Finite Group Databasebeta

← #13141151

Subgroups — #13141151

Order 1728.

797 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88552C211yesmaximal, characteristic
2C2×#3757⋊C3C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic
4C2×#88449C211yesmaximal, characteristic, complemented
5C2×#88447C211yesmaximal, characteristic, complemented
6#11720334C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591702—31nomaximal
9#10591701—31nomaximal
10C24×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#88552C414yescomplemented
18#3757⋊C3—22no—
19#3757⋊C3—22no—
20#88449—22no—
21#88447—22no—
22C2×#3757D311yescharacteristic
23C2×#3757D311yescharacteristic
24C2×C3×#217—31no—
25C2×C3×#217—31no—
26C2×C3×#217—31no—
27C2×C3×#217—31no—
28C2×C3×#217—61no—
29C2×Dic3⋊Dic3—31no—
30C2×Dic3⋊Dic3—31no—
31C2×Dic32—31no—
32C2×Dic32—31no—
33#63579—31no—
34#63579—31no—
35#63579—31no—
36#63579—31no—
37C23×H(3,𝔽3)—21no—
38C23×H(3,𝔽3)—21no—
39C23×H(3,𝔽3)—21no—
40C23×H(3,𝔽3)—21no—
41C23×H(3,𝔽3)—21no—
42C23×H(3,𝔽3)—21no—
43C23×H(3,𝔽3)C2×C412yes—
44C23×H(3,𝔽3)C2311yescharacteristic
45C2×#1084—21no—
46C2×#1084—21no—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084C2×C412yes—
50C2×#1084C2×C412yes—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3—21no—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3—21no—
55C2×(C3⋊Dic3)⋊C3—21no—
56C2×(C3⋊Dic3)⋊C3—21no—
57C2×(C3⋊Dic3)⋊C3—21no—
58C2×(C3⋊Dic3)⋊C3—21no—
59C2×(C3⋊Dic3)⋊C3—21no—
60C2×(C3⋊Dic3)⋊C3—21no—
61C2×(C3⋊Dic3)⋊C3D412yes—
62C2×(C3⋊Dic3)⋊C3D412yes—
63#4951—91no—
64#4945—91no—
65#4944—91no—
66C24×C32—21no—
67C24×C32C2×D311yescharacteristic
68C24×C32C2×D311yescharacteristic
69C22×C3⋊Dic3C2×D311yescharacteristic
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—61no—
82C22×C3×Dic3—61no—
83#3757—22no—
84#3757—22no—
85C3×#217—34no—
86C3×#217—34no—
87C3×#217—62no—
88C3×#217—62no—
89C3×#217—64no—
90Dic3⋊Dic3—62no—
91Dic3⋊Dic3—62no—
92Dic32—62no—
93Dic32—62no—
94C22×H(3,𝔽3)—22no—
95C22×H(3,𝔽3)—22no—
96C22×H(3,𝔽3)—22no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)—41no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)—41no—
101C22×H(3,𝔽3)C22×C411yescharacteristic
102C22×H(3,𝔽3)C2×D411yescharacteristic
103C22×H(3,𝔽3)C2×D411yescharacteristic
104C22×H(3,𝔽3)#4111yescharacteristic
105C22×H(3,𝔽3)#4111yescharacteristic
106C22×H(3,𝔽3)#4111yescharacteristic
107C22×H(3,𝔽3)#4111yescharacteristic
108#1084—22no—
109#1084—22no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3—22no—
112(C3⋊Dic3)⋊C3—41no—
113(C3⋊Dic3)⋊C3—41no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×#217—31no—
118C2×#217—31no—
119C2×#217—31no—
120C2×#217—31no—
121C2×#217—61no—
122C2×#217—91no—
123C2×#217—91no—
124C2×C4⋊Dic3—91no—
125C2×Dic3⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×C4×Dic3—91no—
128C2×C4×Dic3—91no—
129C2×C4×Dic3—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C3⋊#95—91no—
133C3⋊#95—91no—
134C3⋊#95—91no—
135C3⋊#95—91no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—21no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—21no—
142C23×C32—21no—
143C23×C32—21no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—22no—
151C23×C32—21no—
152C23×C32—41no—
153C23×C32—41no—
154C23×C32—41no—
155C23×C32—41no—
156C23×C32C4×D312yes—
157C23×C32C4×D312yes—
158C23×C32C22×D311yescharacteristic
159C23×C32C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3#6712yes—
171C2×C3⋊Dic3#6712yes—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3—121no—
200C2×C3×Dic3—121no—
201C2×C3×Dic3—121no—
202#388—271noSylow
203C2×H(3,𝔽3)—42no—
204C2×H(3,𝔽3)#11711yescharacteristic
205C2×H(3,𝔽3)C4×D411yescharacteristic
206C2×H(3,𝔽3)C4×D411yescharacteristic
207C2×H(3,𝔽3)#12111yescharacteristic
208C2×H(3,𝔽3)#12311yescharacteristic
209C2×H(3,𝔽3)#12411yescharacteristic
210C2×H(3,𝔽3)#12611yescharacteristic
211C24×C3—31no—
212C24×C3—31no—
213C24×C3—61no—
214C24×C3D3211yescharacteristic
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4—91no—
221C22×Dic3—31no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C22×Dic3—31no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—61no—
230C22×Dic3—61no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C22×Dic3—91no—
235C3×C22⋊C4—94no—
236C3×C22⋊C4—182no—
237C3×C22⋊C4—182no—
238#217—34no—
239#217—34no—
240#217—64no—
241#217—62no—
242#217—62no—
243#217—182no—
244#217—182no—
245C4⋊Dic3—182no—
246Dic3⋊C4—182no—
247Dic3⋊C4—182no—
248C4×Dic3—182no—
249C4×Dic3—182no—
250C4×Dic3—182no—
251C22×C32—24no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—42no—
270C22×C32—42no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32—41no—
276C22×C32—41no—
277C22×C32—41no—
278C22×C32—41no—
279C22×C32C2×C4×D311yescharacteristic
280C22×C32C2×C4×D311yescharacteristic
281C22×C32#23511yescharacteristic
282C22×C32#23511yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32#23711yescharacteristic
286C22×C32#23711yescharacteristic
287C22×C32#23711yescharacteristic
288C22×C32#23711yescharacteristic
289C22×C32#23711yescharacteristic
290C22×C32#23711yescharacteristic
291C22×C32C2×#6711yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C3⋊Dic3—22no—
294C3⋊Dic3—22no—
295C3⋊Dic3—41no—
296C3⋊Dic3—41no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—122no—
304C3×Dic3—122no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C2×C4⋊C4—271no—
308C2×C22⋊C4—91no—
309C2×C22⋊C4—91no—
310C2×C22⋊C4—91no—
311C2×C42—271no—
312#95—271no—
313#95—271no—
314H(3,𝔽3)#38811yescharacteristic, complemented, Sylow
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—32no—
322C23×C3—32no—
323C23×C3—31no—
324C23×C3—31no—
325C23×C3—61no—
326C23×C3—61no—
327C23×C3—61no—
328C23×C3—61no—
329C23×C3—61no—
330C23×C3—61no—
331C23×C3—61no—
332C23×C3—61no—
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—61no—
336C23×C3—61no—
337C23×C3—62no—
338C23×C3—62no—
339C23×C3—62no—
340C23×C3—61no—
341C23×C3—121no—
342C23×C3—121no—
343C23×C3—121no—
344C23×C3—121no—
345C23×C3#62412yes—
346C23×C3C2×D3211yescharacteristic
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4—181no—
356C2×C3×C4—181no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×C3×C4—181no—
360C2×C3×C4—181no—
361C2×C3×C4—181no—
362C2×C3×C4—181no—
363C2×C3×C4—181no—
364C2×C3×C4—181no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—62no—
372C2×Dic3—62no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—92no—
392C2×Dic3—92no—
393C2×Dic3—121no—
394C2×Dic3—121no—
395C2×Dic3—121no—
396C2×Dic3—121no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×Dic3—181no—
402C2×Dic3—181no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—181no—
406C2×Dic3—181no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—44no—
415C2×C32—42no—
416C2×C32—42no—
417C2×C32#87711yescharacteristic
418C2×C32#87711yescharacteristic
419C2×C32#87911yescharacteristic
420C2×C32#87911yescharacteristic
421C2×C32#67⋊C411yescharacteristic
422C2×C32#67⋊C411yescharacteristic
423C2×C32#88311yescharacteristic
424C2×C32#88511yescharacteristic
425C2×C32C4×#6711yescharacteristic
426C2×C32C4×#6711yescharacteristic
427C2×C32#93511yescharacteristic
428C2×C32#93611yescharacteristic
429C2×C32#93811yescharacteristic
430C2×C32#93911yescharacteristic
431C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C22×C4—91no—
433C22×C4—91no—
434C22×C4—91no—
435C22×C4—91no—
436C22×C4—91no—
437C22×C4—91no—
438C4⋊C4—542no—
439C22⋊C4—94no—
440C22⋊C4—182no—
441C22⋊C4—182no—
442C42—542no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—31no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—31no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—31no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—62no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—64no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—61no—
476C22×C3—61no—
477C22×C3—61no—
478C22×C3—122no—
479C22×C3—122no—
480C22×C3—121no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—121no—
484C22×C3—121no—
485C22×C3—121no—
486C22×C3—121no—
487C22×C3—121no—
488C22×C3—122no—
489C22×C3—122no—
490C22×C3—122no—
491C22×C3—122no—
492C22×C3#380411yescharacteristic
493C22×C3#380411yescharacteristic
494C22×C3#380511yescharacteristic
495C22×C3#380511yescharacteristic
496C22×C3C2×#62411yescharacteristic
497C22×C3D3×#6711yescharacteristic
498C22×C3D3×#6711yescharacteristic
499C3×C4—182no—
500C3×C4—182no—
501C3×C4—182no—
502C3×C4—182no—
503C3×C4—361no—
504C3×C4—361no—
505Dic3—62no—
506Dic3—62no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—122no—
512Dic3—122no—
513Dic3—121no—
514Dic3—121no—
515Dic3—182no—
516Dic3—182no—
517Dic3—361no—
518Dic3—361no—
519C32—21no—
520C32#494411yescharacteristic, complemented
521C32#494511yescharacteristic, complemented
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#619912yes—
529C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C2×C4—92no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4—181no—
542C2×C4—181no—
543C2×C4—181no—
544C2×C4—181no—
545C2×C4—181no—
546C2×C4—181no—
547C2×C4—181no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—31no—
553C2×C3—31no—
554C2×C3—31no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—31no—
561C2×C3—31no—
562C2×C3—42no—
563C2×C3—61no—
564C2×C3—61no—
565C2×C3—61no—
566C2×C3—61no—
567C2×C3—61no—
568C2×C3—61no—
569C2×C3—61no—
570C2×C3—122no—
571C2×C3—122no—
572C2×C3—124no—
573C2×C3#6395211yescharacteristic
574C2×C3#6395211yescharacteristic
575C2×C3#6395611yescharacteristic
576C2×C3#6395711yescharacteristic
577C2×C3#6395911yescharacteristic
578C2×C3#6396911yescharacteristic
579C2×C3#6397011yescharacteristic
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22#8867111yescharacteristic
588C22#8867111yescharacteristic
589C22#8867211yescharacteristic
590C22#8867211yescharacteristic
591C22C2×#619911yescharacteristic
592C22#8867611yescharacteristic
593C22#8867611yescharacteristic
594C4—182no—
595C4—182no—
596C4—182no—
597C4—182no—
598C4—361no—
599C4—361no—
600C3—31no—
601C3—31no—
602C3—61no—
603C3#1059170311yescharacteristic
604C2—42no—
605C2#1172129311yescharacteristic
606C2#1172129311yescharacteristic
607C2#1172129711yescharacteristic
608C2#1172129811yescharacteristic
609C2#1172130411yescharacteristic
610C2#1172131111yescharacteristic
611C2#1172131311yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#388
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88449C2semidirect
5. C2×#88447C2semidirect
6. #11720334C2semidirect
7. #11720334C2semidirect
17. #88552C4semidirect
314. H(3,𝔽3)#388semidirect
520. C32#4944semidirect
521. C32#4945semidirect

Maximal subgroups

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

subgroupindex
C2×#885522
C2×#3757⋊C32
C2×#884492
C2×#884472
#117203342
#105917023
#105917013