic3—122no—
624Dic3—122no—
625C32—21no—
626C32—21no—
627C32—21no—
628C32—41no—
629C32#566811yescharacteristic, complemented
630C32#572511yescharacteristic, complemented
631C32#583211yescharacteristic, complemented
632C23—91no—
633C23—92no—
634C23—92no—
635Q8—22no—
636Q8—182no—
637D4—184no—
638D4—182no—
639D4—182no—
640D4—548no—
641D4—544no—
642C2×C4—92no—
643C2×C4—91no—
644C2×C4—94no—
645C2×C4—92no—
646C2×C4—92no—
647C2×C4—184no—
648C2×C4—544no—
649C2×C4—544no—
650C2×C4C2×#111312yes—
651C2×C4C2×#111311yescharacteristic
652C2×C3—22no—
653C2×C3—22no—
654C2×C3—22no—
655C2×C3—21no—
656C2×C3—21no—
657C2×C3—21no—
658C2×C3—42no—
659C2×C3—41no—
660C2×C3—182no—
661C2×C3—362no—
662C2×C3—362no—
663C2×C3D3×#23912yes—
664C2×C3D3×#23912yes—
665C2×C3#6431912yes—
666C2×C3C2×D3×#6711yescharacteristic
667C2×C3C2×D3×#6711yescharacteristic
668C2×C3C2×#381111yescharacteristic
669D3—62no—
670D3—62no—
671D3—122no—
672D3—124no—
673D3—122no—
674D3—122no—
675D3—122no—
676D3—122no—
677D3—242no—
678C22—92no—
679C22—182no—
680C22—184no—
681C22—184no—
682C22—182no—
683C22—182no—
684C22C22×#111311yescharacteristic
685C4—22no—
686C4—94no—
687C4—92no—
688C4—182no—
689C4—182no—
690C4#8904912yes—
691C3—21no—
692C3—21no—
693C3—21no—
694C3—41no—
695C3#1059535911yescharacteristic, complemented
696C3#1059536011yescharacteristic, complemented
697C3#1059536311yescharacteristic, complemented
698C2—182no—
699C2—362no—
700C2—362no—
701C2#1172462612yes—
702C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#532
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×#3832C2semidirect
2. C2×#89041C2semidirect
3. C3×#64156C2semidirect
4. C3×#64153C2semidirect
5. #11723697C2semidirect
6. #11723687C2semidirect
7. #11723684C2semidirect
8. C4×#6294C2semidirect
9. #11723677C2semidirect
32. C2×#3832D3semidirect
39. #64156D3semidirect
40. #64153D3semidirect
76. C3×C4×C3⋊D3D4semidirect
169. C3×C3⋊Dic3#41semidirect
241. C4×C3⋊D3#67semidirect
410. C3⋊Dic3#239semidirect
433. C33#532semidirect
629. C32#5668semidirect
630. C32#5725semidirect
631. C32#5832semidirect
695. C3#10595359semidirect
696. C3#10595360semidirect
697. C3#10595363semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×#890412
C3×#641562
C3×#641532
#117236972
#117236872
#117236842
C4×#62942
#117236772
#105953633
#105953603
#105953593