Finite Group Databasebeta

← #13140213

Subgroups — #13140213

Order 1728.

783 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
1C2×#88542C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic, complemented
3C2×#88451C211yesmaximal, characteristic
4C2×#88451C211yesmaximal, characteristic
5C2×#88449C211yesmaximal, characteristic, complemented
6C2×#88448C211yesmaximal, characteristic, complemented
7#63658⋊C3C211yesmaximal, characteristic, complemented
8#10591309—31nomaximal
9#10591306—31nomaximal
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#88542—22no—
18#3750⋊C3—22no—
19#88451—22no—
20#88451—22no—
21#88449—22no—
22#88448—22no—
23C2×#3750D311yescharacteristic, complemented
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×(C2×D3)⋊C4—61no—
27C2×C3×Dic3⋊C4—31no—
28C2×#3724—31no—
29C2×Dic3⋊Dic3—31no—
30C2×Dic3⋊Dic3—31no—
31C2×Dic3⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C2×(C2×D3)⋊Dic3—31no—
34C2×(C2×D3)⋊Dic3—31no—
35C2×(C2×D3)⋊Dic3—31no—
36#63658D311yescharacteristic, complemented
37C3×C3⋊#95—31no—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×#272—21no—
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272—21no—
44C22×#272—21no—
45C22×#272Q812yes—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084D412yes—
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)⋊C3—21no—
62C2×(C3⋊Dic3)⋊C3D412yes—
63C2×(C3⋊Dic3)⋊C3D412yes—
64#4994—91no—
65#4948—91no—
66#4676—91no—
67C23×C3×D3—31no—
68C23×C3×D3—31no—
69C23×C3×D3—61no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—61no—
84#3750—22no—
85C3×(C2×D3)⋊C4—62no—
86C3×(C2×D3)⋊C4—62no—
87C3×(C2×D3)⋊C4—64no—
88C3×Dic3⋊C4—62no—
89#3724—62no—
90Dic3⋊Dic3—62no—
91Dic3⋊Dic3—62no—
92Dic3⋊Dic3—62no—
93(C2×D3)⋊Dic3—62no—
94(C2×D3)⋊Dic3—62no—
95(C2×D3)⋊Dic3—62no—
96(C2×D3)⋊Dic3—62no—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C22×H(3,𝔽3)C2×D411yescharacteristic
100C22×H(3,𝔽3)C2×Q811yescharacteristic
101C22×H(3,𝔽3)#4111yescharacteristic
102C22×H(3,𝔽3)#4111yescharacteristic
103C22×H(3,𝔽3)#4111yescharacteristic
104C2×#272—22no—
105C2×#272—22no—
106C2×#272—22no—
107C2×#272—41no—
108C2×#272—41no—
109C2×#272—41no—
110C2×#272—41no—
111C4×H(3,𝔽3)—41no—
112#1084—22no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3—41no—
116(C3⋊Dic3)⋊C3—41no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C22⋊C4—91no—
119C2×#217—91no—
120C2×#217—91no—
121C2×(C2×D3)⋊C4—31no—
122C2×(C2×D3)⋊C4—31no—
123C2×(C2×D3)⋊C4—61no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—91no—
126C2×(C2×D3)⋊C4—91no—
127C2×(C2×D3)⋊C4—91no—
128C2×C4⋊Dic3—91no—
129C2×C4⋊Dic3—91no—
130C2×Dic3⋊C4—31no—
131C2×Dic3⋊C4—91no—
132C2×Dic3⋊C4—91no—
133C2×Dic3⋊C4—91no—
134C2×Dic3⋊C4—91no—
135C2×Dic3⋊C4—91no—
136C3×#95—91no—
137C3⋊#95—31no—
138C3⋊#95—91no—
139C23×C32—21no—
140C23×C32C22×D311yescharacteristic
141C23×C32C22×D311yescharacteristic
142C22×C3×D3—32no—
143C22×C3×D3—32no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—121no—
160C22×C3×D3—121no—
161C22×C3×D3—121no—
162C22×C3×D3—121no—
163C2×C32×C4—22no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3D1212yes—
183C2×C3⋊Dic3#6712yes—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—62no—
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—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—121no—
204C2×C3×Dic3—121no—
205#396—271noSylow
206C2×H(3,𝔽3)#12111yescharacteristic
207C2×H(3,𝔽3)#12111yescharacteristic
208C2×H(3,𝔽3)#12111yescharacteristic
209C2×H(3,𝔽3)C22⋊Q811yescharacteristic
210C2×H(3,𝔽3)C22⋊Q811yescharacteristic
211C2×H(3,𝔽3)C22⋊Q811yescharacteristic
212C2×H(3,𝔽3)#12611yescharacteristic
213#272—42no—
214C24×C3—91no—
215C23×D3—31no—
216C23×D3—31no—
217C23×D3—61no—
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—61no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4D3211yescharacteristic
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—31no—
230C22×Dic3—31no—
231C22×Dic3—31no—
232C22×Dic3—31no—
233C22×Dic3—61no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236C22×Dic3—91no—
237C22×Dic3—91no—
238C3×C4⋊C4—182no—
239C3×C22⋊C4—182no—
240#217—182no—
241#217—182no—
242(C2×D3)⋊C4—62no—
243(C2×D3)⋊C4—62no—
244(C2×D3)⋊C4—64no—
245(C2×D3)⋊C4—182no—
246(C2×D3)⋊C4—182no—
247(C2×D3)⋊C4—182no—
248(C2×D3)⋊C4—182no—
249C4⋊Dic3—182no—
250C4⋊Dic3—182no—
251Dic3⋊C4—62no—
252Dic3⋊C4—182no—
253Dic3⋊C4—182no—
254Dic3⋊C4—182no—
255Dic3⋊C4—182no—
256Dic3⋊C4—182no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32C2×D1211yescharacteristic
265C22×C32#23511yescharacteristic
266C22×C32#23511yescharacteristic
267C22×C32#23511yescharacteristic
268C22×C32#23511yescharacteristic
269C22×C32D3×D411yescharacteristic
270C22×C32D3×D411yescharacteristic
271C22×C32D3×D411yescharacteristic
272C22×C32D3×D411yescharacteristic
273C22×C32#23711yescharacteristic
274C22×C32D3×Q811yescharacteristic
275C22×C32D3×Q811yescharacteristic
276C22×C32#23911yescharacteristic
277C22×C32C2×#6711yescharacteristic
278C2×C3×D3—64no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—122no—
286C2×C3×D3—122no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—122no—
290C2×C3×D3—122no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C32×C4—42no—
300C32×C4—41no—
301C32×C4—41no—
302C3⋊Dic3—22no—
303C3⋊Dic3—22no—
304C3⋊Dic3—41no—
305C3⋊Dic3—41no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310C3×Dic3—122no—
311C3×Dic3—121no—
312C3×Dic3—121no—
313C2×C4⋊C4—91no—
314C2×C4⋊C4—271no—
315C2×C4⋊C4—271no—
316C2×C22⋊C4—91no—
317C2×C22⋊C4—271no—
318C2×C22⋊C4—271no—
319#95—91no—
320H(3,𝔽3)#39611yescharacteristic, complemented, Sylow
321C23×C3—31no—
322C23×C3—31no—
323C23×C3—61no—
324C23×C3—92no—
325C23×C3—181no—
326C23×C3—181no—
327C23×C3—181no—
328C23×C3—181no—
329C23×C3—181no—
330C23×C3—181no—
331C23×C3C2×D3211yescharacteristic
332C22×D3—32no—
333C22×D3—32no—
334C22×D3—62no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—61no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—61no—
341C22×D3—61no—
342C22×D3—61no—
343C22×D3—61no—
344C22×D3—61no—
345C22×D3—61no—
346C22×D3—61no—
347C22×D3—62no—
348C22×D3—62no—
349C22×D3—121no—
350C22×D3—121no—
351C22×D3—121no—
352C22×D3—121no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—62no—
363C2×C3×C4—92no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—121no—
367C2×C3×C4—121no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4—181no—
371C2×C3×C4—181no—
372C2×C3×C4—181no—
373C2×C3×C4—181no—
374C2×C3×C4—181no—
375C2×C3×C4—181no—
376C2×C3×C4—181no—
377C2×C3×C4—181no—
378C2×C3×C4—181no—
379C2×C3×C4—181no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—32no—
384C2×Dic3—62no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—92no—
400C2×Dic3—92no—
401C2×Dic3—121no—
402C2×Dic3—121no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—181no—
406C2×Dic3—181no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—181no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32#87611yescharacteristic
421C2×C32#88411yescharacteristic
422C2×C32#88411yescharacteristic
423C2×C32#88411yescharacteristic
424C2×C32#89511yescharacteristic
425C2×C32#89611yescharacteristic
426C2×C32#89611yescharacteristic
427C2×C32#89611yescharacteristic
428C2×C32#89611yescharacteristic
429C2×C32#89711yescharacteristic
430C2×C32#89811yescharacteristic
431C2×C32#93011yescharacteristic
432C2×C32#93911yescharacteristic
433C2×C32#94611yescharacteristic
434C3×D3—124no—
435C3×D3—122no—
436C3×D3—122no—
437C24—91no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—91no—
442C22×C4—91no—
443C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C4⋊C4—182no—
445C4⋊C4—542no—
446C4⋊C4—542no—
447C22⋊C4—182no—
448C22⋊C4—542no—
449C22⋊C4—542no—
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—31no—
461C22×C3—31no—
462C22×C3—31no—
463C22×C3—31no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—182no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—361no—
475C22×C3—361no—
476C22×C3—361no—
477C22×C3—361no—
478C22×C3#379711yescharacteristic
479C22×C3#379811yescharacteristic
480C22×C3#379911yescharacteristic
481C22×C3D3×D1211yescharacteristic
482C22×C3#380411yescharacteristic
483C22×C3D3×#6711yescharacteristic
484C22×C3#381111yescharacteristic
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—64no—
492C2×D3—122no—
493C2×D3—122no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—121no—
497C2×D3—121no—
498C2×D3—121no—
499C2×D3—121no—
500C2×D3—121no—
501C2×D3—121no—
502C2×D3—122no—
503C2×D3—122no—
504C2×D3—122no—
505C2×D3—122no—
506C3×C4—41no—
507C3×C4—121no—
508C3×C4—121no—
509C3×C4—122no—
510C3×C4—182no—
511C3×C4—182no—
512C3×C4—182no—
513C3×C4—361no—
514C3×C4—361no—
515Dic3—62no—
516Dic3—62no—
517Dic3—62no—
518Dic3—62no—
519Dic3—122no—
520Dic3—121no—
521Dic3—121no—
522Dic3—182no—
523Dic3—182no—
524Dic3—361no—
525Dic3—361no—
526C32—21no—
527C32#467611yescharacteristic, complemented
528C32#499411yescharacteristic, complemented
529C23—92no—
530C23—181no—
531C23—181no—
532C23—181no—
533C23—181no—
534C23—181no—
535C23—181no—
536C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—92no—
541C2×C4—92no—
542C2×C4—92no—
543C2×C4—181no—
544C2×C4—181no—
545C2×C4—181no—
546C2×C4—181no—
547C2×C4—181no—
548C2×C4—181no—
549C2×C4—181no—
550C2×C4—181no—
551C2×C4—181no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4—181no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—31no—
561C2×C3—31no—
562C2×C3—31no—
563C2×C3—31no—
564C2×C3—31no—
565C2×C3—31no—
566C2×C3—31no—
567C2×C3—31no—
568C2×C3—31no—
569C2×C3—61no—
570C2×C3—61no—
571C2×C3—61no—
572C2×C3—61no—
573C2×C3—61no—
574C2×C3—61no—
575C2×C3—61no—
576C2×C3—362no—
577C2×C3#6386511yescharacteristic
578C2×C3#6386811yescharacteristic
579C2×C3#6388811yescharacteristic
580C2×C3#6389511yescharacteristic
581C2×C3#6390611yescharacteristic
582C2×C3#6391011yescharacteristic
583C2×C3#6397111yescharacteristic
584D3—122no—
585D3—122no—
586D3—124no—
587C22—182no—
588C22—182no—
589C22—182no—
590C22—361no—
591C22—361no—
592C22—361no—
593C22—361no—
594C22#8865011yescharacteristic
595C22#8865111yescharacteristic
596C22#8865211yescharacteristic
597C22#8865411yescharacteristic
598C22#8867111yescharacteristic
599C22#8867611yescharacteristic
600C22#8867711yescharacteristic
601C4—41no—
602C4—182no—
603C4—182no—
604C4—182no—
605C4—361no—
606C4—361no—
607C3—31no—
608C3—31no—
609C3—61no—
610C3#1059134711yescharacteristic
611C2—362no—
612C2#1172109211yescharacteristic
613C2#1172109611yescharacteristic
614C2#1172114011yescharacteristic
615C2#1172114811yescharacteristic
616C2#1172115011yescharacteristic
617C2#1172115711yescharacteristic
618C2#1172131211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#396
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3750⋊C3C2semidirect
5. C2×#88449C2semidirect
6. C2×#88448C2semidirect
7. #63658⋊C3C2semidirect
23. C2×#3750D3semidirect
36. #63658D3semidirect
320. H(3,𝔽3)#396semidirect
527. C32#4676semidirect
528. C32#4994semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×#3750⋊C32
C2×#884512
C2×#884492
C2×#884482
#63658⋊C32
#105913093
#105913063