Finite Group Databasebeta

← #10800174

Subgroups — #10800174

Order 768.

1186 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5538C211yesmaximal, characteristic
2C2×Dic3×M16C211yesmaximal, characteristic, complemented
3C2×#4718C211yesmaximal, characteristic, complemented
4C3×#1870C211yesmaximal, characteristic, complemented
5#71996C211yesmaximal, characteristic, complemented
6#71611C211yesmaximal, characteristic, complemented
7#69953C211yesmaximal, characteristic, complemented
8#69916C212yesmaximal, complemented
9#69907C212yesmaximal, complemented
10#68619C212yesmaximal, complemented
11#68601C212yesmaximal, complemented
12#21378—31nomaximal, Sylow
13C22×C3×M16C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#939C411yescharacteristic
16C2×#935C411yescharacteristic
17C2×C4×#67C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22#5538—24no—
23#5538—24no—
24C2×#898C411yescharacteristic
25C2×#889C411yescharacteristic
26C2×#886C411yescharacteristic
27C2×#885C411yescharacteristic
28C2×#879C411yescharacteristic
29C2×#878C411yescharacteristic
30C2×#872C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×C22⋊C8C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×#830C2212yes—
35C2×#830C2211yescharacteristic
36Dic3×M16—24no—
37C2×(C2×D3)⋊C8C2212yes—
38C2×(C2×D3)⋊C8C2212yes—
39C2×#815C2212yes—
40C2×Dic3⋊C8C2212yes—
41C2×C8×Dic3C2212yes—
42C2×C8×Dic3C2212yes—
43#4718—24no—
44C3×#336C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4524C2212yes—
48C2×#529—31no—
49C2×#431—31no—
50C2×C4×M16—31no—
51#1984—32no—
52#1954—32no—
53#1870—31no—
54#1870—31no—
55#1870—31no—
56#1870D311yescharacteristic, complemented
57#1750—32no—
58#1750—32no—
59C22×C3×D4C2×C411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67C2×C411yescharacteristic
63C22×#67C2×C411yescharacteristic
64C23×Dic3C2×C411yescharacteristic
65C23×Dic3C2×C411yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C22×Dic6C2×C411yescharacteristic
68C2×C3×M16—22no—
69C2×C3×M16—24no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4C2×C411yescharacteristic
75C2×C3×C22⋊C4C2×C411yescharacteristic
76C2×C3×C22⋊C4C2×C411yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217C2×C411yescharacteristic
79C2×#217C2×C411yescharacteristic
80C2×#217C2×C411yescharacteristic
81C2×#217C2×C411yescharacteristic
82#939—24no—
83#935—24no—
84C4×#67—22no—
85C4×#67—22no—
86C4×#67—22no—
87C4×#67—22no—
88C2×(C2×D3)⋊C4C2×C411yescharacteristic
89C2×(C2×D3)⋊C4C2×C411yescharacteristic
90C2×(C2×D3)⋊C4C2×C411yescharacteristic
91C2×(C2×D3)⋊C4C2×C411yescharacteristic
92#926—22no—
93#926—22no—
94C2×C4⋊Dic3C2×C411yescharacteristic
95C2×C4⋊Dic3C2×C411yescharacteristic
96C2×C4⋊Dic3C2×C411yescharacteristic
97C2×Dic3⋊C4C2×C411yescharacteristic
98C2×Dic3⋊C4C2×C411yescharacteristic
99C2×Dic3⋊C4C2×C411yescharacteristic
100C2×Dic3⋊C4C2×C411yescharacteristic
101C2×Dic3⋊C4C2×C411yescharacteristic
102C2×Dic3⋊C4C2×C411yescharacteristic
103C2×C4×Dic3C2×C411yescharacteristic
104C2×C4×Dic3C2×C411yescharacteristic
105C2×C4×Dic3D414yes—
106C2×C4×Dic3D414yes—
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C2×#208—22no—
111C2×#208—24no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114#898—24no—
115#889—24no—
116#886—24no—
117#885—24no—
118#879—24no—
119#878—24no—
120#872—22no—
121#872—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×C22⋊C8—22no—
125C3×C22⋊C8—24no—
126#830—22no—
127#830—24no—
128(C2×D3)⋊C8—24no—
129(C2×D3)⋊C8—24no—
130#815—24no—
131Dic3⋊C8—24no—
132C8×Dic3—24no—
133C8×Dic3—24no—
134C22×M16—31no—
135C22×M16C2×D311yescharacteristic
136#529—64no—
137#529—64no—
138C2×#126—31no—
139C2×#126—31no—
140C2×#125—31no—
141C2×#124—31no—
142C2×#123—31no—
143C2×#123—31no—
144C2×C22⋊Q8—31no—
145C2×#121—31no—
146C2×C4×Q8—31no—
147C2×C4×D4—31no—
148C2×C4×D4—31no—
149C2×C4×D4C2×D311yescharacteristic
150C2×#117—31no—
151C2×#117—31no—
152C22×C42—31no—
153#431—64no—
154C2×C4⋊C8—32no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—32no—
157C2×C22⋊C8—32no—
158C2×C22⋊C8—32no—
159C2×C22⋊C8C2×D311yescharacteristic
160C2×C22⋊C8C2×D312yes—
161C4×M16—64no—
162C2×#97—32no—
163C2×C4×C8—32no—
164C2×C4×C8—32no—
165#336—32no—
166#336—32no—
167#336—32no—
168#336C2×D312yes—
169C24×C3C22×C411yescharacteristic
170C24×C3C22×C411yescharacteristic
171C23×D3C22×C411yescharacteristic
172C2×C3×D4—22no—
173C2×C3×D4—41no—
174C2×C3×D4—41no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4C22×C411yescharacteristic
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2411yescharacteristic
188C2×#67—22no—
189C2×#67—22no—
190C2×#67—24no—
191C2×#67—41no—
192C2×#67—41no—
193C22×Dic3—24no—
194C22×Dic3—41no—
195C22×Dic3—41no—
196C22×Dic3C22⋊C414yes—
197C22×Dic3C22⋊C414yes—
198C22×Dic3C22×C411yescharacteristic
199C22×Dic3C22×C411yescharacteristic
200C22×Dic3C22×C411yescharacteristic
201C22×Dic3C22×C411yescharacteristic
202C22×Dic3C22×C411yescharacteristic
203C22×Dic3C22×C411yescharacteristic
204C22×Dic3C22×C411yescharacteristic
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×Dic6—22no—
210C2×Dic6—24no—
211C3×M16—28no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8#4114yes—
218C3×C4⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C42—21no—
222C3×C42—21no—
223#217—22no—
224#217—22no—
225#217—22no—
226#217—22no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—22no—
230(C2×D3)⋊C4—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3—22no—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236Dic3⋊C4—22no—
237Dic3⋊C4—22no—
238Dic3⋊C4—22no—
239Dic3⋊C4—22no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3C22⋊C414yes—
243C4×Dic3C22⋊C414yes—
244C4×Dic3C2×D412yes—
245C4×Dic3C2×D412yes—
246C4×Dic3C2×D412yes—
247C4×Dic3C2×D412yes—
248#208—28no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8#4114yes—
255C22×Q8—31no—
256C22×D4—31no—
257C22×D4—31no—
258C22×D4C4×D311yescharacteristic
259C23×C4—31no—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4C22×D311yescharacteristic
263C23×C4C22×D311yescharacteristic
264C2×M16—22no—
265C2×M16—24no—
266C2×M16—64no—
267C2×M16—62no—
268C22×C8—32no—
269C22×C8—32no—
270C22×C8C22×D312yes—
271C22×C8C22×D312yes—
272#126—64no—
273#126—64no—
274#125—64no—
275#124—64no—
276#123—64no—
277#123—64no—
278C22⋊Q8—64no—
279#121—64no—
280C4×Q8—62no—
281C4×Q8—62no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—62no—
285C4×D4—62no—
286C4×D4—62no—
287C4×D4—62no—
288#117—62no—
289#117—62no—
290#117—62no—
291#117—62no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4C4×D311yescharacteristic
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4C4×D311yescharacteristic
311C2×C22⋊C4C4×D311yescharacteristic
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—34no—
315C2×C42—34no—
316C2×C42—31no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42C22×D311yescharacteristic
320C4⋊C8—64no—
321C22⋊C8—22no—
322C22⋊C8—24no—
323C22⋊C8—64no—
324C22⋊C8—64no—
325C22⋊C8—62no—
326C22⋊C8—64no—
327#97—64no—
328C4×C8—64no—
329C4×C8—64no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3C2×C22⋊C412yes—
340C23×C3C23×C411yescharacteristic
341C22×D3—21no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—21no—
345C22×D3—21no—
346C3×D4—44no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4C2×C22⋊C412yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4112yes—
365C2×C3×C4C2×#4112yes—
366#67—44no—
367#67—44no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—28no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—44no—
382C2×Dic3C2×C22⋊C412yes—
383C2×Dic3C2×C22⋊C412yes—
384C2×Dic3C2×C22⋊C412yes—
385C2×Dic3C2×C22⋊C412yes—
386C2×Dic3#13114yes—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—42no—
390Dic6—44no—
391C3×C8—24no—
392C3×C8—42no—
393C3⋊C8—24no—
394C3⋊C8—42no—
395C24—31no—
396C24C2×C4×D311yescharacteristic
397C24C2×C4×D311yescharacteristic
398C2×Q8—62no—
399C2×Q8—64no—
400C2×D4—22no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—62no—
404C2×D4—64no—
405C2×D4—62no—
406C2×D4—121no—
407C2×D4—121no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—31no—
415C22×C4—34no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—34no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—64no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—121no—
429C22×C4—121no—
430C22×C4C2×C4×D311yescharacteristic
431C22×C4C2×C4×D311yescharacteristic
432C22×C4C2×C4×D311yescharacteristic
433C22×C4C2×C4×D311yescharacteristic
434C22×C4D3×D412yes—
435C22×C4D3×D412yes—
436C22×C4C23×D311yescharacteristic
437M16—28no—
438M16—68no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—34no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8#23714yes—
451C4⋊C4—22no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—62no—
457C4⋊C4—62no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C42—21no—
472C42—21no—
473C42—34no—
474C42—34no—
475C42—32no—
476C42—32no—
477C42—32no—
478C42—32no—
479C42—61no—
480C42—61no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3C22×C22⋊C411yescharacteristic
492C22×C3C4×#4112yes—
493C22×C3C2×#13112yes—
494C22×C3#56811yescharacteristic
495C22×C3#56811yescharacteristic
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—41no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4#52511yescharacteristic
506C3×C4#52511yescharacteristic
507C3×C4#53212yes—
508Dic3—22no—
509Dic3—22no—
510Dic3—22no—
511Dic3—41no—
512Dic3—41no—
513Dic3—41no—
514Dic3#40814yes—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—62no—
528C23—62no—
529C23D3×C22⋊C412yes—
530C23C22×C4×D311yescharacteristic
531Q8—124no—
532D4—44no—
533D4—124no—
534D4—124no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—32no—
547C2×C4—34no—
548C2×C4—32no—
549C2×C4—32no—
550C2×C4—32no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—68no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—124no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—122no—
571C2×C4D3×C22⋊C412yes—
572C2×C4C2×D3×D411yescharacteristic
573C2×C4C2×D3×D411yescharacteristic
574C2×C4C2×#23712yes—
575C2×C4D3×#4112yes—
576C8—24no—
577C8—42no—
578C8—64no—
579C8—122no—
580C2×C3—42no—
581C2×C3—81no—
582C2×C3#229711yescharacteristic
583C2×C3C2×#40811yescharacteristic
584C2×C3#287911yescharacteristic
585C2×C3#296912yes—
586C2×C3#296912yes—
587D3—81no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—42no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—121no—
599C22—121no—
600C22—122no—
601C22—122no—
602C22—121no—
603C22C2×D3×C22⋊C411yescharacteristic
604C22C4×#23712yes—
605C22#574511yescharacteristic
606C22#574511yescharacteristic
607C22D3×#13112yes—
608C4—22no—
609C4—34no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—62no—
614C4—62no—
615C4—62no—
616C4—121no—
617C4—121no—
618C4—121no—
619C4#558611yescharacteristic
620C4#559612yes—
621C4#565611yescharacteristic
622C3#2137811yescharacteristic, complemented, Sylow
623C2—42no—
624C2—81no—
625C2—241no—
626C2#7465811yescharacteristic
627C2#7863011yescharacteristic
628C2D3×#40811yescharacteristic
629C2#7945912yes—
630C2#7952012yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1870C2
prime radicalC2×#939C4
cosocleC22×C3×C4C24
hypercenter#1870D3
hypocenterC3#21378
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×M16C2semidirect
3. C2×#4718C2semidirect
4. C3×#1870C2semidirect
5. #71996C2semidirect
6. #71611C2semidirect
7. #69953C2semidirect
8. #69916C2semidirect
9. #69907C2semidirect
10. #68619C2semidirect
11. #68601C2semidirect
56. #1870D3semidirect
622. C3#21378semidirect

Maximal subgroups

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

subgroupindex
C2×#55382
C2×Dic3×M162
C2×#47182
C3×#18702
#719962
#716112
#699532
#699162
#699072
#686192
#686012
#213783