Finite Group Databasebeta

← #13169398

Subgroups — #13169398

Order 1728.

1138 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723671C211yesmaximal, characteristic
2C4×Dic3×C3⋊D3C211yesmaximal, characteristic
3#11723448C212yesmaximal, complemented
4C3×#63865C211yesmaximal, characteristic
5C3×#63837C212yesmaximal, complemented
6#11723035C211yesmaximal, characteristic, complemented
7#11723032C211yesmaximal, characteristic, complemented
8#11722974C212yesmaximal
9#11722944C211yesmaximal, characteristic, complemented
10Dic3×#627C211yesmaximal, characteristic, complemented
11#11722919C212yesmaximal
12#10594386—32nomaximal
13#10593882—31nomaximal
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×#6296C2211yescharacteristic
16C2×Dic3×C3⋊D3C2211yescharacteristic
17C2×Dic3×C3⋊D3C2211yescharacteristic
18C2×C3×#627C2212yes—
19C2×C3×#627C2211yescharacteristic
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750C2212yes—
22C32×Dic3⋊C4C2212yes—
23C32×C4×Dic3C2211yescharacteristic
24#88815C2211yescharacteristic
25#88814C2212yes—
26#88807C2212yes—
27#88806C2212yes—
28#88806C2212yes—
29Dic3⋊(C3⋊Dic3)C2212yes—
30Dic3×C3⋊Dic3C2211yescharacteristic
31Dic3×C3⋊Dic3C2211yescharacteristic
32C3×#3724C2211yescharacteristic
33C3×Dic3⋊Dic3C2212yes—
34C3×#3722C2212yes—
35C3×Dic32C2212yes—
36C3×Dic32C2212yes—
37#64098—32no—
38C42×C3⋊D3—31no—
39C3×#896—32no—
40C3×#887—32no—
41C3×C4×Dic6—32no—
42#63919—31no—
43C4×#627—31no—
44C4×#627—31no—
45#63893—32no—
46#63887—32no—
47#63877—32no—
48#63877—32no—
49C4×D3×Dic3—32no—
50C4×D3×Dic3—61no—
51#63874—32no—
52#63873—31no—
53#63871—32no—
54#63870—32no—
55#63870—32no—
56#63870—62no—
57#63865—31no—
58#63865D311yescharacteristic
59#63851—32no—
60Dic3×Dic6—32no—
61#63839—32no—
62#63839—32no—
63#63837—32no—
64#63837D312yescomplemented
65C22×C3×C3⋊D3C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4×C3⋊D3—22no—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2311yescharacteristic
77#6296—22no—
78Dic3×C3⋊D3—22no—
79Dic3×C3⋊D3Q814yescomplemented
80C3×#627—22no—
81C3×#627—24no—
82#5622—91no—
83#5578—92no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—32no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—32no—
92C2×#627—32no—
93C2×#627—31no—
94C2×#627C2×D312yes—
95C2×#627C2×D311yescharacteristic
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—61no—
99C2×D3×Dic3—61no—
100C32×C4⋊C4—32no—
101C32×C42—31no—
102C4⋊(C3⋊Dic3)—32no—
103#3750—32no—
104#3750—32no—
105#3750—32no—
106#3750C2×D312yes—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—32no—
109C4×C3⋊Dic3—31no—
110C4×C3⋊Dic3—31no—
111C4×C3⋊Dic3—62no—
112C4×C3⋊Dic3—61no—
113C4×C3⋊Dic3—121no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×C4⋊Dic3—32no—
117C3×Dic3⋊C4—22no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—62no—
123C3×Dic3⋊C4C2×D312yes—
124C3×Dic3⋊C4C2×D312yes—
125C3×C4×Dic3—21no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3C2×D312yes—
131#3724—31no—
132#3724—32no—
133#3724—32no—
134#3724—32no—
135#3724—32no—
136#3724—62no—
137#3724—62no—
138#3724C2×D311yescharacteristic
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—62no—
145Dic3⋊Dic3—62no—
146Dic3⋊Dic3C2×D312yes—
147#3722—32no—
148#3722C2×D312yes—
149(C2×D3)⋊Dic3—32no—
150(C2×D3)⋊Dic3—32no—
151Dic32—32no—
152Dic32—32no—
153Dic32—61no—
154Dic32—61no—
155Dic32C2×D312yes—
156Dic32C2×D312yes—
157C22×C33C2411yescharacteristic
158C2×C3×C3⋊D3—21no—
159C2×C3×C3⋊D3—21no—
160C2×C3×C3⋊D3C2×Q812yes—
161C33×C4—21no—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)C2×Q812yes—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3#4112yes—
167C32×Dic3—22no—
168C32×Dic3—22no—
169C32×Dic3C2×Q812yes—
170C32×Dic3#4112yes—
171C3×C22⋊Q8—91no—
172C3×C4×Q8—92no—
173#946—92no—
174Q8×Dic3—91no—
175#944—91no—
176#944—92no—
177#924—91no—
178C2×C4×Dic3—91no—
179#897—92no—
180#896—32no—
181#896—92no—
182#889—92no—
183#889—92no—
184#889—92no—
185#889—182no—
186#888—92no—
187#888—91no—
188#887—32no—
189#887—91no—
190#878—92no—
191#877—92no—
192#872—92no—
193#872—92no—
194C42×D3—92no—
195C42×D3—181no—
196C4×Dic6—32no—
197C4×Dic6—92no—
198C4×Dic6—92no—
199C22×C3⋊D3C22×D311yescharacteristic
200C22×C3×D3—32no—
201C22×C3×D3—61no—
202C2×C32×C4—21no—
203C2×C32×C4—22no—
204C2×C32×C4—31no—
205C2×C32×C4—31no—
206C2×C32×C4—32no—
207C2×C32×C4—32no—
208C2×C32×C4—41no—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D311yescharacteristic
211C2×C3⋊Dic3—32no—
212C2×C3⋊Dic3—31no—
213C2×C3⋊Dic3—32no—
214C2×C3⋊Dic3—31no—
215C2×C3⋊Dic3—61no—
216C2×C3⋊Dic3—61no—
217C2×C3⋊Dic3—62no—
218C2×C3⋊Dic3—62no—
219C2×C3⋊Dic3—121no—
220C2×C3⋊Dic3—121no—
221C2×C3⋊Dic3C22×D312yes—
222C2×C3⋊Dic3C22×D312yes—
223C2×C3⋊Dic3C22×D311yescharacteristic
224C4×C3⋊D3—22no—
225C4×C3⋊D3—34no—
226C4×C3⋊D3—62no—
227C2×C3×Dic3—21no—
228C2×C3×Dic3—21no—
229C2×C3×Dic3—22no—
230C2×C3×Dic3—22no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—61no—
237C2×C3×Dic3—62no—
238C2×C3×Dic3—62no—
239C2×C3×Dic3C22×D312yes—
240C2×C3×Dic3C22×D312yes—
241C2×C3×Dic3C22×D312yes—
242C2×C3×Dic3C22×D312yes—
243C2×C3×Dic3C22×D312yes—
244C2×C3×Dic3C22×D312yes—
245C3×C4×D3—64no—
246C3×C4×D3—64no—
247C3×Dic6—64no—
248C3×Dic6—64no—
249C3×Dic6—64no—
250#627—22no—
251#627—24no—
252#627—64no—
253#627—62no—
254D3×Dic3—38no—
255D3×Dic3—64no—
256D3×Dic3—64no—
257D3×Dic3—64no—
258#533—271noSylow
259C2×C33C22×Q811yescharacteristic
260C2×C33C2×#4111yescharacteristic
261C2×C33C2×#4111yescharacteristic
262C3×C3⋊D3—22no—
263C2×C3×Q8—91no—
264C2×C3×Q8—92no—
265C22×C3×C4—91no—
266C22×Dic3—91no—
267C22×Dic3—91no—
268C2×C4×D3—32no—
269C2×C4×D3—61no—
270C2×C4×D3—92no—
271C2×C4×D3—92no—
272C2×C4×D3—181no—
273C2×C4×D3—181no—
274C2×Dic6—32no—
275C2×Dic6—32no—
276C2×Dic6—32no—
277C2×Dic6—92no—
278C2×Dic6—91no—
279C3×C4⋊C4—32no—
280C3×C4⋊C4—32no—
281C3×C4⋊C4—62no—
282C3×C4⋊C4—91no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4—92no—
285C3×C22⋊C4—92no—
286C3×C42—32no—
287C3×C42—61no—
288C3×C42—92no—
289C3×C42—92no—
290#217—92no—
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—32no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295C4⋊Dic3—32no—
296C4⋊Dic3—91no—
297C4⋊Dic3—92no—
298C4⋊Dic3—92no—
299C4⋊Dic3—92no—
300C4⋊Dic3—182no—
301Dic3⋊C4—32no—
302Dic3⋊C4—32no—
303Dic3⋊C4—32no—
304Dic3⋊C4—32no—
305Dic3⋊C4—62no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—182no—
317Dic3⋊C4—182no—
318Dic3⋊C4—182no—
319Dic3⋊C4D3212yes—
320C4×Dic3—32no—
321C4×Dic3—32no—
322C4×Dic3—32no—
323C4×Dic3—32no—
324C4×Dic3—92no—
325C4×Dic3—92no—
326C4×Dic3—92no—
327C4×Dic3—91no—
328C4×Dic3—91no—
329C4×Dic3—91no—
330C4×Dic3—181no—
331C4×Dic3—181no—
332C4×Dic3—181no—
333C4×Dic3—182no—
334C4×Dic3—361no—
335C4×Dic3D3211yescharacteristic
336C22×C32—22no—
337C22×C32—21no—
338C22×C32—41no—
339C22×C32C23×D312yes—
340C22×C32C23×D311yescharacteristic
341C2×C3⋊D3—21no—
342C2×C3⋊D3—21no—
343C2×C3⋊D3D3×Q812yes—
344C2×C3×D3—34no—
345C2×C3×D3—62no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—62no—
349C2×C3×D3—62no—
350C32×C4—21no—
351C32×C4—22no—
352C32×C4—22no—
353C32×C4—32no—
354C32×C4—32no—
355C32×C4—42no—
356C32×C4—42no—
357C32×C4—62no—
358C32×C4—62no—
359C3⋊Dic3—22no—
360C3⋊Dic3—22no—
361C3⋊Dic3—34no—
362C3⋊Dic3—32no—
363C3⋊Dic3—62no—
364C3⋊Dic3—64no—
365C3⋊Dic3—61no—
366C3⋊Dic3—62no—
367C3⋊Dic3—62no—
368C3⋊Dic3—122no—
369C3⋊Dic3—122no—
370C3⋊Dic3—122no—
371C3⋊Dic3#23712yes—
372C3×Dic3—22no—
373C3×Dic3—22no—
374C3×Dic3—22no—
375C3×Dic3—22no—
376C3×Dic3—22no—
377C3×Dic3—22no—
378C3×Dic3—34no—
379C3×Dic3—42no—
380C3×Dic3—42no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—62no—
386C3×Dic3—122no—
387C3×Dic3—122no—
388C3×Dic3#23514yes—
389C3×Dic3D3×Q814yes—
390C4⋊Q8—271no—
391C4⋊Q8—271no—
392#125—272no—
393C22⋊Q8—91no—
394C22⋊Q8—272no—
395C22⋊Q8—271no—
396C4×Q8—92no—
397C4×Q8—271no—
398C4×Q8—271no—
399#117—272no—
400C2×C42—271no—
401C33#53311yescharacteristic, complemented, Sylow
402C23×C3—91no—
403C22×D3—32no—
404C22×D3—61no—
405C3×Q8—184no—
406C3×Q8—182no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—32no—
410C2×C3×C4—32no—
411C2×C3×C4—32no—
412C2×C3×C4—32no—
413C2×C3×C4—32no—
414C2×C3×C4—32no—
415C2×C3×C4—41no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—62no—
419C2×C3×C4—62no—
420C2×C3×C4—91no—
421C2×C3×C4—92no—
422C2×C3×C4—92no—
423C2×C3×C4—182no—
424C2×C3×C4C2×D3211yescharacteristic
425C2×C3×C4C2×D3212yes—
426C2×Dic3—32no—
427C2×Dic3—32no—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—32no—
431C2×Dic3—61no—
432C2×Dic3—62no—
433C2×Dic3—62no—
434C2×Dic3—92no—
435C2×Dic3—91no—
436C2×Dic3—94no—
437C2×Dic3—92no—
438C2×Dic3—91no—
439C2×Dic3—181no—
440C2×Dic3—181no—
441C2×Dic3—182no—
442C2×Dic3—182no—
443C2×Dic3—182no—
444C2×Dic3—361no—
445C2×Dic3—361no—
446C2×Dic3C2×D3212yes—
447C2×Dic3C2×D3212yes—
448C2×Dic3C2×D3211yescharacteristic
449C2×Dic3C2×D3211yescharacteristic
450C4×D3—64no—
451C4×D3—64no—
452C4×D3—98no—
453C4×D3—184no—
454C4×D3—184no—
455C4×D3—184no—
456Dic6—64no—
457Dic6—64no—
458Dic6—64no—
459Dic6—184no—
460Dic6—182no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—22no—
464C2×C32—21no—
465C2×C32—22no—
466C2×C32—22no—
467C2×C32—41no—
468C2×C32—41no—
469C2×C32—41no—
470C2×C32C2×#23512yes—
471C2×C32C2×#23711yescharacteristic
472C2×C32C2×D3×Q812yes—
473C2×C32C2×D3×Q811yescharacteristic
474C2×C32D3×#4111yescharacteristic
475C2×C32D3×#4112yes—
476C3⋊D3—22no—
477C3×D3—64no—
478C3×D3—64no—
479C2×Q8—92no—
480C2×Q8—91no—
481C2×Q8—271no—
482C22×C4—91no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—32no—
486C4⋊C4—92no—
487C4⋊C4—91no—
488C4⋊C4—92no—
489C4⋊C4—272no—
490C4⋊C4—272no—
491C4⋊C4—272no—
492C4⋊C4—272no—
493C4⋊C4—271no—
494C22⋊C4—92no—
495C22⋊C4—272no—
496C42—31no—
497C42—92no—
498C42—92no—
499C42—271no—
500C42—271no—
501C42—271no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—41no—
505C22×C3—92no—
506C22×C3—181no—
507C22×C3—181no—
508C22×C3C22×D3211yescharacteristic
509C22×C3C22×D3212yes—
510C2×D3—34no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—62no—
514C2×D3—62no—
515C2×D3—62no—
516C3×C4—22no—
517C3×C4—21no—
518C3×C4—22no—
519C3×C4—34no—
520C3×C4—34no—
521C3×C4—42no—
522C3×C4—42no—
523C3×C4—62no—
524C3×C4—62no—
525C3×C4—62no—
526C3×C4—62no—
527C3×C4—62no—
528C3×C4—62no—
529C3×C4—92no—
530C3×C4—122no—
531C3×C4—122no—
532C3×C4—182no—
533C3×C4—182no—
534Dic3—22no—
535Dic3—22no—
536Dic3—34no—
537Dic3—62no—
538Dic3—62no—
539Dic3—62no—
540Dic3—62no—
541Dic3—62no—
542Dic3—92no—
543Dic3—94no—
544Dic3—122no—
545Dic3—122no—
546Dic3—182no—
547Dic3—184no—
548Dic3—182no—
549Dic3—181no—
550Dic3—182no—
551Dic3—362no—
552Dic3—362no—
553Dic3—362no—
554Dic3#379712yes—
555Dic3#379812yes—
556C32—22no—
557C32—21no—
558C32—41no—
559C32#557812yescomplemented
560C32#562211yescharacteristic, complemented
561C23—91no—
562Q8—184no—
563Q8—182no—
564Q8—542no—
565C2×C4—31no—
566C2×C4—31no—
567C2×C4—32no—
568C2×C4—32no—
569C2×C4—91no—
570C2×C4—92no—
571C2×C4—92no—
572C2×C4—182no—
573C2×C4—274no—
574C2×C4—271no—
575C2×C4—271no—
576C2×C4—542no—
577C2×C4D3311yescharacteristic
578C2×C3—21no—
579C2×C3—22no—
580C2×C3—22no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—22no—
584C2×C3—41no—
585C2×C3—41no—
586C2×C3—41no—
587C2×C3—182no—
588C2×C3C2×#379711yescharacteristic
589C2×C3C2×#379811yescharacteristic
590C2×C3D3×#23512yes—
591C2×C3#6430611yescharacteristic
592C2×C3D3×#23712yes—
593C2×C3D32×Q812yes—
594D3—64no—
595D3—64no—
596C22—92no—
597C22—181no—
598C22—181no—
599C22C2×D3311yescharacteristic
600C4—21no—
601C4—32no—
602C4—32no—
603C4—62no—
604C4—62no—
605C4—92no—
606C4—182no—
607C4—182no—
608C4—272no—
609C4—541no—
610C3—21no—
611C3—22no—
612C3—41no—
613C3#1059388211yescharacteristic, complemented
614C3#1059438612yescomplemented
615C2—182no—
616C2D3×#379711yescharacteristic
617C2D3×#379811yescharacteristic
618C2#1172440711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723448C2semidirect
5. C3×#63837C2semidirect
6. #11723035C2semidirect
7. #11723032C2semidirect
9. #11722944C2semidirect
10. Dic3×#627C2semidirect
64. #63837D3semidirect
79. Dic3×C3⋊D3Q8semidirect
401. C33#533semidirect
559. C32#5578semidirect
560. C32#5622semidirect
613. C3#10593882semidirect
614. C3#10594386semidirect

Maximal subgroups

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

subgroupindex
#117236712
C4×Dic3×C3⋊D32
#117234482
C3×#638652
C3×#638372
#117230352
#117230322
#117229742
#117229442
Dic3×#6272
#117229192
#105943863
#105938823