Finite Group Databasebeta

← #13140254

Subgroups — #13140254

Order 1728.

951 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6199C211yesmaximal, characteristic
2C2×#88542C211yesmaximal, characteristic
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×#88451C211yesmaximal, characteristic
5C2×#88449C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591379—31nomaximal
9#10591377—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
17C2×#6199—22no—
18C2×#6199—22no—
19C2×#6199—22no—
20#88542C414yescomplemented
21C4⋊((C3⋊Dic3)⋊C3)—22no—
22#88451—22no—
23#88449—22no—
24C22×D3×Dic3—31no—
25C22×D3×Dic3—31no—
26C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×(C2×D3)⋊C4—61no—
30C2×C3×C4⋊Dic3—31no—
31C2×Dic3⋊Dic3—31no—
32C2×Dic3⋊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)C2×C412yescomplemented
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084C2×C412yescomplemented
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)⋊C3—21no—
63C2×(C3⋊Dic3)⋊C3D412yes—
64C2×(C3⋊Dic3)⋊C3D412yes—
65#6199—28no—
66#4989—91no—
67#4950—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×D3×Dic3—62no—
87C2×D3×Dic3—62no—
88C2×D3×Dic3—62no—
89C2×D3×Dic3—62no—
90C2×D3×Dic3—62no—
91C2×D3×Dic3—62no—
92C4⋊(C3⋊Dic3)—22no—
93C3×(C2×D3)⋊C4—34no—
94C3×(C2×D3)⋊C4—34no—
95C3×(C2×D3)⋊C4—64no—
96C3×C4⋊Dic3—62no—
97Dic3⋊Dic3—62no—
98Dic3⋊Dic3—62no—
99(C2×D3)⋊Dic3—62no—
100(C2×D3)⋊Dic3—62no—
101C22×H(3,𝔽3)C22×C411yescharacteristic
102C22×H(3,𝔽3)C2×D411yescharacteristic
103C22×H(3,𝔽3)C2×D411yescharacteristic
104C22×H(3,𝔽3)C2×D411yescharacteristic
105C22×H(3,𝔽3)C2×Q811yescharacteristic
106C22×H(3,𝔽3)#4111yescharacteristic
107C22×H(3,𝔽3)#4111yescharacteristic
108C2×#272—22no—
109C2×#272—22no—
110C2×#272—22no—
111C2×#272—22no—
112C2×#272—22no—
113C2×#272—22no—
114C2×#272C4⋊C414yescomplemented
115C4×H(3,𝔽3)—22no—
116#1084—22no—
117(C3⋊Dic3)⋊C3—22no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—41no—
120(C3⋊Dic3)⋊C3—41no—
121C23×Dic3—91no—
122C22×C4×D3—91no—
123C22×C4×D3—91no—
124C2×C3×C4⋊C4—91no—
125C2×C3×C22⋊C4—91no—
126C2×#217—91no—
127C2×(C2×D3)⋊C4—31no—
128C2×(C2×D3)⋊C4—31no—
129C2×(C2×D3)⋊C4—61no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×C4⋊Dic3—31no—
133C2×C4⋊Dic3—91no—
134C2×C4⋊Dic3—91no—
135C2×Dic3⋊C4—91no—
136C2×Dic3⋊C4—91no—
137C3×#95—91no—
138C3⋊#95—31no—
139C3⋊#95—91no—
140C3⋊#95—91no—
141C3⋊#95—91no—
142C3⋊#95—91no—
143C23×C32—21no—
144C23×C32C22×D311yescharacteristic
145C23×C32C22×D311yescharacteristic
146C22×C3×D3—32no—
147C22×C3×D3—32no—
148C22×C3×D3—32no—
149C22×C3×D3—32no—
150C22×C3×D3—32no—
151C22×C3×D3—32no—
152C22×C3×D3—62no—
153C22×C3×D3—62no—
154C22×C3×D3—62no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C22×C3×D3—121no—
165C22×C3×D3—121no—
166C22×C3×D3—121no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—22no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4C4×D312yescomplemented
175C2×C32×C4C4×D312yescomplemented
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⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3D1212yes—
187C2×C3⋊Dic3#6712yes—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—62no—
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—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—121no—
208C2×C3×Dic3—121no—
209D3×Dic3—68no—
210D3×Dic3—68no—
211#385—271noSylow
212C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
213C2×H(3,𝔽3)C4×D411yescharacteristic
214C2×H(3,𝔽3)C4×D411yescharacteristic
215C2×H(3,𝔽3)#12011yescharacteristic
216C2×H(3,𝔽3)C22⋊Q811yescharacteristic
217C2×H(3,𝔽3)C22⋊Q811yescharacteristic
218C2×H(3,𝔽3)#12311yescharacteristic
219#272—24no—
220C24×C3—91no—
221C23×D3—31no—
222C23×D3—31no—
223C23×D3—61no—
224C22×C3×C4—31no—
225C22×C3×C4—31no—
226C22×C3×C4—61no—
227C22×C3×C4—91no—
228C22×C3×C4—91no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C22×C3×C4D3211yescharacteristic
233C22×Dic3—31no—
234C22×Dic3—31no—
235C22×Dic3—31no—
236C22×Dic3—31no—
237C22×Dic3—31no—
238C22×Dic3—31no—
239C22×Dic3—61no—
240C22×Dic3—91no—
241C22×Dic3—91no—
242C22×Dic3—91no—
243C22×Dic3—91no—
244C22×Dic3—182no—
245C22×Dic3—182no—
246C22×Dic3—182no—
247C2×C4×D3—182no—
248C2×C4×D3—182no—
249C2×C4×D3—182no—
250C2×C4×D3—182no—
251C2×C4×D3—182no—
252C2×C4×D3—182no—
253C3×C4⋊C4—182no—
254C3×C22⋊C4—94no—
255#217—182no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4—34no—
258(C2×D3)⋊C4—64no—
259(C2×D3)⋊C4—182no—
260(C2×D3)⋊C4—182no—
261C4⋊Dic3—62no—
262C4⋊Dic3—182no—
263C4⋊Dic3—182no—
264Dic3⋊C4—182no—
265Dic3⋊C4—182no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32C2×C4×D311yescharacteristic
274C22×C32C2×C4×D311yescharacteristic
275C22×C32C2×D1211yescharacteristic
276C22×C32#23511yescharacteristic
277C22×C32#23511yescharacteristic
278C22×C32D3×D411yescharacteristic
279C22×C32D3×D411yescharacteristic
280C22×C32D3×D411yescharacteristic
281C22×C32D3×D411yescharacteristic
282C22×C32#23711yescharacteristic
283C22×C32D3×Q811yescharacteristic
284C22×C32D3×Q811yescharacteristic
285C22×C32#23911yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C2×C3×D3—34no—
288C2×C3×D3—34no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—122no—
303C2×C3×D3—122no—
304C2×C3×D3—122no—
305C2×C3×D3—122no—
306C2×C3×D3—122no—
307C2×C3×D3—122no—
308C32×C4—22no—
309C32×C4—22no—
310C32×C4—42no—
311C3⋊Dic3—22no—
312C3⋊Dic3—22no—
313C3⋊Dic3—41no—
314C3⋊Dic3—41no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—62no—
318C3×Dic3—62no—
319C3×Dic3—122no—
320C3×Dic3—121no—
321C3×Dic3—121no—
322C23×C4—271no—
323C2×C4⋊C4—91no—
324C2×C4⋊C4—271no—
325C2×C22⋊C4—91no—
326C2×C22⋊C4—271no—
327#95—91no—
328#95—271no—
329H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
330C23×C3—31no—
331C23×C3—31no—
332C23×C3—61no—
333C23×C3—92no—
334C23×C3—92no—
335C23×C3—92no—
336C23×C3—181no—
337C23×C3—181no—
338C23×C3—181no—
339C23×C3—181no—
340C23×C3C2×D3211yescharacteristic
341C22×D3—32no—
342C22×D3—32no—
343C22×D3—32no—
344C22×D3—32no—
345C22×D3—32no—
346C22×D3—32no—
347C22×D3—62no—
348C22×D3—62no—
349C22×D3—62no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—61no—
354C22×D3—61no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—61no—
358C22×D3—121no—
359C22×D3—121no—
360C22×D3—121no—
361C22×D3—121no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—32no—
365C2×C3×C4—32no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—92no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—121no—
375C2×C3×C4—121no—
376C2×C3×C4—181no—
377C2×C3×C4—181no—
378C2×C3×C4—181no—
379C2×C3×C4—181no—
380C2×C3×C4—181no—
381C2×C3×C4—181no—
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#62412yes—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—62no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—61no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—92no—
409C2×Dic3—92no—
410C2×Dic3—121no—
411C2×Dic3—121no—
412C2×Dic3—181no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—181no—
416C2×Dic3—181no—
417C2×Dic3—181no—
418C2×Dic3—181no—
419C2×Dic3—181no—
420C2×Dic3—181no—
421C2×Dic3—181no—
422C2×Dic3—188no—
423C4×D3—188no—
424C4×D3—188no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32C4×D1211yescharacteristic
433C2×C32#67⋊C411yescharacteristic
434C2×C32#88211yescharacteristic
435C2×C32#88311yescharacteristic
436C2×C32D3×C4⋊C411yescharacteristic
437C2×C32D3×C4⋊C411yescharacteristic
438C2×C32#89311yescharacteristic
439C2×C32#89411yescharacteristic
440C2×C32#89611yescharacteristic
441C2×C32#89611yescharacteristic
442C2×C32#89711yescharacteristic
443C2×C32C4×#6711yescharacteristic
444C2×C32#93711yescharacteristic
445C2×C32#94611yescharacteristic
446C3×D3—64no—
447C3×D3—64no—
448C3×D3—124no—
449C24—91no—
450C22×C4—91no—
451C22×C4—91no—
452C22×C4—91no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—542no—
456C22×C4—542no—
457C22×C4—542no—
458C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
459C4⋊C4—182no—
460C4⋊C4—542no—
461C22⋊C4—94no—
462C22⋊C4—542no—
463C22×C3—31no—
464C22×C3—31no—
465C22×C3—31no—
466C22×C3—31no—
467C22×C3—31no—
468C22×C3—31no—
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—61no—
478C22×C3—61no—
479C22×C3—61no—
480C22×C3—61no—
481C22×C3—61no—
482C22×C3—61no—
483C22×C3—61no—
484C22×C3—94no—
485C22×C3—182no—
486C22×C3—182no—
487C22×C3—182no—
488C22×C3—182no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3#379711yescharacteristic
492C22×C3#379911yescharacteristic
493C22×C3D3×D1211yescharacteristic
494C22×C3#380211yescharacteristic
495C22×C3#380411yescharacteristic
496C22×C3C2×#62411yescharacteristic
497C22×C3D3×#6711yescharacteristic
498C2×D3—34no—
499C2×D3—34no—
500C2×D3—64no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—122no—
514C2×D3—122no—
515C2×D3—122no—
516C2×D3—122no—
517C2×D3—122no—
518C2×D3—122no—
519C3×C4—22no—
520C3×C4—62no—
521C3×C4—62no—
522C3×C4—122no—
523C3×C4—182no—
524C3×C4—182no—
525C3×C4—182no—
526C3×C4—361no—
527C3×C4—361no—
528Dic3—62no—
529Dic3—62no—
530Dic3—62no—
531Dic3—62no—
532Dic3—122no—
533Dic3—121no—
534Dic3—121no—
535Dic3—182no—
536Dic3—182no—
537Dic3—361no—
538Dic3—361no—
539C32—21no—
540C32#466811yescharacteristic, complemented
541C32#498911yescharacteristic, complemented
542C23—92no—
543C23—92no—
544C23—92no—
545C23—181no—
546C23—181no—
547C23—181no—
548C23—181no—
549C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—92no—
553C2×C4—92no—
554C2×C4—92no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4—181no—
562C2×C4—181no—
563C2×C4—181no—
564C2×C4—181no—
565C2×C4—181no—
566C2×C4—181no—
567C2×C4—548no—
568C2×C4#619912yescomplemented
569C2×C3—31no—
570C2×C3—31no—
571C2×C3—31no—
572C2×C3—31no—
573C2×C3—31no—
574C2×C3—31no—
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—61no—
584C2×C3—61no—
585C2×C3—61no—
586C2×C3—61no—
587C2×C3—61no—
588C2×C3—61no—
589C2×C3—61no—
590C2×C3—184no—
591C2×C3#6385411yescharacteristic
592C2×C3#6388811yescharacteristic
593C2×C3#6389411yescharacteristic
594C2×C3#6389511yescharacteristic
595C2×C3#6390011yescharacteristic
596C2×C3#6392211yescharacteristic
597C2×C3#6395211yescharacteristic
598D3—64no—
599D3—64no—
600D3—124no—
601C22—94no—
602C22—182no—
603C22—182no—
604C22—182no—
605C22—182no—
606C22—182no—
607C22—182no—
608C22#8865111yescharacteristic
609C22#8865211yescharacteristic
610C22#8865411yescharacteristic
611C22#8865511yescharacteristic
612C22#8867111yescharacteristic
613C22C2×#619911yescharacteristic
614C22#8867611yescharacteristic
615C4—22no—
616C4—182no—
617C4—182no—
618C4—182no—
619C4—361no—
620C4—361no—
621C3—31no—
622C3—31no—
623C3—61no—
624C3#1059137811yescharacteristic
625C2—184no—
626C2#1172110211yescharacteristic
627C2#1172114811yescharacteristic
628C2#1172114911yescharacteristic
629C2#1172115011yescharacteristic
630C2#1172115211yescharacteristic
631C2#1172116211yescharacteristic
632C2#1172129311yescharacteristic
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
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88449C2semidirect
6. #63658⋊C3C2semidirect
7. #11720334C2semidirect
20. #88542C4semidirect
26. C2×C4⋊(C3⋊Dic3)D3semidirect
35. #63658D3semidirect
49. C2×C4×H(3,𝔽3)C2×C4semidirect
52. C2×#1084C2×C4semidirect
114. C2×#272C4⋊C4semidirect
174. C2×C32×C4C4×D3semidirect
175. C2×C32×C4C4×D3semidirect
329. H(3,𝔽3)#385semidirect
540. C32#4668semidirect
541. C32#4989semidirect
568. C2×C4#6199semidirect

Maximal subgroups

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

subgroupindex
C22×#61992
C2×#885422
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884512
C2×#884492
#63658⋊C32
#117203342
#105913793
#105913773