Finite Group Databasebeta

← #13140147

Subgroups — #13140147

Order 1728.

911 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#88540C211yesmaximal, characteristic, complemented
3C2×#88450C211yesmaximal, characteristic
4C2×#88450C211yesmaximal, characteristic
5C2×#88448C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591290—31nomaximal
9#10591277—31nomaximal
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3—22no—
18C2×C4×(C3⋊D3)⋊C3—22no—
19C2×C4×(C3⋊D3)⋊C3—22no—
20#88540—22no—
21#88450—22no—
22#88450C414yescomplemented
23#88448—22no—
24C22×C4×C3⋊D3D311yescharacteristic
25C22×C3×C4×D3—31no—
26C2×C3×Dic3⋊C4—31no—
27C2×C3×Dic3⋊C4—31no—
28C2×C3×Dic3⋊C4—61no—
29C2×#3724—31no—
30C2×Dic3⋊Dic3—31no—
31C2×#3722—31no—
32C2×#3722—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×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—21no—
44C22×(C3⋊D3)⋊C3C2×C412yes—
45C22×(C3⋊D3)⋊C3D412yes—
46C22×(C3⋊D3)⋊C3Q812yes—
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×#1084—21no—
53C2×#1084—21no—
54C2×#1084—21no—
55C2×#1084C2×C412yescomplemented
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)⋊C3C2×C412yescomplemented
64C2×(C3⋊Dic3)⋊C3D412yes—
65C4×(C3⋊D3)⋊C3—28no—
66#5210—91no—
67#4989—91no—
68#4668—91no—
69C23×C3⋊D3C2×D311yescharacteristic
70C23×C3×D3—31no—
71C22×C32×C4—21no—
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C2×C4×C3⋊D3—22no—
78C2×C4×C3⋊D3—22no—
79C2×C4×C3⋊D3—22no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—61no—
88C22×C3×Dic3—61no—
89C2×C3×C4×D3—62no—
90C2×C3×C4×D3—62no—
91C2×C3×C4×D3—62no—
92C3×Dic3⋊C4—64no—
93C3×Dic3⋊C4—62no—
94C3×Dic3⋊C4—62no—
95#3724—62no—
96Dic3⋊Dic3—62no—
97#3722—34no—
98#3722—62no—
99(C2×D3)⋊Dic3—34no—
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×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—22no—
110C2×(C3⋊D3)⋊C3—22no—
111C2×(C3⋊D3)⋊C3—22no—
112C2×(C3⋊D3)⋊C3—22no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3C4⋊C414yescomplemented
115C4×H(3,𝔽3)—22no—
116#1084—22no—
117#1084—41no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—41no—
121C23×C3×C4—91no—
122C22×C4×D3—31no—
123C22×C4×D3—91no—
124C2×C3×C4⋊C4—91no—
125C2×#217—91no—
126C2×#217—91no—
127C2×(C2×D3)⋊C4—91no—
128C2×(C2×D3)⋊C4—91no—
129C2×(C2×D3)⋊C4—91no—
130C2×(C2×D3)⋊C4—91no—
131C2×C4⋊Dic3—91no—
132C2×Dic3⋊C4—31no—
133C2×Dic3⋊C4—31no—
134C2×Dic3⋊C4—61no—
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—21no—
147C22×C3⋊D3—21no—
148C22×C3⋊D3—21no—
149C22×C3⋊D3—21no—
150C22×C3⋊D3Dic612yes—
151C22×C3⋊D3C4×D312yes—
152C22×C3⋊D3#6712yes—
153C22×C3×D3—32no—
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—41no—
166C2×C32×C4—41no—
167C2×C32×C4D1212yes—
168C2×C32×C4#6712yes—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3C4×D312yescomplemented
177C2×C3⋊Dic3#6712yes—
178C4×C3⋊D3—28no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—62no—
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—62no—
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—121no—
205C2×C3×Dic3—121no—
206C3×C4×D3—68no—
207#385—271noSylow
208C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
209C2×H(3,𝔽3)C4×D411yescharacteristic
210C2×H(3,𝔽3)C4×D411yescharacteristic
211C2×H(3,𝔽3)#12011yescharacteristic
212C2×H(3,𝔽3)C22⋊Q811yescharacteristic
213C2×H(3,𝔽3)C22⋊Q811yescharacteristic
214C2×H(3,𝔽3)#12311yescharacteristic
215(C3⋊D3)⋊C3—24no—
216C24×C3—91no—
217C23×D3—31no—
218C23×D3—91no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—61no—
222C22×C3×C4—91no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—182no—
228C22×C3×C4—182no—
229C22×C3×C4—182no—
230C22×C3×C4D3211yescharacteristic
231C22×Dic3—31no—
232C22×Dic3—31no—
233C22×Dic3—31no—
234C22×Dic3—31no—
235C22×Dic3—31no—
236C22×Dic3—31no—
237C22×Dic3—31no—
238C22×Dic3—61no—
239C22×Dic3—61no—
240C22×Dic3—91no—
241C22×Dic3—91no—
242C22×Dic3—91no—
243C2×C4×D3—62no—
244C2×C4×D3—62no—
245C2×C4×D3—62no—
246C2×C4×D3—182no—
247C2×C4×D3—182no—
248C2×C4×D3—182no—
249C3×C4⋊C4—182no—
250#217—94no—
251#217—182no—
252(C2×D3)⋊C4—94no—
253(C2×D3)⋊C4—94no—
254(C2×D3)⋊C4—182no—
255(C2×D3)⋊C4—182no—
256C4⋊Dic3—182no—
257Dic3⋊C4—62no—
258Dic3⋊C4—62no—
259Dic3⋊C4—64no—
260Dic3⋊C4—182no—
261Dic3⋊C4—182no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32C2×Dic611yescharacteristic
270C22×C32C2×C4×D311yescharacteristic
271C22×C32C2×C4×D311yescharacteristic
272C22×C32C2×D1211yescharacteristic
273C22×C32#23511yescharacteristic
274C22×C32#23511yescharacteristic
275C22×C32#23511yescharacteristic
276C22×C32D3×D411yescharacteristic
277C22×C32D3×D411yescharacteristic
278C22×C32#23711yescharacteristic
279C22×C32D3×Q811yescharacteristic
280C22×C32C2×#6711yescharacteristic
281C22×C32C2×#6711yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C2×C3⋊D3—22no—
284C2×C3⋊D3—22no—
285C2×C3⋊D3—22no—
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3⋊D3Dic3⋊C414yescomplemented
290C2×C3×D3—34no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4—42no—
300C3⋊Dic3—22no—
301C3⋊Dic3—22no—
302C3⋊Dic3—41no—
303C3×Dic3—64no—
304C3×Dic3—62no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—122no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C23×C4—91no—
313C2×C4⋊C4—91no—
314C2×C4⋊C4—271no—
315C2×C22⋊C4—271no—
316C2×C22⋊C4—271no—
317#95—91no—
318#95—271no—
319H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
320C23×C3—31no—
321C23×C3—31no—
322C23×C3—61no—
323C23×C3—92no—
324C23×C3—92no—
325C23×C3—92no—
326C23×C3—181no—
327C23×C3—181no—
328C23×C3—181no—
329C23×C3—181no—
330C23×C3C2×D3211yescharacteristic
331C22×D3—32no—
332C22×D3—32no—
333C22×D3—32no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—61no—
338C22×D3—92no—
339C22×D3—92no—
340C22×D3—92no—
341C22×D3—181no—
342C22×D3—181no—
343C22×D3—181no—
344C22×D3—181no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—62no—
354C2×C3×C4—92no—
355C2×C3×C4—92no—
356C2×C3×C4—92no—
357C2×C3×C4—121no—
358C2×C3×C4—121no—
359C2×C3×C4—188no—
360C2×C3×C4—181no—
361C2×C3×C4—181no—
362C2×C3×C4—181no—
363C2×C3×C4—181no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4—181no—
371C2×C3×C4—181no—
372C2×C3×C4#62612yes—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—62no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—61no—
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—62no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400C2×Dic3—121no—
401C2×Dic3—121no—
402C2×Dic3—181no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—181no—
406C2×Dic3—181no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C4×D3—68no—
410C4×D3—188no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32C4×D1211yescharacteristic
419C2×C32#67⋊C411yescharacteristic
420C2×C32#88211yescharacteristic
421C2×C32#88311yescharacteristic
422C2×C32D3×C4⋊C411yescharacteristic
423C2×C32#89611yescharacteristic
424C2×C32#89711yescharacteristic
425C2×C32C2×Dic3⋊C411yescharacteristic
426C2×C32#92411yescharacteristic
427C2×C32#92411yescharacteristic
428C2×C32C4×#6711yescharacteristic
429C2×C32C4×#6711yescharacteristic
430C2×C32#92911yescharacteristic
431C2×C32#95311yescharacteristic
432C3⋊D3—24no—
433C3×D3—64no—
434C24—91no—
435C22×C4—91no—
436C22×C4—91no—
437C22×C4—91no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—182no—
441C22×C4—182no—
442C22×C4—182no—
443C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C4⋊C4—182no—
445C4⋊C4—542no—
446C22⋊C4—274no—
447C22⋊C4—542no—
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—31no—
461C22×C3—31no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—94no—
470C22×C3—182no—
471C22×C3—182no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3D3×Dic611yescharacteristic
477C22×C3#379511yescharacteristic
478C22×C3#379811yescharacteristic
479C22×C3C4×D3211yescharacteristic
480C22×C3C2×#62611yescharacteristic
481C22×C3D3×#6711yescharacteristic
482C22×C3D3×#6711yescharacteristic
483C2×D3—34no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—94no—
491C2×D3—182no—
492C2×D3—182no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—182no—
496C2×D3—182no—
497C3×C4—22no—
498C3×C4—62no—
499C3×C4—62no—
500C3×C4—122no—
501C3×C4—182no—
502C3×C4—182no—
503C3×C4—182no—
504C3×C4—361no—
505C3×C4—361no—
506Dic3—62no—
507Dic3—62no—
508Dic3—64no—
509Dic3—62no—
510Dic3—62no—
511Dic3—121no—
512Dic3—121no—
513Dic3—122no—
514Dic3—121no—
515Dic3—182no—
516Dic3—182no—
517Dic3—361no—
518C32—21no—
519C32#466811yescharacteristic, complemented
520C32#521011yescharacteristic, complemented
521C23—92no—
522C23—92no—
523C23—92no—
524C23—181no—
525C23—181no—
526C23—181no—
527C23—181no—
528C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—188no—
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#620012yes—
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—61no—
563C2×C3—61no—
564C2×C3—61no—
565C2×C3—61no—
566C2×C3—61no—
567C2×C3—61no—
568C2×C3—61no—
569C2×C3—184no—
570C2×C3#6385111yescharacteristic
571C2×C3#6385711yescharacteristic
572C2×C3D3×Dic3⋊C411yescharacteristic
573C2×C3#6387911yescharacteristic
574C2×C3C4×#62611yescharacteristic
575C2×C3#6390511yescharacteristic
576C2×C3#6397711yescharacteristic
577D3—64no—
578D3—184no—
579C22—94no—
580C22—182no—
581C22—182no—
582C22—182no—
583C22—182no—
584C22—182no—
585C22—182no—
586C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
587C22#8864811yescharacteristic
588C22#8865011yescharacteristic
589C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
590C22C2×#620011yescharacteristic
591C22#8867611yescharacteristic
592C22#8867611yescharacteristic
593C4—22no—
594C4—182no—
595C4—182no—
596C4—182no—
597C4—361no—
598C4—361no—
599C3—31no—
600C3—31no—
601C3—61no—
602C3#1059127811yescharacteristic
603C2—184no—
604C2#1172110111yescharacteristic
605C2#1172110711yescharacteristic
606C2#1172111911yescharacteristic
607C2#1172112211yescharacteristic
608C2#1172113911yescharacteristic
609C2C4×#620011yescharacteristic
610C2#1172132111yescharacteristic
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×(C3⋊D3)⋊C3C22
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×#88540C2semidirect
5. C2×#88448C2semidirect
6. #63658⋊C3C2semidirect
7. #11720334C2semidirect
22. #88450C4semidirect
35. #63658D3semidirect
55. C2×#1084C2×C4semidirect
63. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
114. C2×(C3⋊D3)⋊C3C4⋊C4semidirect
176. C2×C3⋊Dic3C4×D3semidirect
289. C2×C3⋊D3Dic3⋊C4semidirect
319. H(3,𝔽3)#385semidirect
519. C32#4668semidirect
520. C32#5210semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×#885402
C2×#884502
C2×#884482
#63658⋊C32
#117203342
#105912903
#105912773