Finite Group Databasebeta

← #13140221

Subgroups — #13140221

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×#88542C211yesmaximal, characteristic
2C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#88451C211yesmaximal, characteristic
4C2×#88451C211yesmaximal, characteristic
5C2×#88448C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591287—31nomaximal
9#10591279—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—
18C4×(C3⋊Dic3)⋊C3—22no—
19#88451—22no—
20#88451C414yescomplemented
21#88448—22no—
22C2×C4×C3⋊Dic3D311yescharacteristic, complemented
23C2×C3×(C2×D3)⋊C4—31no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×(C2×D3)⋊C4—61no—
26C2×C3×C4×Dic3—31no—
27C2×#3724—31no—
28C2×Dic3⋊Dic3—31no—
29C2×(C2×D3)⋊Dic3—31no—
30C2×(C2×D3)⋊Dic3—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33#63658D311yescharacteristic, complemented
34C3×C3⋊#95—31no—
35#63579—31no—
36#63579—31no—
37C23×H(3,𝔽3)C2311yescharacteristic
38C22×#272—21no—
39C22×#272—21no—
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272—21no—
44C22×#272C2×C412yes—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)D412yes—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
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)⋊C3C2×C412yes—
61C2×(C3⋊Dic3)⋊C3C2×C412yes—
62C2×(C3⋊Dic3)⋊C3D412yes—
63#4944—91no—
64#4940—91no—
65#4670—91no—
66C23×C3×D3—31no—
67C23×C3×D3—31no—
68C23×C3×D3—61no—
69C22×C32×C4—21no—
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—61no—
83C4×C3⋊Dic3—22no—
84C3×(C2×D3)⋊C4—64no—
85C3×(C2×D3)⋊C4—62no—
86C3×(C2×D3)⋊C4—62no—
87C3×C4×Dic3—62no—
88#3724—62no—
89Dic3⋊Dic3—62no—
90(C2×D3)⋊Dic3—34no—
91(C2×D3)⋊Dic3—34no—
92(C2×D3)⋊Dic3—62no—
93(C2×D3)⋊Dic3—62no—
94C22×H(3,𝔽3)C22×C411yescharacteristic
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C22×H(3,𝔽3)#4111yescharacteristic
98C22×H(3,𝔽3)#4111yescharacteristic
99C22×H(3,𝔽3)#4111yescharacteristic
100C22×H(3,𝔽3)#4111yescharacteristic
101C2×#272—22no—
102C2×#272—22no—
103C2×#272—22no—
104C2×#272—41no—
105C2×#272—41no—
106C2×#272—41no—
107C2×#272—41no—
108C4×H(3,𝔽3)—22no—
109#1084—41no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3—22no—
112(C3⋊Dic3)⋊C3—22no—
113(C3⋊Dic3)⋊C3—41no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C42—91no—
116C2×#217—91no—
117C2×#217—91no—
118C2×(C2×D3)⋊C4—31no—
119C2×(C2×D3)⋊C4—31no—
120C2×(C2×D3)⋊C4—61no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×(C2×D3)⋊C4—91no—
125C2×C4⋊Dic3—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×C4×Dic3—31no—
129C2×C4×Dic3—91no—
130C3×#95—91no—
131C3⋊#95—31no—
132C3⋊#95—91no—
133C3⋊#95—91no—
134C3⋊#95—91no—
135C3⋊#95—91no—
136C23×C32—21no—
137C23×C32C22×D311yescharacteristic
138C23×C32C22×D311yescharacteristic
139C22×C3×D3—32no—
140C22×C3×D3—32no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
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—121no—
157C22×C3×D3—121no—
158C22×C3×D3—121no—
159C22×C3×D3—121no—
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⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3C4×D312yes—
179C2×C3⋊Dic3C4×D312yes—
180C2×C3⋊Dic3D1212yes—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
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—62no—
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—121no—
201C2×C3×Dic3—121no—
202#388—271noSylow
203C2×H(3,𝔽3)#11711yescharacteristic
204C2×H(3,𝔽3)C4×D411yescharacteristic
205C2×H(3,𝔽3)C4×D411yescharacteristic
206C2×H(3,𝔽3)#12111yescharacteristic
207C2×H(3,𝔽3)#12311yescharacteristic
208C2×H(3,𝔽3)#12411yescharacteristic
209C2×H(3,𝔽3)#12611yescharacteristic
210#272—42no—
211C24×C3—91no—
212C23×D3—31no—
213C23×D3—31no—
214C23×D3—61no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4—61no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4D3211yescharacteristic
224C22×Dic3—31no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—31no—
230C22×Dic3—61no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C22×Dic3—91no—
235C3×C22⋊C4—182no—
236C3×C42—182no—
237#217—94no—
238#217—182no—
239(C2×D3)⋊C4—62no—
240(C2×D3)⋊C4—62no—
241(C2×D3)⋊C4—64no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—182no—
245(C2×D3)⋊C4—182no—
246C4⋊Dic3—182no—
247Dic3⋊C4—182no—
248Dic3⋊C4—182no—
249C4×Dic3—62no—
250C4×Dic3—182no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32C2×C4×D311yescharacteristic
259C22×C32C2×C4×D311yescharacteristic
260C22×C32C2×D1211yescharacteristic
261C22×C32C2×D1211yescharacteristic
262C22×C32#23511yescharacteristic
263C22×C32#23511yescharacteristic
264C22×C32#23511yescharacteristic
265C22×C32#23511yescharacteristic
266C22×C32#23511yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32#23711yescharacteristic
269C22×C32#23711yescharacteristic
270C22×C32#23911yescharacteristic
271C22×C32C2×#6711yescharacteristic
272C2×C3×D3—64no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—122no—
280C2×C3×D3—122no—
281C2×C3×D3—122no—
282C2×C3×D3—122no—
283C2×C3×D3—122no—
284C2×C3×D3—122no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C32×C4—22no—
294C32×C4—22no—
295C32×C4—42no—
296C3⋊Dic3—22no—
297C3⋊Dic3—22no—
298C3⋊Dic3—22no—
299C3⋊Dic3—41no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—122no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C2×C4⋊C4—271no—
308C2×C22⋊C4—91no—
309C2×C22⋊C4—271no—
310C2×C22⋊C4—271no—
311C2×C42—91no—
312#95—91no—
313#95—271no—
314H(3,𝔽3)#38811yescharacteristic, complemented, Sylow
315C23×C3—31no—
316C23×C3—31no—
317C23×C3—61no—
318C23×C3—92no—
319C23×C3—181no—
320C23×C3—181no—
321C23×C3—181no—
322C23×C3—181no—
323C23×C3—181no—
324C23×C3—181no—
325C23×C3C2×D3211yescharacteristic
326C22×D3—32no—
327C22×D3—32no—
328C22×D3—62no—
329C22×D3—61no—
330C22×D3—61no—
331C22×D3—61no—
332C22×D3—61no—
333C22×D3—61no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—61no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—61no—
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—121no—
344C22×D3—121no—
345C22×D3—121no—
346C22×D3—121no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—32no—
350C2×C3×C4—32no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—92no—
359C2×C3×C4—121no—
360C2×C3×C4—121no—
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—181no—
373C2×C3×C4#62612yes—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
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—62no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—92no—
394C2×Dic3—92no—
395C2×Dic3—92no—
396C2×Dic3—121no—
397C2×Dic3—121no—
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#87211yescharacteristic
415C2×C32C4×D1211yescharacteristic
416C2×C32C4×D1211yescharacteristic
417C2×C32#87511yescharacteristic
418C2×C32#87611yescharacteristic
419C2×C32#67⋊C411yescharacteristic
420C2×C32#88511yescharacteristic
421C2×C32#88611yescharacteristic
422C2×C32#89211yescharacteristic
423C2×C32#89511yescharacteristic
424C2×C32#89811yescharacteristic
425C2×C32C4×#6711yescharacteristic
426C2×C32#92911yescharacteristic
427C2×C32#93011yescharacteristic
428C3×D3—124no—
429C3×D3—122no—
430C3×D3—122no—
431C24—91no—
432C22×C4—91no—
433C22×C4—91no—
434C22×C4—91no—
435C22×C4—91no—
436C22×C4—91no—
437C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
438C4⋊C4—542no—
439C22⋊C4—182no—
440C22⋊C4—274no—
441C22⋊C4—542no—
442C42—182no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—31no—
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—61no—
458C22×C3—61no—
459C22×C3—61no—
460C22×C3—61no—
461C22×C3—61no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—361no—
468C22×C3—361no—
469C22×C3—361no—
470C22×C3—361no—
471C22×C3#379811yescharacteristic
472C22×C3#379911yescharacteristic
473C22×C3C4×D3211yescharacteristic
474C22×C3D3×D1211yescharacteristic
475C22×C3#380411yescharacteristic
476C22×C3#380411yescharacteristic
477C22×C3C2×#62611yescharacteristic
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—64no—
485C2×D3—122no—
486C2×D3—122no—
487C2×D3—121no—
488C2×D3—121no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C2×D3—122no—
496C2×D3—122no—
497C2×D3—122no—
498C2×D3—122no—
499C3×C4—22no—
500C3×C4—62no—
501C3×C4—62no—
502C3×C4—122no—
503C3×C4—182no—
504C3×C4—182no—
505C3×C4—182no—
506C3×C4—361no—
507C3×C4—361no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—121no—
512Dic3—121no—
513Dic3—122no—
514Dic3—121no—
515Dic3—182no—
516Dic3—182no—
517Dic3—182no—
518Dic3—361no—
519C32—21no—
520C32#467011yescharacteristic, complemented
521C32#494011yescharacteristic, complemented
522C23—92no—
523C23—181no—
524C23—181no—
525C23—181no—
526C23—181no—
527C23—181no—
528C23—181no—
529C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—92no—
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—362no—
570C2×C3#6383611yescharacteristic
571C2×C3#6385211yescharacteristic
572C2×C3#6386811yescharacteristic
573C2×C3#6388011yescharacteristic
574C2×C3C4×#62611yescharacteristic
575C2×C3#6390911yescharacteristic
576C2×C3#6396611yescharacteristic
577D3—122no—
578D3—122no—
579D3—124no—
580C22—182no—
581C22—182no—
582C22—182no—
583C22—361no—
584C22—361no—
585C22—361no—
586C22—361no—
587C22#8865011yescharacteristic
588C22#8865211yescharacteristic
589C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
590C22#8865411yescharacteristic
591C22#8867111yescharacteristic
592C22#8867111yescharacteristic
593C22C2×#620011yescharacteristic
594C4—22no—
595C4—182no—
596C4—182no—
597C4—182no—
598C4—361no—
599C4—361no—
600C3—31no—
601C3—31no—
602C3—61no—
603C3#1059119411yescharacteristic
604C2—362no—
605C2#1172107611yescharacteristic
606C2#1172109211yescharacteristic
607C2#1172110311yescharacteristic
608C2#1172112311yescharacteristic
609C2C4×#620011yescharacteristic
610C2#1172115611yescharacteristic
611C2#1172130111yescharacteristic
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)#388
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×C4×(C3⋊Dic3)⋊C3C2semidirect
5. C2×#88448C2semidirect
6. #63658⋊C3C2semidirect
7. #11720334C2semidirect
20. #88451C4semidirect
22. C2×C4×C3⋊Dic3D3semidirect
33. #63658D3semidirect
314. H(3,𝔽3)#388semidirect
520. C32#4670semidirect
521. C32#4940semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×C4×(C3⋊Dic3)⋊C32
C2×#884512
C2×#884482
#63658⋊C32
#117203342
#105912873
#105912793