Finite Group Databasebeta

← #13178231

Subgroups — #13178231

Order 1728.

1276 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64091C211yesmaximal, characteristic
2C3×#64088C212yesmaximal, complemented
3#11723724C211yesmaximal, characteristic, complemented
4#11723723C212yesmaximal, complemented
5#11723718C211yesmaximal, characteristic, complemented
6#11723692C212yesmaximal, complemented
7#11723691C212yesmaximal, complemented
8#11723686C212yesmaximal, complemented
9#11723683C212yesmaximal, complemented
10#10594165—31nomaximal
11#10594115—32nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C2×C3×#638C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2212yes—
15C2×#6296C2211yescharacteristic, complemented
16C2×#6294C2212yes—
17C2×#6294C2212yes—
18C2×#6294C2211yescharacteristic, complemented
19C33×C22⋊C4C2211yescharacteristic
20C3×#3757C2212yes—
21C3×#3757C2212yes—
22C3×#3752C2212yes—
23C3×C4⋊(C3⋊Dic3)C2212yescomplemented
24C3×#3750C2212yescomplemented
25C3×#3750C2212yescomplemented
26C3×C4×C3⋊Dic3C2212yescomplemented
27#88815C2212yescomplemented
28#88814C2212yes—
29#88814C2212yes—
30#88814C2211yescharacteristic, complemented
31#88814C2212yescomplemented
32#88813C2211yescharacteristic, complemented
33#64091D311yescharacteristic
34#64088D312yescomplemented
35C3×#882—32no—
36C3×#882—61no—
37C3×#879—32no—
38C3×#879—32no—
39C3×#879—62no—
40#63975—31no—
41#63974—32no—
42#63969—32no—
43D3×#217—32no—
44D3×#217—32no—
45#63959—32no—
46#63953—31no—
47#63899—32no—
48#63898—32no—
49#63897—32no—
50#63895—32no—
51#63894—32no—
52#63893—32no—
53Dic3×D12—32no—
54#63884—32no—
55#63884—32no—
56#63878—32no—
57#63877—32no—
58#63836—32no—
59C23×C33C2311yescharacteristic
60C22×C3×C3⋊D3—24no—
61C22×C3×C3⋊D3—41no—
62C22×C3×C3⋊D3C2311yescharacteristic
63C22×C3×C3⋊D3C2312yes—
64C2×C33×C4C2312yes—
65C3×#638—41no—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×C4⋊(C3⋊D3)—24no—
72#6296—41no—
73#6294—24no—
74#6294—42no—
75#6294—41no—
76#5705—92no—
77#5587—91no—
78C23×C3⋊D3C2×D311yescharacteristic
79C23×C3×D3—32no—
80C23×C3×D3—61no—
81C2×#638C2×D311yescharacteristic
82C2×C4⋊(C3⋊D3)C2×D312yes—
83C2×C3×#67—32no—
84C2×C3×#67—61no—
85C2×C3×D12—32no—
86C2×C3×D12—32no—
87C2×C3×D12—62no—
88C2×#627—32no—
89C2×#627—31no—
90C2×#624—32no—
91C2×#624—31no—
92C2×#624—32no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—32no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C32×C22⋊C4—22no—
99C32×C22⋊C4—21no—
100C32×C22⋊C4—41no—
101C32×C22⋊C4C2×D311yescharacteristic
102C32×C22⋊C4C2×D312yes—
103#3757C2×D312yes—
104#3757C2×D312yes—
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)C2×D312yescomplemented
107#3750C2×D312yescomplemented
108#3750C2×D312yescomplemented
109C4×C3⋊Dic3C2×D312yescomplemented
110C3×#217—32no—
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—32no—
114C3×#217—62no—
115C3×#217—62no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—62no—
119C3×C4⋊Dic3—32no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—62no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—62no—
127C3×Dic3⋊C4—62no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—62no—
131#3724—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134#3722—32no—
135#3722—32no—
136#3722—32no—
137#3722—31no—
138(C2×D3)⋊Dic3—32no—
139(C2×D3)⋊Dic3—32no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142(C2×D3)⋊Dic3—32no—
143(C2×D3)⋊Dic3—32no—
144(C2×D3)⋊Dic3—32no—
145Dic32—31no—
146Dic32—32no—
147C22×C33—22no—
148C22×C33—21no—
149C22×C33C2411yescharacteristic
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—44no—
155C2×C3×C3⋊D3#4114yes—
156C33×C4—22no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3—22no—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3—22no—
161C3×C3⋊Dic3—21no—
162C3×#126—92no—
163C3×#120—91no—
164C2×#217—92no—
165#936—91no—
166#935—92no—
167#935—91no—
168D4×Dic3—92no—
169#897—92no—
170#896—92no—
171#894—92no—
172#894—92no—
173#893—92no—
174#892—92no—
175#885—92no—
176#883—92no—
177#882—32no—
178#882—61no—
179D3×C22⋊C4—92no—
180D3×C22⋊C4—92no—
181#879—32no—
182#879—32no—
183#879—62no—
184#878—92no—
185#877—92no—
186C4×D12—92no—
187#872—92no—
188C23×C32—21no—
189C23×C32—22no—
190C23×C32—41no—
191C23×C32C22×D312yes—
192C23×C32C22×D311yescharacteristic
193C22×C3⋊D3—24no—
194C22×C3⋊D3—41no—
195C22×C3⋊D3C22×D311yescharacteristic
196C22×C3⋊D3C22×D312yes—
197C22×C3×D3—32no—
198C22×C3×D3—32no—
199C22×C3×D3—32no—
200C22×C3×D3—62no—
201C22×C3×D3—61no—
202C22×C3×D3—64no—
203C22×C3×D3—64no—
204C22×C3×D3—124no—
205C22×C3×D3—122no—
206C22×C3×D3—122no—
207C2×C32×C4—22no—
208C2×C32×C4—22no—
209C2×C32×C4—22no—
210C2×C32×C4—42no—
211C2×C32×C4C22×D312yes—
212C2×C32×C4C22×D312yes—
213C2×C32×C4C22×D312yes—
214#638—41no—
215C2×C3⋊Dic3C22×D312yes—
216C2×C3⋊Dic3C22×D311yescharacteristic
217C2×C3⋊Dic3C22×D312yes—
218C2×C3⋊Dic3C22×D312yes—
219C2×C3⋊Dic3C22×D312yes—
220C4⋊(C3⋊D3)—24no—
221C3×#67—122no—
222C3×#67—122no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—62no—
233C2×C3×Dic3—62no—
234C2×C3×Dic3—62no—
235C2×C3×Dic3—62no—
236C2×C3×Dic3—61no—
237C3×D12—64no—
238C3×D12—64no—
239C3×D12—124no—
240#627—121no—
241#627—122no—
242#624—64no—
243#624—121no—
244#624—122no—
245D3×Dic3—64no—
246D3×Dic3—64no—
247D3×Dic3—122no—
248D3×Dic3—122no—
249D3×Dic3—122no—
250#551—271noSylow
251C2×C33—41no—
252C2×C33C2×#4112yes—
253C2×C33H(5,𝔽2)11yescharacteristic
254C3×C3⋊D3—24no—
255C3×C3⋊D3—41no—
256C24×C3—91no—
257C23×D3—32no—
258C23×D3—61no—
259C2×C3×D4—91no—
260C2×C3×D4—92no—
261C2×#67—32no—
262C2×#67—61no—
263C22×Dic3—92no—
264C22×Dic3—92no—
265C22×Dic3—91no—
266C2×D12—32no—
267C2×D12—32no—
268C2×D12—62no—
269C2×C4×D3—92no—
270C2×C4×D3—92no—
271C2×C4×D3—92no—
272C2×C4×D3—92no—
273C2×C4×D3—92no—
274C2×Dic6—91no—
275C2×Dic6—92no—
276C3×C4⋊C4—92no—
277C3×C4⋊C4—92no—
278C3×C4⋊C4—92no—
279C3×C22⋊C4—22no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—41no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4—92no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4D3211yescharacteristic
286C3×C22⋊C4D3212yes—
287C3×C42—92no—
288#217—32no—
289#217—32no—
290#217—32no—
291#217—32no—
292#217—62no—
293#217—62no—
294#217—92no—
295#217—92no—
296#217—92no—
297#217—91no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—32no—
300(C2×D3)⋊C4—62no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—92no—
308C4⋊Dic3—32no—
309C4⋊Dic3—32no—
310C4⋊Dic3—62no—
311C4⋊Dic3—92no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—62no—
317Dic3⋊C4—62no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320C4×Dic3—32no—
321C4×Dic3—32no—
322C4×Dic3—62no—
323C4×Dic3—92no—
324C4×Dic3—91no—
325C22×C32—21no—
326C22×C32—22no—
327C22×C32—22no—
328C22×C32—22no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—22no—
332C22×C32—21no—
333C22×C32—41no—
334C22×C32—42no—
335C22×C32—42no—
336C22×C32—42no—
337C22×C32—42no—
338C22×C32—42no—
339C22×C32—82no—
340C22×C32C23×D311yescharacteristic
341C22×C32C23×D312yes—
342C2×C3⋊D3—22no—
343C2×C3⋊D3—22no—
344C2×C3⋊D3—22no—
345C2×C3⋊D3—21no—
346C2×C3⋊D3—44no—
347C2×C3⋊D3#23714yes—
348C2×C3×D3—34no—
349C2×C3×D3—34no—
350C2×C3×D3—64no—
351C2×C3×D3—62no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—62no—
355C2×C3×D3—62no—
356C2×C3×D3—62no—
357C2×C3×D3—62no—
358C2×C3×D3—122no—
359C2×C3×D3—122no—
360C2×C3×D3—122no—
361C2×C3×D3—121no—
362C2×C3×D3—128no—
363C2×C3×D3—128no—
364C32×C4—22no—
365C32×C4—22no—
366C32×C4—22no—
367C32×C4—42no—
368C32×C4—42no—
369C32×C4—42no—
370C32×C4—82no—
371C3⋊Dic3—21no—
372C3⋊Dic3—22no—
373C3⋊Dic3—22no—
374C3⋊Dic3—22no—
375C3⋊Dic3—22no—
376C3×Dic3—62no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—122no—
386C3×Dic3—122no—
387C3×Dic3—122no—
388C3×Dic3—122no—
389C3×Dic3—121no—
390#126—92no—
391#124—271no—
392#123—271no—
393#123—272no—
394C22⋊Q8—272no—
395#120—91no—
396C4×D4—272no—
397#117—272no—
398C2×C22⋊C4—272no—
399C33#55111yescharacteristic, complemented, Sylow
400C23×C3—21no—
401C23×C3—22no—
402C23×C3—41no—
403C23×C3—92no—
404C23×C3—91no—
405C23×C3—184no—
406C23×C3—361no—
407C23×C3C2×D3211yescharacteristic
408C23×C3C2×D3212yes—
409C22×D3—32no—
410C22×D3—32no—
411C22×D3—32no—
412C22×D3—62no—
413C22×D3—61no—
414C22×D3—64no—
415C22×D3—64no—
416C22×D3—124no—
417C22×D3—122no—
418C22×D3—122no—
419C3×D4—184no—
420C3×D4—361no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—42no—
425C2×C3×C4—91no—
426C2×C3×C4—92no—
427C2×C3×C4—92no—
428C2×C3×C4—92no—
429C2×C3×C4—92no—
430C2×C3×C4C2×D3212yes—
431C2×C3×C4C2×D3212yes—
432C2×C3×C4C2×D3212yes—
433#67—122no—
434#67—122no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—32no—
441C2×Dic3—32no—
442C2×Dic3—32no—
443C2×Dic3—32no—
444C2×Dic3—62no—
445C2×Dic3—62no—
446C2×Dic3—62no—
447C2×Dic3—62no—
448C2×Dic3—61no—
449C2×Dic3—184no—
450C2×Dic3—362no—
451C2×Dic3—361no—
452D12—64no—
453D12—64no—
454D12—124no—
455C4×D3—184no—
456C4×D3—184no—
457C4×D3—362no—
458C4×D3—362no—
459C4×D3—362no—
460Dic6—362no—
461Dic6—361no—
462C2×C32—22no—
463C2×C32—21no—
464C2×C32—22no—
465C2×C32—22no—
466C2×C32—22no—
467C2×C32—42no—
468C2×C32—41no—
469C2×C32—41no—
470C2×C32—42no—
471C2×C32—42no—
472C2×C32—82no—
473C2×C32—82no—
474C2×C32C2×#23712yes—
475C2×C32#100411yescharacteristic
476C2×C32D3×#4112yes—
477C2×C32D3×#4112yes—
478C2×C32#100912yes—
479C3⋊D3—24no—
480C3⋊D3—41no—
481C3×D3—64no—
482C3×D3—64no—
483C3×D3—124no—
484C3×D3—122no—
485C3×D3—241no—
486C24—91no—
487C2×Q8—271no—
488C2×D4—91no—
489C2×D4—92no—
490C22×C4—272no—
491C22×C4—272no—
492C22×C4—271no—
493C4⋊C4—92no—
494C4⋊C4—92no—
495C4⋊C4—92no—
496C4⋊C4—272no—
497C22⋊C4—92no—
498C22⋊C4—92no—
499C22⋊C4—92no—
500C22⋊C4—272no—
501C22⋊C4—272no—
502C22⋊C4—272no—
503C22⋊C4—271no—
504C22⋊C4#111311yescharacteristic
505C42—92no—
506C42—271no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—21no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—21no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—41no—
521C22×C3—82no—
522C22×C3—94no—
523C22×C3—182no—
524C22×C3—182no—
525C22×C3—181no—
526C22×C3—182no—
527C22×C3—364no—
528C22×C3C22×D3212yes—
529C22×C3C22×D3211yescharacteristic
530C2×D3—34no—
531C2×D3—34no—
532C2×D3—64no—
533C2×D3—62no—
534C2×D3—62no—
535C2×D3—62no—
536C2×D3—62no—
537C2×D3—62no—
538C2×D3—62no—
539C2×D3—62no—
540C2×D3—122no—
541C2×D3—122no—
542C2×D3—122no—
543C2×D3—121no—
544C2×D3—128no—
545C2×D3—128no—
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—22no—
549C3×C4—42no—
550C3×C4—42no—
551C3×C4—42no—
552C3×C4—82no—
553C3×C4—182no—
554C3×C4—182no—
555C3×C4—181no—
556C3×C4—182no—
557C3×C4—182no—
558Dic3—62no—
559Dic3—62no—
560Dic3—62no—
561Dic3—62no—
562Dic3—62no—
563Dic3—62no—
564Dic3—62no—
565Dic3—62no—
566Dic3—62no—
567Dic3—121no—
568Dic3—122no—
569Dic3—122no—
570Dic3—122no—
571Dic3—122no—
572C32—22no—
573C32—21no—
574C32—41no—
575C32#558711yescharacteristic, complemented
576C32#570512yescomplemented
577C23—92no—
578C23—91no—
579C23—184no—
580C23—361no—
581C23C2×#111311yescharacteristic
582Q8—1081no—
583D4—184no—
584D4—361no—
585C2×C4—92no—
586C2×C4—92no—
587C2×C4—91no—
588C2×C4—92no—
589C2×C4—92no—
590C2×C4—544no—
591C2×C4—1081no—
592C2×C4—1082no—
593C2×C4C2×#111312yes—
594C2×C3—22no—
595C2×C3—21no—
596C2×C3—22no—
597C2×C3—22no—
598C2×C3—22no—
599C2×C3—42no—
600C2×C3—42no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—42no—
604C2×C3—82no—
605C2×C3—82no—
606C2×C3—184no—
607C2×C3—361no—
608C2×C3#6430712yes—
609C2×C3#6430811yescharacteristic
610C2×C3D3×#23712yes—
611C2×C3D3×#23712yes—
612C2×C3#6431212yes—
613D3—64no—
614D3—64no—
615D3—124no—
616D3—122no—
617D3—241no—
618C22—21no—
619C22—22no—
620C22—94no—
621C22—182no—
622C22—182no—
623C22—181no—
624C22—182no—
625C22—364no—
626C22C22×#111311yescharacteristic
627C4—22no—
628C4—182no—
629C4—182no—
630C4—181no—
631C4—182no—
632C4—182no—
633C3—21no—
634C3—22no—
635C3—41no—
636C3#1059411512yescomplemented
637C3#1059416511yescharacteristic, complemented
638C2—41no—
639C2—184no—
640C2—361no—
641C2#1172461911yescharacteristic
642C2#1172462212yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC3×#64091C2
hypercenterC22⋊C4#1113
hypocenterC33#551
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64088C2semidirect
3. #11723724C2semidirect
4. #11723723C2semidirect
5. #11723718C2semidirect
6. #11723692C2semidirect
7. #11723691C2semidirect
8. #11723686C2semidirect
9. #11723683C2semidirect
15. C2×#6296C22semidirect
18. C2×#6294C22semidirect
23. C3×C4⋊(C3⋊Dic3)C22semidirect
24. C3×#3750C22semidirect
25. C3×#3750C22semidirect
26. C3×C4×C3⋊Dic3C22semidirect
27. #88815C22semidirect
30. #88814C22semidirect
31. #88814C22semidirect
32. #88813C22semidirect
34. #64088D3semidirect
106. C4⋊(C3⋊Dic3)C2×D3semidirect
107. #3750C2×D3semidirect
108. #3750C2×D3semidirect
109. C4×C3⋊Dic3C2×D3semidirect
399. C33#551semidirect
575. C32#5587semidirect
576. C32#5705semidirect
636. C3#10594115semidirect
637. C3#10594165semidirect

Maximal subgroups

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

subgroupindex
C3×#640912
C3×#640882
#117237242
#117237232
#117237182
#117236922
#117236912
#117236862
#117236832
#105941653
#105941153