Finite Group Databasebeta

← #13178294

Subgroups — #13178294

Order 1728.

1356 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64094C212yesmaximal, complemented
2C3×#64091C211yesmaximal, characteristic, complemented
3#11723729C211yesmaximal, characteristic, complemented
4#11723727C212yesmaximal, complemented
5#11723718C211yesmaximal, characteristic, complemented
6#11723701C212yesmaximal, complemented
7#11723677C212yesmaximal, complemented
8#11723672C212yesmaximal, complemented
9#11723670C212yesmaximal, complemented
10#10594311—31nomaximal
11#10594309—32nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic, complemented
13C22×#1113C2211yescharacteristic, complemented
14C2×C3×#638C2211yescharacteristic
15C2×C3×#638C2212yes—
16C2×C3×C4⋊(C3⋊D3)C2212yes—
17C2×C3×#634C2212yescomplemented
18C2×#6296C2211yescharacteristic, complemented
19C2×#6295C2212yes—
20C2×#6295C2212yes—
21C33×C22⋊C4C2211yescharacteristic, complemented
22C3×#3757C2212yescomplemented
23C3×#3752C2212yescomplemented
24C3×#3752C2212yescomplemented
25C3×#3752C2212yescomplemented
26C3×C4×C3⋊Dic3C2212yescomplemented
27#88814C2212yescomplemented
28#88814C2212yescomplemented
29#88814C2212yescomplemented
30#88814C2211yescharacteristic, complemented
31#88814C2212yescomplemented
32#88813C2211yescharacteristic, complemented
33#64094D312yescomplemented
34#64091D311yescharacteristic, complemented
35C3×#885—32no—
36C3×#885—32no—
37C3×#885—62no—
38C3×#882—32no—
39C3×#882—61no—
40#63983—32no—
41#63983—31no—
42#63981—32no—
43#63977—32no—
44#63976—32no—
45#63959—32no—
46#63953—31no—
47#63923—32no—
48#63922—32no—
49#63922—32no—
50#63915—32no—
51#63913—32no—
52#63907—32no—
53#63907—32no—
54#63864—32no—
55#63863—32no—
56#63860—32no—
57#63852—32no—
58#63852—32no—
59C23×C33C2311yescharacteristic
60C22×C3×C3⋊D3—42no—
61C22×C3×C3⋊D3—41no—
62C22×C3×C3⋊D3C2312yes—
63C22×C3×C3⋊D3C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C2×#1113—41no—
66C2×#1113—42no—
67C2×C33×C4C2312yes—
68C3×#638—42no—
69C3×#638—41no—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C3×C4⋊(C3⋊D3)—42no—
75C3×#634—42no—
76#6296—41no—
77#6295—42no—
78#6295—42no—
79#5683—92no—
80#5594—91no—
81C23×C3⋊D3C2×D311yescharacteristic, complemented
82C23×C3×D3—32no—
83C23×C3×D3—61no—
84C22×D32—32no—
85C22×D32—31no—
86C2×#638C2×D311yescharacteristic
87C2×#638C2×D312yes—
88C2×C4⋊(C3⋊D3)C2×D312yes—
89C2×#634C2×D312yescomplemented
90C2×C3×#67—32no—
91C2×C3×#67—32no—
92C2×C3×#67—32no—
93C2×C3×#67—61no—
94C2×C3×#67—62no—
95C2×C3×D12—32no—
96C2×C3×D12—32no—
97C2×C3×D12—62no—
98C2×C3×Dic6—32no—
99C2×C3×Dic6—32no—
100C2×C3×Dic6—62no—
101C2×#627—31no—
102C2×#627—32no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—32no—
106C2×#626—32no—
107C2×#625—32no—
108C2×#625—32no—
109C32×C22⋊C4—21no—
110C32×C22⋊C4—22no—
111C32×C22⋊C4—41no—
112C32×C22⋊C4C2×D312yescomplemented
113C32×C22⋊C4C2×D311yescharacteristic, complemented
114#3757C2×D312yescomplemented
115#3752C2×D312yescomplemented
116#3752C2×D312yescomplemented
117#3752C2×D312yescomplemented
118C4×C3⋊Dic3C2×D312yescomplemented
119C3×#217—32no—
120C3×#217—32no—
121C3×#217—62no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—32no—
126C3×(C2×D3)⋊C4—32no—
127C3×(C2×D3)⋊C4—32no—
128C3×(C2×D3)⋊C4—62no—
129C3×(C2×D3)⋊C4—62no—
130C3×(C2×D3)⋊C4—62no—
131C3×C4×Dic3—32no—
132C3×C4×Dic3—32no—
133C3×C4×Dic3—62no—
134#3722—32no—
135#3722—32no—
136#3722—31no—
137#3722—32no—
138#3722—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—
145(C2×D3)⋊Dic3—32no—
146(C2×D3)⋊Dic3—32no—
147(C2×D3)⋊Dic3—32no—
148Dic32—31no—
149Dic32—32no—
150C22×C33—21no—
151C22×C33—22no—
152C22×C33C2411yescharacteristic
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—22no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—22no—
160C2×C3×C3⋊D3—21no—
161C2×C3×C3⋊D3—44no—
162#1113—44no—
163C33×C4—22no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—21no—
168C3×#124—92no—
169C3×#120—91no—
170#953—92no—
171#947—92no—
172#947—92no—
173#937—92no—
174#937—92no—
175#937—92no—
176#937—91no—
177#936—92no—
178#936—91no—
179#885—32no—
180#885—32no—
181#885—62no—
182#885—92no—
183#885—92no—
184#885—92no—
185#885—92no—
186#885—92no—
187#882—32no—
188#882—61no—
189#882—92no—
190#882—92no—
191#882—92no—
192#875—92no—
193#875—92no—
194C23×C32—21no—
195C23×C32—22no—
196C23×C32—41no—
197C23×C32C22×D312yes—
198C23×C32C22×D311yescharacteristic
199C22×C3⋊D3—42no—
200C22×C3⋊D3—41no—
201C22×C3⋊D3C22×D311yescharacteristic
202C22×C3⋊D3C22×D312yes—
203C22×C3⋊D3C22×D312yes—
204C22×C3×D3—32no—
205C22×C3×D3—32no—
206C22×C3×D3—32no—
207C22×C3×D3—32no—
208C22×C3×D3—32no—
209C22×C3×D3—62no—
210C22×C3×D3—62no—
211C22×C3×D3—61no—
212C22×C3×D3—122no—
213C22×C3×D3—122no—
214C22×C3×D3—122no—
215C22×C3×D3—124no—
216C22×C3×D3—122no—
217C2×D32—122no—
218C2×D32—121no—
219C2×D32—122no—
220C2×D32—122no—
221C2×D32—122no—
222C2×C32×C4—22no—
223C2×C32×C4—22no—
224C2×C32×C4—22no—
225C2×C32×C4—42no—
226C2×C32×C4C22×D312yes—
227C2×C32×C4C22×D312yes—
228C2×C32×C4C22×D312yes—
229#638—42no—
230#638—41no—
231C2×C3⋊Dic3C22×D311yescharacteristic
232C2×C3⋊Dic3C22×D312yes—
233C2×C3⋊Dic3C22×D312yes—
234C2×C3⋊Dic3C22×D312yes—
235C4⋊(C3⋊D3)—42no—
236#634—42no—
237C3×#67—122no—
238C3×#67—122no—
239C3×#67—122no—
240C3×#67—124no—
241C3×#67—122no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—32no—
244C2×C3×Dic3—32no—
245C2×C3×Dic3—32no—
246C2×C3×Dic3—32no—
247C2×C3×Dic3—32no—
248C2×C3×Dic3—32no—
249C2×C3×Dic3—62no—
250C2×C3×Dic3—62no—
251C2×C3×Dic3—61no—
252C2×C3×Dic3—62no—
253C3×D12—122no—
254C3×D12—122no—
255C3×D12—124no—
256C3×Dic6—122no—
257C3×Dic6—122no—
258C3×Dic6—124no—
259#627—121no—
260#627—122no—
261#626—122no—
262#626—122no—
263#626—122no—
264#626—122no—
265#625—122no—
266#625—122no—
267H(3,𝔽4)—271noSylow
268C2×C33—41no—
269C2×C33H(5,𝔽2)12yes—
270C2×C33H(5,𝔽2)11yescharacteristic
271C3×C3⋊D3—42no—
272C3×C3⋊D3—41no—
273C3×C3⋊D3—42no—
274C24×C3—91no—
275C23×D3—32no—
276C23×D3—61no—
277C23×D3—92no—
278C23×D3—91no—
279C2×C3×Q8—92no—
280C2×C3×D4—91no—
281C2×C3×D4—92no—
282C2×C3×D4—92no—
283C2×#67—32no—
284C2×#67—32no—
285C2×#67—32no—
286C2×#67—61no—
287C2×#67—62no—
288C2×#67—92no—
289C2×#67—92no—
290C2×#67—92no—
291C2×#67—92no—
292C2×D12—32no—
293C2×D12—32no—
294C2×D12—62no—
295C2×D12—92no—
296C2×D12—92no—
297C2×Dic6—32no—
298C2×Dic6—32no—
299C2×Dic6—62no—
300C2×Dic6—92no—
301C2×Dic6—91no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—22no—
304C3×C22⋊C4—41no—
305C3×C22⋊C4—92no—
306C3×C22⋊C4—92no—
307C3×C22⋊C4—92no—
308C3×C22⋊C4—92no—
309C3×C22⋊C4D3212yescomplemented
310C3×C22⋊C4D3211yescharacteristic, complemented
311C3×C42—92no—
312#217—32no—
313#217—32no—
314#217—62no—
315#217—92no—
316#217—92no—
317#217—92no—
318#217—92no—
319#217—91no—
320(C2×D3)⋊C4—32no—
321(C2×D3)⋊C4—32no—
322(C2×D3)⋊C4—32no—
323(C2×D3)⋊C4—32no—
324(C2×D3)⋊C4—32no—
325(C2×D3)⋊C4—32no—
326(C2×D3)⋊C4—62no—
327(C2×D3)⋊C4—62no—
328(C2×D3)⋊C4—62no—
329(C2×D3)⋊C4—92no—
330(C2×D3)⋊C4—92no—
331(C2×D3)⋊C4—92no—
332(C2×D3)⋊C4—92no—
333(C2×D3)⋊C4—92no—
334(C2×D3)⋊C4—92no—
335(C2×D3)⋊C4—92no—
336(C2×D3)⋊C4—92no—
337(C2×D3)⋊C4—92no—
338C4×Dic3—32no—
339C4×Dic3—32no—
340C4×Dic3—62no—
341C4×Dic3—91no—
342C4×Dic3—92no—
343C22×C32—21no—
344C22×C32—22no—
345C22×C32—22no—
346C22×C32—22no—
347C22×C32—22no—
348C22×C32—22no—
349C22×C32—22no—
350C22×C32—21no—
351C22×C32—41no—
352C22×C32—42no—
353C22×C32—42no—
354C22×C32—42no—
355C22×C32—42no—
356C22×C32—42no—
357C22×C32—82no—
358C22×C32C23×D311yescharacteristic
359C22×C32C23×D312yes—
360C2×C3⋊D3—22no—
361C2×C3⋊D3—22no—
362C2×C3⋊D3—22no—
363C2×C3⋊D3—22no—
364C2×C3⋊D3—21no—
365C2×C3⋊D3—22no—
366C2×C3⋊D3—22no—
367C2×C3⋊D3—22no—
368C2×C3⋊D3—44no—
369C2×C3×D3—64no—
370C2×C3×D3—62no—
371C2×C3×D3—62no—
372C2×C3×D3—62no—
373C2×C3×D3—62no—
374C2×C3×D3—62no—
375C2×C3×D3—62no—
376C2×C3×D3—62no—
377C2×C3×D3—62no—
378C2×C3×D3—62no—
379C2×C3×D3—62no—
380C2×C3×D3—62no—
381C2×C3×D3—62no—
382C2×C3×D3—62no—
383C2×C3×D3—62no—
384C2×C3×D3—62no—
385C2×C3×D3—122no—
386C2×C3×D3—122no—
387C2×C3×D3—122no—
388C2×C3×D3—121no—
389C2×C3×D3—122no—
390C2×C3×D3—122no—
391C2×C3×D3—122no—
392C2×C3×D3—128no—
393C2×C3×D3—128no—
394D32—128no—
395D32—124no—
396C32×C4—22no—
397C32×C4—22no—
398C32×C4—22no—
399C32×C4—42no—
400C32×C4—42no—
401C32×C4—42no—
402C32×C4—82no—
403C3⋊Dic3—22no—
404C3⋊Dic3—22no—
405C3⋊Dic3—21no—
406C3⋊Dic3—22no—
407C3×Dic3—64no—
408C3×Dic3—62no—
409C3×Dic3—62no—
410C3×Dic3—62no—
411C3×Dic3—62no—
412C3×Dic3—62no—
413C3×Dic3—62no—
414C3×Dic3—62no—
415C3×Dic3—122no—
416C3×Dic3—122no—
417C3×Dic3—121no—
418#124—92no—
419#124—272no—
420#124—272no—
421#124—271no—
422#124—272no—
423#120—91no—
424#120—272no—
425#120—272no—
426#120—271no—
427C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
428C23×C3—21no—
429C23×C3—22no—
430C23×C3—41no—
431C23×C3—91no—
432C23×C3—92no—
433C23×C3—92no—
434C23×C3—361no—
435C23×C3—362no—
436C23×C3C2×D3211yescharacteristic
437C23×C3C2×D3212yes—
438C22×D3—32no—
439C22×D3—32no—
440C22×D3—32no—
441C22×D3—32no—
442C22×D3—32no—
443C22×D3—62no—
444C22×D3—62no—
445C22×D3—61no—
446C22×D3—122no—
447C22×D3—122no—
448C22×D3—122no—
449C22×D3—124no—
450C22×D3—122no—
451C22×D3—362no—
452C22×D3—361no—
453C22×D3—362no—
454C22×D3—362no—
455C22×D3—362no—
456C3×Q8—362no—
457C3×D4—362no—
458C3×D4—361no—
459C3×D4—362no—
460C2×C3×C4—22no—
461C2×C3×C4—22no—
462C2×C3×C4—22no—
463C2×C3×C4—42no—
464C2×C3×C4—91no—
465C2×C3×C4—92no—
466C2×C3×C4—92no—
467C2×C3×C4—92no—
468C2×C3×C4C2×D3212yes—
469C2×C3×C4C2×D3212yes—
470C2×C3×C4C2×D3212yes—
471#67—122no—
472#67—122no—
473#67—122no—
474#67—124no—
475#67—122no—
476#67—362no—
477#67—362no—
478#67—362no—
479#67—362no—
480C2×Dic3—32no—
481C2×Dic3—32no—
482C2×Dic3—32no—
483C2×Dic3—32no—
484C2×Dic3—32no—
485C2×Dic3—32no—
486C2×Dic3—32no—
487C2×Dic3—62no—
488C2×Dic3—62no—
489C2×Dic3—61no—
490C2×Dic3—62no—
491D12—122no—
492D12—122no—
493D12—124no—
494D12—362no—
495D12—362no—
496Dic6—122no—
497Dic6—122no—
498Dic6—124no—
499Dic6—362no—
500Dic6—361no—
501C2×C32—22no—
502C2×C32—21no—
503C2×C32—22no—
504C2×C32—22no—
505C2×C32—22no—
506C2×C32—42no—
507C2×C32—41no—
508C2×C32—41no—
509C2×C32—42no—
510C2×C32—42no—
511C2×C32—82no—
512C2×C32—82no—
513C2×C32#100411yescharacteristic
514C2×C32#100412yes—
515C2×C32#100412yes—
516C2×C32#100912yes—
517C2×C32#100912yes—
518C3⋊D3—42no—
519C3⋊D3—42no—
520C3⋊D3—41no—
521C3×D3—124no—
522C3×D3—122no—
523C3×D3—122no—
524C3×D3—122no—
525C3×D3—122no—
526C3×D3—122no—
527C3×D3—242no—
528C3×D3—241no—
529C24—91no—
530C24—271no—
531C2×Q8—92no—
532C2×Q8—271no—
533C2×D4—91no—
534C2×D4—92no—
535C2×D4—92no—
536C2×D4—272no—
537C2×D4—272no—
538C22⋊C4—92no—
539C22⋊C4—92no—
540C22⋊C4—92no—
541C22⋊C4—92no—
542C22⋊C4—272no—
543C22⋊C4—272no—
544C22⋊C4—271no—
545C22⋊C4—272no—
546C22⋊C4—272no—
547C22⋊C4#111311yescharacteristic, complemented
548C42—92no—
549C42—271no—
550C22×C3—22no—
551C22×C3—22no—
552C22×C3—21no—
553C22×C3—22no—
554C22×C3—22no—
555C22×C3—22no—
556C22×C3—22no—
557C22×C3—21no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3—42no—
563C22×C3—41no—
564C22×C3—82no—
565C22×C3—182no—
566C22×C3—182no—
567C22×C3—181no—
568C22×C3—182no—
569C22×C3—182no—
570C22×C3—182no—
571C22×C3—182no—
572C22×C3—182no—
573C22×C3—364no—
574C22×C3C22×D3212yes—
575C22×C3C22×D3211yescharacteristic
576C2×D3—64no—
577C2×D3—62no—
578C2×D3—62no—
579C2×D3—62no—
580C2×D3—62no—
581C2×D3—62no—
582C2×D3—62no—
583C2×D3—62no—
584C2×D3—62no—
585C2×D3—62no—
586C2×D3—62no—
587C2×D3—62no—
588C2×D3—62no—
589C2×D3—62no—
590C2×D3—62no—
591C2×D3—62no—
592C2×D3—122no—
593C2×D3—122no—
594C2×D3—122no—
595C2×D3—121no—
596C2×D3—122no—
597C2×D3—122no—
598C2×D3—122no—
599C2×D3—128no—
600C2×D3—128no—
601C2×D3—368no—
602C2×D3—364no—
603C3×C4—22no—
604C3×C4—22no—
605C3×C4—22no—
606C3×C4—42no—
607C3×C4—42no—
608C3×C4—42no—
609C3×C4—82no—
610C3×C4—182no—
611C3×C4—181no—
612C3×C4—182no—
613C3×C4—182no—
614Dic3—64no—
615Dic3—62no—
616Dic3—62no—
617Dic3—62no—
618Dic3—62no—
619Dic3—62no—
620Dic3—62no—
621Dic3—62no—
622Dic3—122no—
623Dic3—121no—
624Dic3—122no—
625C32—22no—
626C32—21no—
627C32—41no—
628C32#559411yescharacteristic, complemented
629C32#568312yescomplemented
630C23—92no—
631C23—92no—
632C23—91no—
633C23—361no—
634C23—362no—
635C23—1081no—
636C23—1082no—
637C23C2×#111311yescharacteristic
638Q8—362no—
639Q8—1081no—
640D4—362no—
641D4—361no—
642D4—362no—
643D4—1082no—
644D4—1082no—
645C2×C4—92no—
646C2×C4—92no—
647C2×C4—91no—
648C2×C4—92no—
649C2×C4C2×#111312yes—
650C2×C3—21no—
651C2×C3—22no—
652C2×C3—22no—
653C2×C3—22no—
654C2×C3—22no—
655C2×C3—42no—
656C2×C3—42no—
657C2×C3—41no—
658C2×C3—41no—
659C2×C3—42no—
660C2×C3—82no—
661C2×C3—82no—
662C2×C3—362no—
663C2×C3—361no—
664C2×C3—362no—
665C2×C3#6430712yes—
666C2×C3#6430811yescharacteristic
667C2×C3#6431312yes—
668C2×C3#6431412yes—
669C2×C3#6431412yes—
670D3—124no—
671D3—122no—
672D3—122no—
673D3—122no—
674D3—122no—
675D3—122no—
676D3—241no—
677D3—242no—
678C22—21no—
679C22—22no—
680C22—181no—
681C22—182no—
682C22—182no—
683C22—182no—
684C22—182no—
685C22—182no—
686C22—182no—
687C22—182no—
688C22—364no—
689C22—1084no—
690C22C22×#111311yescharacteristic
691C4—22no—
692C4—181no—
693C4—182no—
694C4—182no—
695C4—182no—
696C3—21no—
697C3—22no—
698C3—41no—
699C3#1059430912yescomplemented
700C3#1059431111yescharacteristic, complemented
701C2—41no—
702C2—362no—
703C2—361no—
704C2—362no—
705C2#1172461911yescharacteristic
706C2#1172462312yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4#1113
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64094C2semidirect
2. C3×#64091C2semidirect
3. #11723729C2semidirect
4. #11723727C2semidirect
5. #11723718C2semidirect
6. #11723701C2semidirect
7. #11723677C2semidirect
8. #11723672C2semidirect
9. #11723670C2semidirect
12. C23×C3×C3⋊D3C22semidirect
13. C22×#1113C22semidirect
17. C2×C3×#634C22semidirect
18. C2×#6296C22semidirect
21. C33×C22⋊C4C22semidirect
22. C3×#3757C22semidirect
23. C3×#3752C22semidirect
24. C3×#3752C22semidirect
25. C3×#3752C22semidirect
26. C3×C4×C3⋊Dic3C22semidirect
27. #88814C22semidirect
28. #88814C22semidirect
29. #88814C22semidirect
30. #88814C22semidirect
31. #88814C22semidirect
32. #88813C22semidirect
33. #64094D3semidirect
34. #64091D3semidirect
81. C23×C3⋊D3C2×D3semidirect
89. C2×#634C2×D3semidirect
112. C32×C22⋊C4C2×D3semidirect
113. C32×C22⋊C4C2×D3semidirect
114. #3757C2×D3semidirect
115. #3752C2×D3semidirect
116. #3752C2×D3semidirect
117. #3752C2×D3semidirect
118. C4×C3⋊Dic3C2×D3semidirect
309. C3×C22⋊C4D32semidirect
310. C3×C22⋊C4D32semidirect
427. C33H(3,𝔽4)semidirect
547. C22⋊C4#1113semidirect
628. C32#5594semidirect
629. C32#5683semidirect
699. C3#10594309semidirect
700. C3#10594311semidirect

Maximal subgroups

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

subgroupindex
C3×#640942
C3×#640912
#117237292
#117237272
#117237182
#117237012
#117236772
#117236722
#117236702
#105943113
#105943093