Finite Group Databasebeta

← #10728697

Subgroups — #10728697

Order 768.

1742 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic
2C2×#4979C211yesmaximal, characteristic
3C2×#4973C211yesmaximal, characteristic
4C2×#4656C211yesmaximal, characteristic
5C3×#1441C211yesmaximal, characteristic
6#71566C211yesmaximal, characteristic
7#69424C212yesmaximal
8#69420C212yesmaximal
9#69407C212yesmaximal
10#69405C212yesmaximal
11#69402C211yesmaximal, characteristic
12#14881—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C4⋊Dic3C2212yes—
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C2×#887—24no—
25C2×#887—24no—
26C2×#887—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4979—24no—
40#4973—24no—
41#4656—24no—
42C22×C4×Q8—31no—
43C2×#395—31no—
44C2×#395—31no—
45C2×#391—31no—
46#1446—32no—
47#1444—32no—
48#1441—31no—
49#1441D311yescharacteristic
50#1436—32no—
51#1436—32no—
52#1434—31no—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—24no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—42no—
75C22×Dic6Q814yescomplemented
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—42no—
80C2×C3×C4⋊C4D414yes—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—42no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—42no—
92C2×Dic3⋊C4D414yes—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3D414yes—
102#887—232no—
103C3×#95—22no—
104C3×#95—24no—
105C3×#95—42no—
106C3⋊#95—22no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—42no—
114C3⋊#95Q818yescomplemented
115C23×Q8—31no—
116C23×Q8—31no—
117C2×C4×Q8—64no—
118C2×C4×Q8—64no—
119C2×C4×Q8—64no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4C2×D311yescharacteristic
125C22×C4⋊C4C2×D311yescharacteristic
126C22×C42—31no—
127C22×C42—31no—
128C22×C42—31no—
129#395—64no—
130#395—64no—
131#391—64no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95C2×D311yescharacteristic
142C2×#95C2×D312yes—
143C2×#95C2×D312yes—
144C24×C3C2411yescharacteristic
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×Q814yes—
174C22×C3×C4C2×Q812yes—
175C22×C3×C4#4112yes—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×Q814yes—
208C22×Dic3C2×Q812yes—
209C22×Dic3C2×Q812yes—
210C22×Dic3C2×Q814yes—
211C22×Dic3#4114yes—
212C22×Dic3#4112yes—
213C22×Dic3#4112yes—
214C2×Dic6—28no—
215C2×Dic6—28no—
216C2×Dic6—28no—
217C2×Dic6—28no—
218C2×Dic6—28no—
219C2×Dic6—28no—
220C2×Dic6—28no—
221C2×Dic6—28no—
222C2×Dic6—48no—
223C2×Dic6—48no—
224C2×Dic6—44no—
225C2×Dic6—44no—
226C3×C4⋊C4—28no—
227C3×C4⋊C4—44no—
228C4⋊Dic3—48no—
229Dic3⋊C4—28no—
230Dic3⋊C4—28no—
231Dic3⋊C4—48no—
232C4×Dic3—28no—
233C4×Dic3—28no—
234C4×Dic3—28no—
235C22×Q8—34no—
236C22×Q8—62no—
237C22×Q8—62no—
238C22×Q8—62no—
239C22×Q8—62no—
240C22×Q8—64no—
241C22×Q8—62no—
242C22×Q8—62no—
243C22×Q8—62no—
244C22×Q8—62no—
245C22×Q8—62no—
246C22×Q8—122no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4—32no—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D312yes—
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D311yescharacteristic
258C4×Q8—632no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—122no—
274C2×C4⋊C4—122no—
275C2×C4⋊C4D1214yes—
276C2×C42—34no—
277C2×C42—62no—
278C2×C42—62no—
279C2×C42—62no—
280C2×C42—62no—
281C2×C42—62no—
282C2×C42—62no—
283C2×C42—62no—
284C2×C42—62no—
285#95—22no—
286#95—24no—
287#95—38no—
288#95—42no—
289#95—64no—
290#95—64no—
291#95—64no—
292#95—64no—
293#95—64no—
294#95—62no—
295#95—64no—
296#95—122no—
297C23×C3C22×D411yescharacteristic
298C23×C3C22×D411yescharacteristic
299C23×C3C22×D411yescharacteristic
300C23×C3C22×Q811yescharacteristic
301C23×C3C22×Q812yes—
302C23×C3C2×#4111yescharacteristic
303C23×C3C2×#4111yescharacteristic
304C23×C3C2×#4111yescharacteristic
305C23×C3C2×#4112yes—
306C23×C3H(5,𝔽2)12yes—
307C23×C32−1+412yes—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4#12014yes—
338C2×C3×C4C22⋊Q814yes—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3C22⋊Q814yes—
375C2×Dic3C22⋊Q814yes—
376C2×Dic3C4⋊Q814yes—
377Dic6—232no—
378Dic6—416no—
379C24C23×D311yescharacteristic
380C2×Q8—68no—
381C2×Q8—68no—
382C2×Q8—68no—
383C2×Q8—68no—
384C2×Q8—68no—
385C2×Q8—68no—
386C2×Q8—68no—
387C2×Q8—68no—
388C2×Q8—128no—
389C2×Q8—128no—
390C2×Q8—124no—
391C2×Q8—124no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—34no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—34no—
425C22×C4—34no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4C2×D1212yes—
454C22×C4C2×D1212yes—
455C22×C4C2×D1212yes—
456C22×C4D3×D412yes—
457C22×C4D3×D412yes—
458C22×C4#23712yes—
459C22×C4D3×Q814yes—
460C22×C4D3×Q812yes—
461C4⋊C4—28no—
462C4⋊C4—44no—
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—128no—
466C4⋊C4—128no—
467C42—68no—
468C42—68no—
469C42—68no—
470C22×C3C2×#12011yescharacteristic
471C22×C3C2×C22⋊Q811yescharacteristic
472C22×C3C2×C22⋊Q811yescharacteristic
473C22×C3C2×C22⋊Q811yescharacteristic
474C22×C3#52511yescharacteristic
475C22×C3#52511yescharacteristic
476C22×C3#52511yescharacteristic
477C22×C3#52511yescharacteristic
478C22×C3C2×C4⋊Q811yescharacteristic
479C22×C3#53312yes—
480C22×C3#53312yes—
481C22×C3#53511yescharacteristic
482C22×C3#53611yescharacteristic
483C22×C3#54212yes—
484C22×C3#54212yes—
485C22×C3#54612yes—
486C22×C3#54612yes—
487C22×C3D4×Q812yes—
488C22×C3D4×Q812yes—
489C22×C3#55412yes—
490C22×C3#55612yes—
491C22×C3#55712yes—
492C22×C3Q8212yes—
493C3×C4—24no—
494C3×C4—24no—
495C3×C4—42no—
496C3×C4—44no—
497C3×C4—82no—
498Dic3—24no—
499Dic3—24no—
500Dic3—24no—
501Dic3—44no—
502Dic3—42no—
503Dic3—44no—
504C23C22×D1211yescharacteristic
505C23C2×D3×D411yescharacteristic
506C23C2×D3×D411yescharacteristic
507C23C2×#23711yescharacteristic
508C23#100412yes—
509C23C2×D3×Q812yes—
510C23C2×D3×Q811yescharacteristic
511C23D3×#4111yescharacteristic
512C23D3×#4112yes—
513C23D3×#4111yescharacteristic
514C23#101012yes—
515Q8—632no—
516Q8—1216no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—34no—
533C2×C4—34no—
534C2×C4—34no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—62no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—121no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—121no—
583C2×C4—122no—
584C2×C4#88214yes—
585C2×C4#89714yes—
586C2×C3#239911yescharacteristic
587C2×C3#240411yescharacteristic
588C2×C3#243411yescharacteristic
589C2×C3#245112yes—
590C2×C3#245112yes—
591C2×C3#246512yes—
592C2×C3#246512yes—
593C2×C3#256711yescharacteristic
594C2×C3#258111yescharacteristic
595C2×C3#263211yescharacteristic
596C2×C3#284811yescharacteristic
597C22C2×#88211yescharacteristic
598C22C2×#89711yescharacteristic
599C22#552411yescharacteristic
600C22#555412yes—
601C22Q8×D1212yes—
602C22#559211yescharacteristic
603C22#560312yes—
604C22#560611yescharacteristic
605C22#560611yescharacteristic
606C22#561012yes—
607C22#562212yes—
608C22D3×C22⋊Q811yescharacteristic
609C22D3×C22⋊Q811yescharacteristic
610C22#563212yes—
611C22#563612yes—
612C22#565611yescharacteristic
613C22#566212yes—
614C22#567511yescharacteristic
615C22#568512yes—
616C22#568812yes—
617C22#570212yes—
618C22#572212yes—
619C22D3×C4⋊Q811yescharacteristic
620C4—24no—
621C4—24no—
622C4—42no—
623C4—44no—
624C4—64no—
625C4—64no—
626C4—64no—
627C4—82no—
628C4—124no—
629C4—124no—
630C4—122no—
631C3#1488111yescharacteristic, complemented, Sylow
632C2#7404911yescharacteristic
633C2#7488511yescharacteristic
634C2#7532912yes—
635C2#7600912yes—
636C2#7649911yescharacteristic
637C2#7697711yescharacteristic
638C2#7718911yescharacteristic
639C2#7729612yes—
640C2#7774011yescharacteristic
641C2#7774812yes—
642C2#7819111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1441C2
hypercenter#1441D3
hypocenterC3#14881
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
75. C22×Dic6Q8semidirect
114. C3⋊#95Q8semidirect
631. C3#14881semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C2×#49792
C2×#49732
C2×#46562
C3×#14412
#715662
#694242
#694202
#694072
#694052
#694022
#148813