Finite Group Databasebeta

← #13169294

Subgroups — #13169294

Order 1728.

1554 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic, complemented
2C2×C3×#3796C211yesmaximal, characteristic, complemented
3#11723529C211yesmaximal, characteristic, complemented
4C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
5#11723451C211yesmaximal, characteristic, complemented
6C3×#63864C211yesmaximal, characteristic, complemented
7C3×#63860C211yesmaximal, characteristic, complemented
8#11722967C214yesmaximal, complemented
9#11722936C214yesmaximal, complemented
10#10595359—31nomaximal
11#10595153—31nomaximal
12#10593971—31nomaximal
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×D12C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×#6295C2212yes—
19C2×#6294C2212yes—
20#89041—24no—
21C2×#6290C2214yes—
22C2×Dic3×C3⋊D3C2212yes—
23C2×C3×#626C2212yes—
24C2×C3×D3×Dic3C2212yes—
25C3×#3796—24no—
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C32×C4×Dic3C2211yescharacteristic
28#88806C2214yes—
29Dic3×C3⋊Dic3C2212yes—
30C3×#3722C2214yes—
31C3×(C2×D3)⋊Dic3C2214yes—
32C2×C3×#239—31no—
33C2×C3×#237—31no—
34C2×#3799—31no—
35C2×#3796—31no—
36C2×#3796D311yescharacteristic, complemented
37C2×#3795—31no—
38#64153—31no—
39#64149—31no—
40C42×C3⋊D3—31no—
41C3×#947—31no—
42C3×#936—31no—
43C3×#875—31no—
44C3×#875—31no—
45#63971—34no—
46#63952—34no—
47#63914—31no—
48#63914—31no—
49#63914—61no—
50#63912—34no—
51#63902—34no—
52#63885—34no—
53#63879—34no—
54C4×D3×Dic3—31no—
55C4×D3×Dic3—31no—
56C4×D3×Dic3—61no—
57#63873—31no—
58#63873—31no—
59#63873—61no—
60#63864D311yescharacteristic, complemented
61#63860D311yescharacteristic, complemented
62C22×C3×C3⋊D3C2312yes—
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2312yes—
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C3×C4⋊(C3⋊D3)—22no—
70C3×C4×C3⋊D3D414yescomplemented
71C3×#634—22no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2312yes—
74C32×D12—22no—
75C32×Dic6—22no—
76#6295—24no—
77#6294—24no—
78#6290—28no—
79Dic3×C3⋊D3—24no—
80C3×#626—24no—
81C3×D3×Dic3—24no—
82#5832—91no—
83#5668—91no—
84#5668—91no—
85C2×C32×Q8—31no—
86C2×C32×D4—31no—
87C2×#638—34no—
88C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
89C2×C4×C3⋊D3—32no—
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×#634C2×D311yescharacteristic
92C2×C3×#67—32no—
93C22×C3×Dic3—32no—
94C3×#239—64no—
95C3×#237—64no—
96C2×C3×D12—21no—
97C2×C3×D12—32no—
98C2×C3×D12—31no—
99C2×C3×D12—31no—
100C2×C3×D12—61no—
101C2×C3×D12C2×D311yescharacteristic
102C2×C3×D12C2×D311yescharacteristic
103C2×C3×C4×D3—31no—
104C2×C3×C4×D3—31no—
105C2×C3×C4×D3—32no—
106C2×C3×C4×D3—61no—
107C2×C3×Dic6—21no—
108C2×C3×Dic6—31no—
109C2×C3×Dic6—31no—
110C2×C3×Dic6—61no—
111C2×C3×Dic6C2×D311yescharacteristic
112C2×C3×Dic6C2×D311yescharacteristic
113C2×#626—32no—
114C2×#626—32no—
115C2×#626C2×D312yes—
116C2×#625—34no—
117C2×#625—34no—
118C2×#625—32no—
119C2×#625—64no—
120C2×#624—32no—
121C2×D3×Dic3—32no—
122C2×D3×Dic3—32no—
123C2×D3×Dic3—32no—
124C2×D3×Dic3—32no—
125C2×D3×Dic3—62no—
126C2×D3×Dic3C2×D312yes—
127#3799—64no—
128#3796—24no—
129#3796—64no—
130#3795—64no—
131C32×C42—31no—
132#3750—34no—
133C4×C3⋊Dic3—31no—
134C4×C3⋊Dic3—31no—
135C4×C3⋊Dic3—31no—
136C4×C3⋊Dic3—32no—
137C4×C3⋊Dic3—61no—
138C4×C3⋊Dic3—61no—
139C4×C3⋊Dic3—61no—
140C4×C3⋊Dic3—121no—
141C3×#217—34no—
142C3×(C2×D3)⋊C4—34no—
143C3×(C2×D3)⋊C4—34no—
144C3×(C2×D3)⋊C4—34no—
145C3×C4×Dic3—21no—
146C3×C4×Dic3C2×D311yescharacteristic
147C3×C4×Dic3C2×D311yescharacteristic
148#3724—34no—
149#3724—34no—
150#3724—64no—
151#3722C2×D314yes—
152(C2×D3)⋊Dic3C2×D314yes—
153Dic32—32no—
154Dic32—32no—
155Dic32—62no—
156C22×C33C2411yescharacteristic
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—24no—
160C2×C3×C3⋊D3C2×D412yes—
161C2×C32×D3—22no—
162C2×C32×D3—24no—
163C33×C4C2×D412yes—
164C3⋊(C3⋊Dic3)#4114yescomplemented
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3C2×D412yes—
167C32×Dic3—22no—
168C32×Dic3—22no—
169C2×C3×#41—91no—
170C2×#239—31no—
171C2×#239—91no—
172C2×#237—31no—
173C2×#237—91no—
174C2×#235—91no—
175C3×#124—91no—
176C3×#124—91no—
177#947—31no—
178#944—91no—
179#944—91no—
180#944—91no—
181#944—181no—
182#940—91no—
183#940—91no—
184#940—91no—
185#940—181no—
186#939—94no—
187#936—31no—
188C4×#67—94no—
189C2×C4×Dic3—91no—
190#884—94no—
191#884—94no—
192#67⋊C4—94no—
193#67⋊C4—94no—
194#875—31no—
195#875—31no—
196C42×D3—91no—
197C42×D3—91no—
198C42×D3—181no—
199C23×C32—32no—
200C22×C3⋊D3C22×D312yes—
201C22×C3⋊D3C22×D311yescharacteristic
202C22×C3×D3—22no—
203C22×C3×D3—31no—
204C22×C3×D3—31no—
205C22×C3×D3—32no—
206C22×C3×D3—32no—
207C22×C3×D3—61no—
208C22×C3×D3—62no—
209C22×C3×D3C22×D312yes—
210C22×C3×D3C22×D312yes—
211C32×Q8—62no—
212C32×D4—62no—
213C2×C32×C4—21no—
214C2×C32×C4—21no—
215C2×C32×C4—21no—
216C2×C32×C4—32no—
217C2×C32×C4—32no—
218C2×C32×C4—41no—
219C2×C32×C4C22×D311yescharacteristic
220C2×C32×C4C22×D311yescharacteristic
221C2×C32×C4C22×D311yescharacteristic
222#638—68no—
223C2×C3⋊Dic3—32no—
224C2×C3⋊Dic3—32no—
225C2×C3⋊Dic3—32no—
226C2×C3⋊Dic3—62no—
227C2×C3⋊Dic3—62no—
228C2×C3⋊Dic3—62no—
229C2×C3⋊Dic3—122no—
230C2×C3⋊Dic3C22×D312yes—
231C2×C3⋊Dic3C22×D311yescharacteristic
232C4⋊(C3⋊D3)—22no—
233C4×C3⋊D3—64no—
234C4×C3⋊D3#6714yescomplemented
235#634—22no—
236C3×#67—64no—
237C2×C3×Dic3—22no—
238C2×C3×Dic3—22no—
239C2×C3×Dic3—31no—
240C2×C3×Dic3—31no—
241C2×C3×Dic3—32no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—61no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—64no—
246C2×C3×Dic3C22×D312yes—
247C2×C3×Dic3C22×D312yes—
248C2×C3×Dic3C22×D312yes—
249C2×C3×Dic3C22×D312yes—
250C3×D12—22no—
251C3×D12—22no—
252C3×D12—42no—
253C3×D12—64no—
254C3×D12—62no—
255C3×D12—62no—
256C3×D12—122no—
257C3×C4×D3—34no—
258C3×C4×D3—34no—
259C3×C4×D3—64no—
260C3×C4×D3—64no—
261C3×Dic6—22no—
262C3×Dic6—22no—
263C3×Dic6—42no—
264C3×Dic6—62no—
265C3×Dic6—62no—
266C3×Dic6—122no—
267#626—24no—
268#626—64no—
269#626—64no—
270#625—64no—
271#625—68no—
272#625—68no—
273#625—128no—
274#624—64no—
275D3×Dic3—24no—
276D3×Dic3—64no—
277D3×Dic3—64no—
278D3×Dic3—68no—
279D3×Dic3—64no—
280D3×Dic3—64no—
281#532—271noSylow
282C2×C33C22×D411yescharacteristic
283C2×C33C2×#4112yes—
284C3×C3⋊D3—22no—
285C3×C3⋊D3—42no—
286C32×D3—42no—
287C3×#41—184no—
288C2×C3×Q8—31no—
289C2×C3×Q8—31no—
290C2×C3×Q8—61no—
291C2×C3×Q8—91no—
292C2×C3×D4—31no—
293C2×C3×D4—31no—
294C2×C3×D4—61no—
295C2×C3×D4—92no—
296C2×C3×D4—91no—
297C22×C3×C4—92no—
298C22×C3×C4—91no—
299C2×#67—32no—
300C2×#67—92no—
301C2×#67—92no—
302C2×#67—94no—
303C2×#67—94no—
304C2×#67—94no—
305C2×#67—184no—
306C22×Dic3—32no—
307C22×Dic3—92no—
308C22×Dic3—92no—
309#239—64no—
310#239—184no—
311#237—64no—
312#237—184no—
313#235—184no—
314C2×D12—32no—
315C2×D12—31no—
316C2×D12—31no—
317C2×D12—61no—
318C2×D12—92no—
319C2×D12D3211yescharacteristic
320C2×C4×D3—31no—
321C2×C4×D3—31no—
322C2×C4×D3—32no—
323C2×C4×D3—61no—
324C2×C4×D3—92no—
325C2×C4×D3—92no—
326C2×C4×D3—92no—
327C2×C4×D3—92no—
328C2×C4×D3—182no—
329C2×Dic6—31no—
330C2×Dic6—31no—
331C2×Dic6—61no—
332C2×Dic6D3211yescharacteristic
333C3×C22⋊C4—94no—
334C3×C22⋊C4—94no—
335C3×C42—31no—
336C3×C42—31no—
337C3×C42—61no—
338#217—34no—
339(C2×D3)⋊C4—34no—
340(C2×D3)⋊C4—34no—
341(C2×D3)⋊C4—34no—
342Dic3⋊C4—94no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—184no—
346C4×Dic3—92no—
347C4×Dic3—92no—
348C4×Dic3—91no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—92no—
352C4×Dic3—182no—
353C4×Dic3—181no—
354C4×Dic3—181no—
355C4×Dic3—181no—
356C4×Dic3—361no—
357C4×Dic3D3211yescharacteristic
358C22×C32—21no—
359C22×C32—21no—
360C22×C32—21no—
361C22×C32—41no—
362C22×C32—62no—
363C22×C32—64no—
364C22×C32C23×D311yescharacteristic
365C22×C32C23×D311yescharacteristic
366C22×C32C23×D311yescharacteristic
367C2×C3⋊D3—22no—
368C2×C3⋊D3—22no—
369C2×C3⋊D3—24no—
370C2×C3⋊D3C2×#6712yes—
371C2×C3×D3—22no—
372C2×C3×D3—24no—
373C2×C3×D3—24no—
374C2×C3×D3—22no—
375C2×C3×D3—32no—
376C2×C3×D3—32no—
377C2×C3×D3—44no—
378C2×C3×D3—42no—
379C2×C3×D3—62no—
380C2×C3×D3—64no—
381C2×C3×D3—62no—
382C2×C3×D3—62no—
383C2×C3×D3—62no—
384C2×C3×D3—64no—
385C2×C3×D3—64no—
386C2×C3×D3—62no—
387C2×C3×D3—124no—
388C2×C3×D3—122no—
389C32×C4—22no—
390C32×C4—22no—
391C32×C4—22no—
392C32×C4—42no—
393C32×C4—62no—
394C32×C4—62no—
395C32×C4D3×D412yes—
396C32×C4D3×D412yes—
397C32×C4C2×#6712yes—
398C3⋊Dic3—22no—
399C3⋊Dic3—34no—
400C3⋊Dic3—34no—
401C3⋊Dic3—34no—
402C3⋊Dic3—64no—
403C3⋊Dic3—64no—
404C3⋊Dic3—64no—
405C3⋊Dic3—124no—
406C3⋊Dic3C2×#6712yes—
407C3×Dic3—24no—
408C3×Dic3—22no—
409C3×Dic3—22no—
410C3×Dic3—22no—
411C3×Dic3—22no—
412C3×Dic3—32no—
413C3×Dic3—32no—
414C3×Dic3—42no—
415C3×Dic3—62no—
416C3×Dic3—62no—
417C3×Dic3—62no—
418C3×Dic3—122no—
419C2×#41—91no—
420C2×#41—271no—
421C4⋊Q8—271no—
422#127—271no—
423#124—91no—
424#124—91no—
425#121—274no—
426C4×D4—274no—
427C2×C42—271no—
428C33#53211yescharacteristic, complemented, Sylow
429C23×C3—32no—
430C23×C3—32no—
431C23×C3—62no—
432C23×C3—91no—
433C23×C3—92no—
434C22×D3—31no—
435C22×D3—31no—
436C22×D3—32no—
437C22×D3—32no—
438C22×D3—61no—
439C22×D3—62no—
440C22×D3C2×D3212yes—
441C3×Q8—62no—
442C3×Q8—62no—
443C3×Q8—122no—
444C3×Q8—182no—
445C3×D4—62no—
446C3×D4—62no—
447C3×D4—122no—
448C3×D4—184no—
449C3×D4—182no—
450C2×C3×C4—21no—
451C2×C3×C4—21no—
452C2×C3×C4—21no—
453C2×C3×C4—32no—
454C2×C3×C4—32no—
455C2×C3×C4—32no—
456C2×C3×C4—32no—
457C2×C3×C4—41no—
458C2×C3×C4—62no—
459C2×C3×C4—62no—
460C2×C3×C4—91no—
461C2×C3×C4—94no—
462C2×C3×C4—92no—
463C2×C3×C4—184no—
464C2×C3×C4C2×D3211yescharacteristic
465C2×C3×C4C2×D3211yescharacteristic
466C2×C3×C4C2×D3211yescharacteristic
467#67—64no—
468#67—184no—
469#67—184no—
470#67—188no—
471#67—188no—
472#67—188no—
473#67—368no—
474C2×Dic3—31no—
475C2×Dic3—31no—
476C2×Dic3—32no—
477C2×Dic3—32no—
478C2×Dic3—61no—
479C2×Dic3—62no—
480C2×Dic3—64no—
481C2×Dic3—92no—
482C2×Dic3—92no—
483C2×Dic3—92no—
484C2×Dic3—182no—
485C2×Dic3—182no—
486C2×Dic3—182no—
487C2×Dic3—184no—
488C2×Dic3—184no—
489C2×Dic3—362no—
490C2×Dic3C2×D3212yes—
491C2×Dic3C2×D3212yes—
492D12—22no—
493D12—64no—
494D12—62no—
495D12—62no—
496D12—122no—
497D12—184no—
498C4×D3—34no—
499C4×D3—34no—
500C4×D3—64no—
501C4×D3—64no—
502C4×D3—184no—
503C4×D3—184no—
504C4×D3—184no—
505C4×D3—184no—
506C4×D3—188no—
507Dic6—22no—
508Dic6—62no—
509Dic6—62no—
510Dic6—122no—
511C2×C32—21no—
512C2×C32—22no—
513C2×C32—22no—
514C2×C32—22no—
515C2×C32—21no—
516C2×C32—21no—
517C2×C32—41no—
518C2×C32—42no—
519C2×C32—122no—
520C2×C32C2×D3×D411yescharacteristic
521C2×C32C2×D3×D411yescharacteristic
522C2×C32D3×#4112yes—
523C2×C32D3×#4112yes—
524C2×C32D3×#4112yes—
525C2×C32C22×#6711yescharacteristic
526C3⋊D3—22no—
527C3⋊D3—42no—
528C3×D3—42no—
529C3×D3—42no—
530C3×D3—64no—
531C3×D3—62no—
532C3×D3—62no—
533C3×D3—82no—
534C3×D3—122no—
535C3×D3—122no—
536C3×D3—242no—
537#41—184no—
538#41—544no—
539C2×Q8—31no—
540C2×Q8—91no—
541C2×D4—31no—
542C2×D4—92no—
543C2×D4—91no—
544C2×D4—272no—
545C2×D4—274no—
546C22×C4—92no—
547C22×C4—91no—
548C22×C4—272no—
549C22×C4—272no—
550C4⋊C4—274no—
551C22⋊C4—94no—
552C22⋊C4—94no—
553C42—31no—
554C42—272no—
555C42—271no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—41no—
560C22×C3—62no—
561C22×C3—64no—
562C22×C3—64no—
563C22×C3—62no—
564C22×C3—92no—
565C22×C3—124no—
566C22×C3—122no—
567C22×C3—182no—
568C22×C3—184no—
569C22×C3—182no—
570C22×C3C22×D3211yescharacteristic
571C22×C3C22×D3211yescharacteristic
572C22×C3C22×D3211yescharacteristic
573C2×D3—24no—
574C2×D3—22no—
575C2×D3—32no—
576C2×D3—32no—
577C2×D3—62no—
578C2×D3—64no—
579C2×D3—62no—
580C2×D3—62no—
581C2×D3—62no—
582C2×D3—64no—
583C2×D3—64no—
584C2×D3—62no—
585C2×D3—122no—
586C2×D3—124no—
587C3×C4—22no—
588C3×C4—22no—
589C3×C4—22no—
590C3×C4—42no—
591C3×C4—64no—
592C3×C4—62no—
593C3×C4—62no—
594C3×C4—62no—
595C3×C4—62no—
596C3×C4—92no—
597C3×C4—122no—
598C3×C4—182no—
599C3×C4#380212yes—
600C3×C4D3×#6712yes—
601C3×C4D3×#6712yes—
602Dic3—22no—
603Dic3—22no—
604Dic3—32no—
605Dic3—32no—
606Dic3—62no—
607Dic3—62no—
608Dic3—62no—
609Dic3—94no—
610Dic3—94no—
611Dic3—94no—
612Dic3—122no—
613Dic3—184no—
614Dic3—184no—
615Dic3—184no—
616Dic3—364no—
617C32—21no—
618C32—21no—
619C32—21no—
620C32—41no—
621C32#566811yescharacteristic, complemented
622C32#566811yescharacteristic, complemented
623C32#583211yescharacteristic, complemented
624C23—32no—
625C23—91no—
626C23—92no—
627Q8—62no—
628Q8—182no—
629D4—62no—
630D4—184no—
631D4—182no—
632D4—544no—
633D4—548no—
634C2×C4—32no—
635C2×C4—32no—
636C2×C4—91no—
637C2×C4—94no—
638C2×C4—92no—
639C2×C4—184no—
640C2×C4—272no—
641C2×C4—544no—
642C2×C4—544no—
643C2×C4D3311yescharacteristic
644C2×C3—22no—
645C2×C3—22no—
646C2×C3—22no—
647C2×C3—21no—
648C2×C3—21no—
649C2×C3—21no—
650C2×C3—42no—
651C2×C3—41no—
652C2×C3—122no—
653C2×C3—122no—
654C2×C3—182no—
655C2×C3—242no—
656C2×C3—362no—
657C2×C3C2×#380211yescharacteristic
658C2×C3#6430612yes—
659C2×C3#6431212yes—
660C2×C3#6431912yes—
661C2×C3C2×D3×#6711yescharacteristic
662C2×C3C2×D3×#6711yescharacteristic
663D3—42no—
664D3—64no—
665D3—62no—
666D3—62no—
667D3—122no—
668D3—122no—
669D3—242no—
670C22—64no—
671C22—62no—
672C22—92no—
673C22—182no—
674C22—184no—
675C22—182no—
676C22C2×D3311yescharacteristic
677C4—62no—
678C4—62no—
679C4—92no—
680C4—182no—
681C4—274no—
682C4#8895312yes—
683C3—21no—
684C3—21no—
685C3—21no—
686C3—41no—
687C3#1059397111yescharacteristic, complemented
688C3#1059515311yescharacteristic, complemented
689C3#1059535911yescharacteristic, complemented
690C2—122no—
691C2—182no—
692C2—362no—
693C2#1172437612yes—
694C2C2×#8895311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89041C2semidirect
2. C2×C3×#3796C2semidirect
3. #11723529C2semidirect
4. C4×Dic3×C3⋊D3C2semidirect
5. #11723451C2semidirect
6. C3×#63864C2semidirect
7. C3×#63860C2semidirect
8. #11722967C2semidirect
9. #11722936C2semidirect
36. C2×#3796D3semidirect
60. #63864D3semidirect
61. #63860D3semidirect
70. C3×C4×C3⋊D3D4semidirect
164. C3⋊(C3⋊Dic3)#41semidirect
234. C4×C3⋊D3#67semidirect
428. C33#532semidirect
621. C32#5668semidirect
622. C32#5668semidirect
623. C32#5832semidirect
687. C3#10593971semidirect
688. C3#10595153semidirect
689. C3#10595359semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C2×C3×#37962
#117235292
C4×Dic3×C3⋊D32
#117234512
C3×#638642
C3×#638602
#117229672
#117229362
#105953593
#105951533
#105939713