Finite Group Databasebeta

← #13140208

Subgroups — #13140208

Order 1728.

1055 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#272C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic, complemented
3C2×#88451C211yesmaximal, characteristic
4C2×#88451C211yesmaximal, characteristic
5C2×#88448C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591278—31nomaximal
9#10591277—31nomaximal
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C2×C4×#272—22no—
14C2×C4×#272—22no—
15C2×C4×#272—22no—
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20#3750⋊C3—22no—
21#88451—22no—
22#88451C414yescomplemented
23#88448—22no—
24C22×C3×C4×D3—31no—
25C22×C3×C4×D3—31no—
26C22×C3×C4×D3—61no—
27C2×#3750D311yescharacteristic, complemented
28C2×C3×Dic3⋊C4—31no—
29C2×#3724—31no—
30C2×Dic3⋊Dic3—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C2×(C2×D3)⋊Dic3—31no—
34C2×(C2×D3)⋊Dic3—31no—
35#63658D311yescharacteristic, complemented
36C3×C3⋊#95—31no—
37#63579—31no—
38#63579—31no—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272—21no—
44C22×#272C2×C412yes—
45C22×#272D412yes—
46C22×#272Q812yes—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)D412yes—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084D412yes—
53C4×#272—28no—
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)⋊C3—21no—
63C2×(C3⋊Dic3)⋊C3—21no—
64C2×(C3⋊Dic3)⋊C3C2×C412yescomplemented
65C2×(C3⋊Dic3)⋊C3C2×C412yescomplemented
66#5210—91no—
67#4989—91no—
68#4668—91no—
69C23×C3×D3—31no—
70C23×C3×D3—31no—
71C23×C3×D3—61no—
72C22×C32×C4—21no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—61no—
86C2×C3×C4×D3—62no—
87C2×C3×C4×D3—62no—
88C2×C3×C4×D3—62no—
89C2×C3×C4×D3—62no—
90C2×C3×C4×D3—64no—
91C2×C3×C4×D3—64no—
92C2×C3×C4×D3—64no—
93C2×C3×C4×D3—62no—
94C2×C3×C4×D3—62no—
95#3750—22no—
96C3×Dic3⋊C4—62no—
97#3724—62no—
98Dic3⋊Dic3—62no—
99(C2×D3)⋊Dic3—34no—
100(C2×D3)⋊Dic3—34no—
101(C2×D3)⋊Dic3—62no—
102(C2×D3)⋊Dic3—62no—
103C22×H(3,𝔽3)C22×C411yescharacteristic
104C22×H(3,𝔽3)C2×D411yescharacteristic
105C22×H(3,𝔽3)C2×D411yescharacteristic
106C22×H(3,𝔽3)C2×D411yescharacteristic
107C22×H(3,𝔽3)C2×Q811yescharacteristic
108C22×H(3,𝔽3)#4111yescharacteristic
109C22×H(3,𝔽3)#4111yescharacteristic
110C2×#272—22no—
111C2×#272—22no—
112C2×#272—22no—
113C2×#272—22no—
114C2×#272—22no—
115C2×#272—22no—
116C2×#272C4⋊C414yescomplemented
117C4×H(3,𝔽3)—22no—
118#1084—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—22no—
121(C3⋊Dic3)⋊C3—41no—
122(C3⋊Dic3)⋊C3—41no—
123C23×C3×C4—91no—
124C22×C4×D3—31no—
125C22×C4×D3—31no—
126C22×C4×D3—61no—
127C2×C3×C4⋊C4—91no—
128C2×#217—91no—
129C2×#217—91no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×(C2×D3)⋊C4—91no—
133C2×(C2×D3)⋊C4—91no—
134C2×C4⋊Dic3—91no—
135C2×Dic3⋊C4—31no—
136C2×Dic3⋊C4—91no—
137C2×Dic3⋊C4—91no—
138C2×Dic3⋊C4—91no—
139C3×#95—91no—
140C3⋊#95—31no—
141C3⋊#95—91no—
142C3⋊#95—91no—
143C3⋊#95—91no—
144C3⋊#95—91no—
145C23×C32—21no—
146C23×C32C22×D311yescharacteristic
147C23×C32C22×D311yescharacteristic
148C22×C3×D3—32no—
149C22×C3×D3—32no—
150C22×C3×D3—32no—
151C22×C3×D3—32no—
152C22×C3×D3—32no—
153C22×C3×D3—32no—
154C22×C3×D3—62no—
155C22×C3×D3—62no—
156C22×C3×D3—62no—
157C22×C3×D3—62no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C22×C3×D3—62no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—61no—
168C22×C3×D3—61no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4D1212yes—
177C2×C32×C4#6712yes—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3C4×D312yescomplemented
189C2×C3⋊Dic3C4×D312yescomplemented
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
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—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—61no—
211C3×C4×D3—616no—
212C3×C4×D3—68no—
213C3×C4×D3—68no—
214#385—271noSylow
215C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
216C2×H(3,𝔽3)C4×D411yescharacteristic
217C2×H(3,𝔽3)C4×D411yescharacteristic
218C2×H(3,𝔽3)#12011yescharacteristic
219C2×H(3,𝔽3)C22⋊Q811yescharacteristic
220C2×H(3,𝔽3)C22⋊Q811yescharacteristic
221C2×H(3,𝔽3)#12311yescharacteristic
222#272—24no—
223C24×C3—91no—
224C23×D3—31no—
225C23×D3—31no—
226C23×D3—61no—
227C22×C3×C4—31no—
228C22×C3×C4—31no—
229C22×C3×C4—61no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C22×C3×C4—91no—
235C22×C3×C4—182no—
236C22×C3×C4—182no—
237C22×C3×C4—182no—
238C22×C3×C4D3211yescharacteristic
239C22×Dic3—31no—
240C22×Dic3—31no—
241C22×Dic3—31no—
242C22×Dic3—31no—
243C22×Dic3—31no—
244C22×Dic3—31no—
245C22×Dic3—61no—
246C22×Dic3—91no—
247C22×Dic3—91no—
248C22×Dic3—91no—
249C22×Dic3—91no—
250C2×C4×D3—62no—
251C2×C4×D3—62no—
252C2×C4×D3—62no—
253C2×C4×D3—62no—
254C2×C4×D3—62no—
255C2×C4×D3—62no—
256C2×C4×D3—64no—
257C2×C4×D3—64no—
258C2×C4×D3—64no—
259C3×C4⋊C4—182no—
260#217—94no—
261#217—182no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—182no—
265(C2×D3)⋊C4—182no—
266C4⋊Dic3—182no—
267Dic3⋊C4—62no—
268Dic3⋊C4—182no—
269Dic3⋊C4—182no—
270Dic3⋊C4—182no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32C2×C4×D311yescharacteristic
279C22×C32C2×C4×D311yescharacteristic
280C22×C32C2×D1211yescharacteristic
281C22×C32#23511yescharacteristic
282C22×C32#23511yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32D3×D411yescharacteristic
286C22×C32D3×D411yescharacteristic
287C22×C32#23711yescharacteristic
288C22×C32D3×Q811yescharacteristic
289C22×C32D3×Q811yescharacteristic
290C22×C32#23911yescharacteristic
291C22×C32C2×#6711yescharacteristic
292C2×C3×D3—34no—
293C2×C3×D3—34no—
294C2×C3×D3—64no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C2×C3×D3—64no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—62no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C32×C4—24no—
314C32×C4—22no—
315C32×C4—22no—
316C3⋊Dic3—22no—
317C3⋊Dic3—22no—
318C3⋊Dic3—41no—
319C3⋊Dic3—41no—
320C3×Dic3—64no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—62no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C23×C4—91no—
328C2×C4⋊C4—91no—
329C2×C4⋊C4—271no—
330C2×C22⋊C4—271no—
331C2×C22⋊C4—271no—
332#95—91no—
333#95—271no—
334H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
335C23×C3—31no—
336C23×C3—31no—
337C23×C3—61no—
338C23×C3—92no—
339C23×C3—92no—
340C23×C3—92no—
341C23×C3—181no—
342C23×C3—181no—
343C23×C3—181no—
344C23×C3—181no—
345C23×C3C2×D3211yescharacteristic
346C22×D3—32no—
347C22×D3—32no—
348C22×D3—32no—
349C22×D3—32no—
350C22×D3—32no—
351C22×D3—32no—
352C22×D3—62no—
353C22×D3—62no—
354C22×D3—62no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—61no—
360C22×D3—61no—
361C22×D3—61no—
362C22×D3—61no—
363C22×D3—62no—
364C22×D3—62no—
365C22×D3—62no—
366C22×D3—62no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—32no—
370C2×C3×C4—32no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—92no—
379C2×C3×C4—92no—
380C2×C3×C4—92no—
381C2×C3×C4—188no—
382C2×C3×C4—181no—
383C2×C3×C4—181no—
384C2×C3×C4—181no—
385C2×C3×C4—181no—
386C2×C3×C4—181no—
387C2×C3×C4—181no—
388C2×C3×C4—181no—
389C2×C3×C4—181no—
390C2×C3×C4—181no—
391C2×C3×C4—181no—
392C2×C3×C4—181no—
393C2×C3×C4—181no—
394C2×C3×C4#62612yes—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—62no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—61no—
404C2×Dic3—62no—
405C2×Dic3—62no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—61no—
416C2×Dic3—92no—
417C2×Dic3—92no—
418C2×Dic3—181no—
419C2×Dic3—181no—
420C2×Dic3—181no—
421C2×Dic3—181no—
422C2×Dic3—181no—
423C2×Dic3—181no—
424C2×Dic3—181no—
425C2×Dic3—181no—
426C2×Dic3—181no—
427C2×Dic3—181no—
428C4×D3—68no—
429C4×D3—68no—
430C4×D3—616no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32C4×D1211yescharacteristic
439C2×C32#67⋊C411yescharacteristic
440C2×C32#88211yescharacteristic
441C2×C32#88311yescharacteristic
442C2×C32D3×C4⋊C411yescharacteristic
443C2×C32D3×C4⋊C411yescharacteristic
444C2×C32#89311yescharacteristic
445C2×C32#89411yescharacteristic
446C2×C32#89611yescharacteristic
447C2×C32#89611yescharacteristic
448C2×C32#89711yescharacteristic
449C2×C32C4×#6711yescharacteristic
450C2×C32#93711yescharacteristic
451C2×C32#94611yescharacteristic
452C3×D3—68no—
453C3×D3—64no—
454C3×D3—64no—
455C24—91no—
456C22×C4—91no—
457C22×C4—91no—
458C22×C4—91no—
459C22×C4—91no—
460C22×C4—91no—
461C22×C4—182no—
462C22×C4—182no—
463C22×C4—182no—
464C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
465C4⋊C4—182no—
466C4⋊C4—542no—
467C22⋊C4—274no—
468C22⋊C4—542no—
469C22×C3—31no—
470C22×C3—31no—
471C22×C3—31no—
472C22×C3—31no—
473C22×C3—31no—
474C22×C3—31no—
475C22×C3—31no—
476C22×C3—31no—
477C22×C3—31no—
478C22×C3—31no—
479C22×C3—31no—
480C22×C3—31no—
481C22×C3—31no—
482C22×C3—31no—
483C22×C3—61no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—61no—
487C22×C3—61no—
488C22×C3—61no—
489C22×C3—61no—
490C22×C3—94no—
491C22×C3—182no—
492C22×C3—182no—
493C22×C3—182no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3#379611yescharacteristic
498C22×C3#379711yescharacteristic
499C22×C3#379811yescharacteristic
500C22×C3C4×D3211yescharacteristic
501C22×C3C2×#62611yescharacteristic
502C22×C3#381111yescharacteristic
503C22×C3#381111yescharacteristic
504C2×D3—34no—
505C2×D3—34no—
506C2×D3—64no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—62no—
514C2×D3—62no—
515C2×D3—62no—
516C2×D3—62no—
517C2×D3—62no—
518C2×D3—62no—
519C2×D3—64no—
520C2×D3—64no—
521C2×D3—64no—
522C2×D3—64no—
523C2×D3—64no—
524C2×D3—64no—
525C3×C4—22no—
526C3×C4—62no—
527C3×C4—62no—
528C3×C4—64no—
529C3×C4—182no—
530C3×C4—182no—
531C3×C4—182no—
532C3×C4—361no—
533C3×C4—361no—
534Dic3—62no—
535Dic3—62no—
536Dic3—64no—
537Dic3—62no—
538Dic3—62no—
539Dic3—121no—
540Dic3—121no—
541Dic3—182no—
542Dic3—182no—
543Dic3—361no—
544Dic3—361no—
545C32—21no—
546C32#466811yescharacteristic, complemented
547C32#498911yescharacteristic, complemented
548C23—92no—
549C23—92no—
550C23—92no—
551C23—181no—
552C23—181no—
553C23—181no—
554C23—181no—
555C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—92no—
559C2×C4—92no—
560C2×C4—92no—
561C2×C4—188no—
562C2×C4—181no—
563C2×C4—181no—
564C2×C4—181no—
565C2×C4—181no—
566C2×C4—181no—
567C2×C4—181no—
568C2×C4—181no—
569C2×C4—181no—
570C2×C4—181no—
571C2×C4—181no—
572C2×C4—181no—
573C2×C4—181no—
574C2×C4#620012yes—
575C2×C3—31no—
576C2×C3—31no—
577C2×C3—31no—
578C2×C3—31no—
579C2×C3—31no—
580C2×C3—31no—
581C2×C3—31no—
582C2×C3—31no—
583C2×C3—31no—
584C2×C3—31no—
585C2×C3—31no—
586C2×C3—31no—
587C2×C3—31no—
588C2×C3—31no—
589C2×C3—61no—
590C2×C3—61no—
591C2×C3—61no—
592C2×C3—61no—
593C2×C3—61no—
594C2×C3—61no—
595C2×C3—61no—
596C2×C3—184no—
597C2×C3#6385311yescharacteristic
598C2×C3#6386511yescharacteristic
599C2×C3#6387111yescharacteristic
600C2×C3#6388111yescharacteristic
601C2×C3#6388311yescharacteristic
602C2×C3C4×#62611yescharacteristic
603C2×C3#6398111yescharacteristic
604D3—64no—
605D3—64no—
606D3—68no—
607C22—94no—
608C22—182no—
609C22—182no—
610C22—182no—
611C22—182no—
612C22—182no—
613C22—182no—
614C22#8864911yescharacteristic
615C22#8865011yescharacteristic
616C22#8865111yescharacteristic
617C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
618C22C2×#620011yescharacteristic
619C22#8867711yescharacteristic
620C22#8867711yescharacteristic
621C4—22no—
622C4—182no—
623C4—182no—
624C4—182no—
625C4—361no—
626C4—361no—
627C3—31no—
628C3—31no—
629C3—61no—
630C3#1059129011yescharacteristic
631C2—184no—
632C2#1172109411yescharacteristic
633C2#1172109611yescharacteristic
634C2#1172110411yescharacteristic
635C2#1172112411yescharacteristic
636C2#1172114711yescharacteristic
637C2C4×#620011yescharacteristic
638C2#1172132311yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#385
LCSG ▷ C22×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×#88448C2semidirect
6. #63658⋊C3C2semidirect
7. #11720334C2semidirect
22. #88451C4semidirect
27. C2×#3750D3semidirect
35. #63658D3semidirect
64. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
65. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
116. C2×#272C4⋊C4semidirect
188. C2×C3⋊Dic3C4×D3semidirect
189. C2×C3⋊Dic3C4×D3semidirect
334. H(3,𝔽3)#385semidirect
546. C32#4668semidirect
547. C32#4989semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#3750⋊C32
C2×#884512
C2×#884482
#63658⋊C32
#117203342
#105912783
#105912773