Finite Group Databasebeta

← #10804110

Subgroups — #10804110

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5828C211yesmaximal, characteristic
2C2×C3×#408C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4#71996C212yesmaximal, complemented
5#71975C212yesmaximal, complemented
6#69955C212yesmaximal, complemented
7#69953C212yesmaximal, complemented
8#69952C212yesmaximal, complemented
9#69911C212yesmaximal, complemented
10#21744—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic, complemented
14C23×C3×C8C2211yescharacteristic, complemented
15#5828—24no—
16#5828—24no—
17C2×#946C412yes—
18C2×#938C412yes—
19C2×#937C411yescharacteristic
20C2×#937C411yescharacteristic
21C2×#935C411yescharacteristic
22C2×#935C411yescharacteristic
23C2×C4×#67C2212yes—
24C2×C4×#67C2212yes—
25C2×#926C2211yescharacteristic
26C22×#208C2211yescharacteristic, complemented
27C23×C3⋊C8C2211yescharacteristic, complemented
28C3×#408—24no—
29C2×C3×C22⋊C8C2212yes—
30C2×C3×C22⋊C8C2212yescomplemented
31#5234—24no—
32C2×#830C2212yes—
33C2×#830C2212yescomplemented
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38#4550C2212yescomplemented
39#4550C2212yescomplemented
40#4550C2212yescomplemented
41#4550C2212yescomplemented
42C2×#525—31no—
43C2×#408—31no—
44C2×#408D311yescharacteristic, complemented
45#1870—32no—
46#1870—32no—
47#1869—32no—
48#1869—32no—
49#1818—32no—
50#1779—32no—
51C24×D3C2×C411yescharacteristic
52C2×C3×#41—41no—
53C2×C3×#41—41no—
54C2×C3×#41—41no—
55C2×C3×#41—41no—
56C2×C3×#41D414yes—
57C2×C3×#41D414yes—
58C22×C3×Q8C2×C411yescharacteristic
59C22×C3×D4C2×C411yescharacteristic
60C22×C3×D4C2×C411yescharacteristic
61C22×C3×D4C2×C411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×#67C2×C412yes—
66C22×#67C2×C412yes—
67C23×Dic3C2×C411yescharacteristic
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3—41no—
71C22×C4×D3—41no—
72C22×C4×D3—41no—
73C22×C4×D3—41no—
74C22×C4×D3C2×C412yes—
75C22×C4×D3C2312yes—
76C2×C3×M16—22no—
77C2×C3×M16—41no—
78C2×C3×M16—41no—
79C22×C3×C8—22no—
80C22×C3×C8—41no—
81C22×C3×C8—41no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2312yes—
84C2×#217C2×C411yescharacteristic
85C2×#217C2×C411yescharacteristic
86C2×#217C2×C412yes—
87C2×#217C2×C412yes—
88#946—28no—
89#938—28no—
90#937—24no—
91#937—24no—
92#935—24no—
93#935—24no—
94C4×#67—24no—
95C4×#67—24no—
96C4×#67—24no—
97C4×#67—24no—
98C2×(C2×D3)⋊C4C2×C412yes—
99C2×(C2×D3)⋊C4C2×C412yes—
100#926—22no—
101#926—22no—
102C2×C4⋊Dic3C2×C412yes—
103C2×Dic3⋊C4C2×C412yes—
104C2×Dic3⋊C4C2×C412yes—
105C2×C4×Dic3C2312yes—
106C2×#208—22no—
107C2×#208—41no—
108C2×#208—41no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8—41no—
111C22×C3⋊C8—41no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114C3×C22⋊C8—24no—
115C3×C22⋊C8—24no—
116#830—24no—
117#830—24no—
118(C2×D3)⋊C8—42no—
119(C2×D3)⋊C8—42no—
120(C2×D3)⋊C8—42no—
121(C2×D3)⋊C8—42no—
122C22×#41C2×D311yescharacteristic
123C24×C4—31no—
124C22×M16—31no—
125C22×M16C2×D311yescharacteristic, complemented
126C23×C8—31no—
127C23×C8C2×D311yescharacteristic, complemented
128#525—64no—
129#525—64no—
130C2×#123—31no—
131C2×#123—31no—
132C2×C22⋊Q8—32no—
133C2×#121—32no—
134C2×#120—31no—
135C2×#120—31no—
136C2×C4×D4—32no—
137C2×C4×D4—32no—
138C2×#117—31no—
139#408—24no—
140#408—64no—
141C2×C22⋊C8—32no—
142C2×C22⋊C8—32no—
143C2×C22⋊C8—32no—
144C2×C22⋊C8—32no—
145C2×C22⋊C8—32no—
146C2×C22⋊C8—32no—
147C2×C22⋊C8C2×D312yes—
148C2×C22⋊C8C2×D312yescomplemented
149#336—32no—
150#336—32no—
151#336—32no—
152#336—32no—
153C24×C3C22×C411yescharacteristic
154C24×C3C22×C411yescharacteristic
155C24×C3C22×C411yescharacteristic
156C23×D3—22no—
157C23×D3—24no—
158C23×D3—41no—
159C23×D3—41no—
160C23×D3—42no—
161C23×D3C22×C412yes—
162C3×#41—44no—
163C3×#41—44no—
164C3×#41—44no—
165C3×#41—44no—
166C2×C3×Q8—41no—
167C2×C3×Q8—41no—
168C2×C3×Q8C22⋊C414yes—
169C2×C3×D4—41no—
170C2×C3×D4—41no—
171C2×C3×D4—41no—
172C2×C3×D4—41no—
173C2×C3×D4—41no—
174C2×C3×D4—41no—
175C2×C3×D4C22⋊C414yes—
176C2×C3×D4C22⋊C414yes—
177C2×C3×D4C22⋊C414yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C22×C411yescharacteristic
191C22×C3×C4C22×C411yescharacteristic
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—24no—
202C2×#67—24no—
203C2×#67—24no—
204C2×#67—42no—
205C2×#67—42no—
206C22×Dic3—22no—
207C22×Dic3—41no—
208C22×Dic3—41no—
209C22×Dic3C22×C412yes—
210C22×Dic3C22×C412yes—
211C22×Dic3C22×C412yes—
212C2×C4×D3—24no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×C4×D3—44no—
224C2×C4×D3—44no—
225C3×M16—44no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—44no—
233#217—22no—
234#217—22no—
235#217—24no—
236#217—24no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239C4⋊Dic3—24no—
240Dic3⋊C4—24no—
241Dic3⋊C4—24no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244#208—44no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—44no—
252C25—31no—
253C2×#41—41no—
254C2×#41—41no—
255C2×#41—41no—
256C2×#41—41no—
257C2×#41D1214yes—
258C2×#41#6714yes—
259C22×Q8C4×D311yescharacteristic
260C22×D4—32no—
261C22×D4—32no—
262C22×D4C4×D311yescharacteristic
263C22×D4C4×D311yescharacteristic
264C22×D4C4×D311yescharacteristic
265C23×C4—31no—
266C23×C4—32no—
267C23×C4—32no—
268C23×C4—62no—
269C23×C4—62no—
270C23×C4—121no—
271C23×C4—121no—
272C23×C4—121no—
273C23×C4—121no—
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D311yescharacteristic
276C23×C4C22×D311yescharacteristic
277C2×M16—22no—
278C2×M16—41no—
279C2×M16—41no—
280C2×M16—62no—
281C2×M16—121no—
282C2×M16—121no—
283C22×C8—22no—
284C22×C8—32no—
285C22×C8—32no—
286C22×C8—41no—
287C22×C8—41no—
288C22×C8—62no—
289C22×C8—121no—
290C22×C8—121no—
291C22×C8C22×D312yes—
292C22×C8C22×D312yes—
293#123—64no—
294#123—64no—
295C22⋊Q8—68no—
296#121—68no—
297#120—64no—
298#120—64no—
299C4×D4—64no—
300C4×D4—64no—
301C4×D4—64no—
302C4×D4—64no—
303#117—62no—
304#117—62no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—32no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—32no—
311C2×C22⋊C4—32no—
312C2×C22⋊C4—32no—
313C2×C22⋊C4—32no—
314C2×C42—32no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—64no—
318C22⋊C8—64no—
319C22⋊C8—122no—
320C22⋊C8—122no—
321C22⋊C8—122no—
322C22⋊C8—122no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3C2×C22⋊C412yes—
336C23×C3C2×C22⋊C412yes—
337C23×C3C2×C22⋊C412yes—
338C23×C3C23×C411yescharacteristic
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—44no—
346C22×D3—48no—
347C22×D3—48no—
348C22×D3—44no—
349C22×D3—44no—
350C22×D3#13114yes—
351C3×Q8—44no—
352C3×D4—44no—
353C3×D4—44no—
354C3×D4—44no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4C2×C22⋊C412yes—
374C2×C3×C4C2×C22⋊C412yes—
375C2×C3×C4C2×C22⋊C412yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382#67—48no—
383#67—48no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—44no—
394C4×D3—48no—
395C4×D3—42no—
396C4×D3—42no—
397C4×D3—42no—
398C4×D3—42no—
399C3×C8—42no—
400C3×C8—42no—
401C3⋊C8—42no—
402C3⋊C8—42no—
403C24—32no—
404C24—64no—
405C24—62no—
406C24—121no—
407C24—121no—
408C24—122no—
409C24C2×C4×D311yescharacteristic
410C24C2×C4×D311yescharacteristic
411C24C2×C4×D311yescharacteristic
412#41—44no—
413#41—44no—
414#41—44no—
415#41—44no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8(C2×D3)⋊C414yes—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—64no—
426C2×D4—64no—
427C2×D4—64no—
428C2×D4—64no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4(C2×D3)⋊C414yes—
432C2×D4(C2×D3)⋊C414yes—
433C2×D4(C2×D3)⋊C414yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—62no—
450C22×C4—64no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4—124no—
460C22×C4—124no—
461C22×C4—124no—
462C22×C4—124no—
463C22×C4—121no—
464C22×C4—121no—
465C22×C4C2×C4×D311yescharacteristic
466C22×C4C2×C4×D311yescharacteristic
467C22×C4C2×C4×D311yescharacteristic
468C22×C4C2×D1212yes—
469C22×C4C2×D1212yes—
470C22×C4C2×D1212yes—
471C22×C4C2×#6712yes—
472C22×C4C2×#6712yes—
473C22×C4C2×#6712yes—
474C22×C4C23×D311yescharacteristic
475M16—44no—
476M16—124no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—44no—
484C2×C8—62no—
485C2×C8—62no—
486C2×C8—62no—
487C2×C8—62no—
488C2×C8—62no—
489C2×C8—62no—
490C2×C8—124no—
491C4⋊C4—64no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C22⋊C4—62no—
495C22⋊C4—62no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—64no—
499C22⋊C4—64no—
500C42—62no—
501C42—62no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—24no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3C22×C22⋊C411yescharacteristic
518C22×C3#51811yescharacteristic
519C22×C3#51811yescharacteristic
520C22×C3C2×#13112yes—
521C22×C3#56812yes—
522C2×D3—24no—
523C2×D3—24no—
524C2×D3—24no—
525C2×D3—416no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4#53411yescharacteristic
534C3×C4#53411yescharacteristic
535C3×C4#53411yescharacteristic
536C3×C4#53411yescharacteristic
537Dic3—42no—
538Dic3—42no—
539Dic3—42no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—34no—
553C23—62no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—62no—
559C23—128no—
560C23—124no—
561C23—124no—
562C23—128no—
563C23—124no—
564C23C2×(C2×D3)⋊C412yes—
565C23C2×(C2×D3)⋊C412yes—
566C23C2×(C2×D3)⋊C412yes—
567C23C22×C4×D311yescharacteristic
568Q8—44no—
569D4—44no—
570D4—44no—
571D4—44no—
572D4—128no—
573D4—128no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—128no—
602C2×C4—124no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4C2×(C2×D3)⋊C412yes—
608C2×C4C2×(C2×D3)⋊C412yes—
609C2×C4C2×(C2×D3)⋊C412yes—
610C2×C4C22×D1211yescharacteristic
611C2×C4#100411yescharacteristic
612C2×C4#100411yescharacteristic
613C2×C4#100411yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4C22×#6711yescharacteristic
616C8—42no—
617C8—42no—
618C8—122no—
619C8—122no—
620C2×C3—42no—
621C2×C3—42no—
622C2×C3—42no—
623C2×C3#230811yescharacteristic
624C2×C3#288012yes—
625C2×C3#297612yes—
626C2×C3#297711yescharacteristic
627C2×C3#297811yescharacteristic
628D3—44no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—64no—
645C22—64no—
646C22—64no—
647C22—1216no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22#548611yescharacteristic
653C22#548611yescharacteristic
654C22D3×#13112yes—
655C22#575012yes—
656C22C22×(C2×D3)⋊C411yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—122no—
661C4—122no—
662C4—122no—
663C4#549311yescharacteristic
664C4#549311yescharacteristic
665C4#580411yescharacteristic
666C4#580411yescharacteristic
667C3#2174411yescharacteristic, complemented, Sylow
668C2—42no—
669C2—42no—
670C2—42no—
671C2—124no—
672C2#7866412yes—
673C2#7866511yescharacteristic
674C2#7866611yescharacteristic
675C2#8212111yescharacteristic
676C2#8379312yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#408C2
prime radicalC2×#5828C2
cosocleC22×C3×C4C24
hypercenterC2×#408D3
hypocenterC3#21744
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C2×C3×#408C2semidirect
3. C2×#5234C2semidirect
4. #71996C2semidirect
5. #71975C2semidirect
6. #69955C2semidirect
7. #69953C2semidirect
8. #69952C2semidirect
9. #69911C2semidirect
13. C22×C3×M16C22semidirect
14. C23×C3×C8C22semidirect
26. C22×#208C22semidirect
27. C23×C3⋊C8C22semidirect
30. C2×C3×C22⋊C8C22semidirect
33. C2×#830C22semidirect
38. #4550C22semidirect
39. #4550C22semidirect
40. #4550C22semidirect
41. #4550C22semidirect
44. C2×#408D3semidirect
125. C22×M16C2×D3semidirect
127. C23×C8C2×D3semidirect
148. C2×C22⋊C8C2×D3semidirect
667. C3#21744semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
C2×C3×#4082
C2×#52342
#719962
#719752
#699552
#699532
#699522
#699112
#217443