Finite Group Databasebeta

← #10804161

Subgroups — #10804161

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#924C211yesmaximal, characteristic
2C2×C3×#407C211yesmaximal, characteristic
3C2×#5207C211yesmaximal, characteristic
4#72003C214yesmaximal, complemented
5#69990C212yesmaximal, complemented
6#69986C214yesmaximal, complemented
7#69973C212yesmaximal, complemented
8#21753—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C23×Dic6C2211yescharacteristic
11C22×C3×M16C2212yes—
12C22×#217C412yes—
13C22×C4⋊Dic3C2211yescharacteristic
14C2×#924—21no—
15C2×#924—21no—
16C2×#924—21no—
17C2×#924—21no—
18C2×#924—21no—
19C2×#924—21no—
20C2×#924—21no—
21C2×#924—21no—
22C2×#924C412yes—
23C2×#924C412yes—
24C2×#924C2212yes—
25C2×#924C2212yes—
26C22×Dic3⋊C4C412yes—
27C22×#208C2212yes—
28C3×#407—21no—
29C3×#407—21no—
30C3×#407C2212yes—
31C3×#407C2212yes—
32C2×C3×C22⋊C8—21no—
33C2×C3×C22⋊C8—21no—
34#5207—21no—
35#5207—21no—
36#5207C2212yes—
37#5207C2212yes—
38C2×#830—21no—
39C2×#830—21no—
40C2×#824C2218yescomplemented
41#4554C2218yescomplemented
42C22×C22⋊Q8—31no—
43C2×#407—31no—
44C2×#407D311yescharacteristic
45#1901—34no—
46#1901—34no—
47#1821—32no—
48#1784—32no—
49C25×C3C2×C411yescharacteristic
50C23×C3×C4—44no—
51C23×C3×C4C2×C411yescharacteristic
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4C2×C411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×Dic3C2×C412yes—
66C23×Dic3C2×C412yes—
67C22×Dic6—21no—
68C22×Dic6—21no—
69C22×Dic6—21no—
70C22×Dic6—21no—
71C22×Dic6—44no—
72C22×Dic6C2×C412yes—
73C22×Dic6C2312yes—
74C2×C3×M16—22no—
75C2×C3×M16—44no—
76C2×C3×M16C2314yes—
77C22×C3×C8—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81C2×#217—22no—
82C2×#217C2×C414yes—
83C2×#217C2×C414yes—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2×C412yes—
89C2×C4⋊Dic3C2312yes—
90#924—22no—
91#924—22no—
92#924—22no—
93#924—22no—
94#924—22no—
95#924—22no—
96#924—22no—
97#924—22no—
98#924—22no—
99#924—22no—
100#924—22no—
101#924—22no—
102#924—22no—
103#924—22no—
104#924—22no—
105#924—22no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4—22no—
108C2×Dic3⋊C4—22no—
109C2×Dic3⋊C4—22no—
110C2×Dic3⋊C4C2×C414yes—
111C2×Dic3⋊C4C2×C414yes—
112C2×#208—22no—
113C2×#208—44no—
114C2×#208C2314yes—
115C22×C3⋊C8—22no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—22no—
119C3×C22⋊C8—22no—
120#830—22no—
121#830—22no—
122#830—22no—
123#830—22no—
124#824—48no—
125C23×Q8—31no—
126C24×C4C2×D311yescharacteristic
127C22×M16—32no—
128C22×M16C2×D312yes—
129C2×C22⋊Q8—32no—
130C2×C22⋊Q8—32no—
131C2×C22⋊Q8—32no—
132C2×C22⋊Q8—32no—
133C2×C22⋊Q8—61no—
134C2×C22⋊Q8—61no—
135C2×C22⋊Q8—61no—
136C2×C22⋊Q8—61no—
137C2×C22⋊Q8—61no—
138C2×C22⋊Q8—61no—
139C2×C22⋊Q8—61no—
140C2×C22⋊Q8—61no—
141C22×C4⋊C4—32no—
142C22×C4⋊C4—31no—
143C22×C22⋊C4—32no—
144C2×#101—38no—
145#407—21no—
146#407—21no—
147#407—32no—
148#407—32no—
149#407—61no—
150#407—61no—
151#407C2×D312yes—
152#407C2×D312yes—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—61no—
156C2×C22⋊C8—61no—
157#343—38no—
158C24×C3—24no—
159C24×C3—42no—
160C24×C3—42no—
161C24×C3C22⋊C411yescharacteristic
162C24×C3C22⋊C411yescharacteristic
163C24×C3C22⋊C411yescharacteristic
164C24×C3C22⋊C411yescharacteristic
165C24×C3C22×C411yescharacteristic
166C24×C3C22×C411yescharacteristic
167C24×C3C22×C411yescharacteristic
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4C22⋊C411yescharacteristic
183C22×C3×C4C22⋊C411yescharacteristic
184C22×C3×C4C22⋊C411yescharacteristic
185C22×C3×C4C22⋊C411yescharacteristic
186C22×C3×C4C22⋊C411yescharacteristic
187C22×C3×C4C22⋊C411yescharacteristic
188C22×C3×C4C22⋊C411yescharacteristic
189C22×C3×C4C22⋊C411yescharacteristic
190C22×C3×C4C22⋊C411yescharacteristic
191C22×C3×C4C22⋊C411yescharacteristic
192C22×C3×C4C22⋊C411yescharacteristic
193C22×C3×C4C22⋊C411yescharacteristic
194C22×C3×C4C22×C411yescharacteristic
195C22×C3×C4C22×C411yescharacteristic
196C22×C3×C4C22×C411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—44no—
215C22×Dic3—44no—
216C22×Dic3C22×C414yes—
217C22×Dic3C22×C414yes—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—28no—
223C2×Dic6—44no—
224C2×Dic6—44no—
225C2×Dic6—44no—
226C2×Dic6—44no—
227C2×Dic6—44no—
228C3×M16—28no—
229C3×M16—44no—
230C2×C3×C8—24no—
231C2×C3×C8—44no—
232#217—24no—
233#217—24no—
234#217—24no—
235#217—24no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240Dic3⋊C4—24no—
241Dic3⋊C4—24no—
242Dic3⋊C4—24no—
243Dic3⋊C4—24no—
244#208—28no—
245#208—44no—
246C2×C3⋊C8—24no—
247C2×C3⋊C8—44no—
248C25C4×D311yescharacteristic
249C22×Q8—32no—
250C22×Q8—32no—
251C22×Q8—61no—
252C22×Q8—61no—
253C22×Q8—61no—
254C22×Q8—61no—
255C22×Q8—124no—
256C23×C4—32no—
257C23×C4—32no—
258C23×C4—44no—
259C23×C4C4×D311yescharacteristic
260C23×C4C4×D311yescharacteristic
261C23×C4C4×D311yescharacteristic
262C23×C4D1211yescharacteristic
263C23×C4D1211yescharacteristic
264C23×C4D1211yescharacteristic
265C23×C4D1211yescharacteristic
266C23×C4#6711yescharacteristic
267C23×C4#6711yescharacteristic
268C23×C4#6711yescharacteristic
269C23×C4#6711yescharacteristic
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C2×M16—22no—
274C2×M16—34no—
275C2×M16—44no—
276C2×M16—62no—
277C2×M16—124no—
278C2×M16C22×D314yes—
279C22×C8—22no—
280C22×C8—62no—
281C22⋊Q8—62no—
282C22⋊Q8—62no—
283C22⋊Q8—62no—
284C22⋊Q8—62no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C22⋊Q8—62no—
288C22⋊Q8—62no—
289C22⋊Q8—62no—
290C22⋊Q8—62no—
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8—62no—
296C22⋊Q8—62no—
297C2×C4⋊C4—34no—
298C2×C4⋊C4—34no—
299C2×C4⋊C4—32no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—62no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C22⋊C4—34no—
310C2×C22⋊C4—34no—
311C2×C22⋊C4—62no—
312C2×C22⋊C4—62no—
313C2×C22⋊C4—62no—
314C2×C22⋊C4—62no—
315#101—128no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—62no—
321C22⋊C8—62no—
322C22⋊C8—62no—
323C22⋊C8—62no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—416no—
345C23×C3#10114yescomplemented
346C23×C3C2×C22⋊C411yescharacteristic
347C23×C3C2×C22⋊C411yescharacteristic
348C23×C3C2×C22⋊C411yescharacteristic
349C23×C3C2×C22⋊C411yescharacteristic
350C23×C3C2×C22⋊C411yescharacteristic
351C23×C3C2×C22⋊C411yescharacteristic
352C23×C3C23×C411yescharacteristic
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
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—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4C2×C22⋊C411yescharacteristic
381C2×C3×C4C2×C22⋊C411yescharacteristic
382C2×C3×C4C2×C22⋊C411yescharacteristic
383C2×C3×C4C2×C22⋊C411yescharacteristic
384C2×C3×C4C2×C22⋊C411yescharacteristic
385C2×C3×C4C2×C22⋊C411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4H(5,𝔽2)14yes—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—28no—
392C2×Dic3—44no—
393C2×Dic3—44no—
394C2×Dic3—44no—
395C2×Dic3—44no—
396C2×Dic3—44no—
397Dic6—44no—
398Dic6—44no—
399Dic6—44no—
400Dic6—44no—
401C3×C8—44no—
402C3⋊C8—44no—
403C24—24no—
404C24—42no—
405C24—42no—
406C24(C2×D3)⋊C411yescharacteristic
407C24(C2×D3)⋊C411yescharacteristic
408C24(C2×D3)⋊C411yescharacteristic
409C24(C2×D3)⋊C411yescharacteristic
410C24C2×C4×D311yescharacteristic
411C24C2×C4×D311yescharacteristic
412C24C2×C4×D311yescharacteristic
413C2×Q8—62no—
414C2×Q8—62no—
415C2×Q8—62no—
416C2×Q8—68no—
417C2×Q8—62no—
418C2×Q8—124no—
419C2×Q8—124no—
420C2×Q8—124no—
421C2×Q8—124no—
422C2×Q8—124no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—34no—
427C22×C4—34no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—124no—
444C22×C4—124no—
445C22×C4(C2×D3)⋊C411yescharacteristic
446C22×C4(C2×D3)⋊C411yescharacteristic
447C22×C4(C2×D3)⋊C411yescharacteristic
448C22×C4(C2×D3)⋊C411yescharacteristic
449C22×C4(C2×D3)⋊C411yescharacteristic
450C22×C4(C2×D3)⋊C411yescharacteristic
451C22×C4(C2×D3)⋊C411yescharacteristic
452C22×C4(C2×D3)⋊C411yescharacteristic
453C22×C4(C2×D3)⋊C411yescharacteristic
454C22×C4(C2×D3)⋊C411yescharacteristic
455C22×C4(C2×D3)⋊C411yescharacteristic
456C22×C4(C2×D3)⋊C411yescharacteristic
457C22×C4C2×C4×D311yescharacteristic
458C22×C4C2×C4×D311yescharacteristic
459C22×C4C2×C4×D311yescharacteristic
460C22×C4C2×D1211yescharacteristic
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4C2×D1211yescharacteristic
464C22×C4C2×D1211yescharacteristic
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C23×D311yescharacteristic
473M16—28no—
474M16—44no—
475M16—68no—
476M16—124no—
477C2×C8—24no—
478C2×C8—44no—
479C2×C8—64no—
480C2×C8—124no—
481C4⋊C4—64no—
482C4⋊C4—64no—
483C4⋊C4—64no—
484C4⋊C4—64no—
485C4⋊C4—62no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4—62no—
489C22⋊C4—64no—
490C22⋊C4—64no—
491C22⋊C4—64no—
492C22⋊C4—64no—
493C22×C3—28no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—44no—
511C22×C3—44no—
512C22×C3—44no—
513C22×C3—44no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3C2×#10112yes—
517C22×C3C2×#10112yes—
518C22×C3C2×#10112yes—
519C22×C3#41312yes—
520C22×C3C22×C22⋊C411yescharacteristic
521C22×C3#51812yes—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4#53412yes—
529C3×C4#53412yes—
530Dic3—44no—
531Dic3—44no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—416no—
553C23#82414yescomplemented
554C23C2×(C2×D3)⋊C411yescharacteristic
555C23C2×(C2×D3)⋊C411yescharacteristic
556C23C2×(C2×D3)⋊C411yescharacteristic
557C23C2×(C2×D3)⋊C411yescharacteristic
558C23C2×(C2×D3)⋊C411yescharacteristic
559C23C2×(C2×D3)⋊C411yescharacteristic
560C23C22×C4×D311yescharacteristic
561Q8—124no—
562Q8—124no—
563Q8—124no—
564Q8—124no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—68no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—124no—
596C2×C4—124no—
597C2×C4—124no—
598C2×C4—124no—
599C2×C4—124no—
600C2×C4C2×(C2×D3)⋊C411yescharacteristic
601C2×C4C2×(C2×D3)⋊C411yescharacteristic
602C2×C4C2×(C2×D3)⋊C411yescharacteristic
603C2×C4C2×(C2×D3)⋊C411yescharacteristic
604C2×C4C2×(C2×D3)⋊C411yescharacteristic
605C2×C4C2×(C2×D3)⋊C411yescharacteristic
606C2×C4C22×D1211yescharacteristic
607C2×C4#100414yes—
608C2×C4C22×#6711yescharacteristic
609C8—44no—
610C8—124no—
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—22no—
614C2×C3—22no—
615C2×C3—22no—
616C2×C3—41no—
617C2×C3—41no—
618C2×C3#230811yescharacteristic
619C2×C3C22×#10111yescharacteristic
620C2×C3C2×#41311yescharacteristic
621C22—28no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22#513212yes—
645C22C2×#82412yes—
646C22C2×#82412yes—
647C22C2×#82412yes—
648C22#548612yes—
649C22C22×(C2×D3)⋊C411yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—124no—
657C4—124no—
658C4#549312yes—
659C4#580412yes—
660C3#2175311yescharacteristic, complemented, Sylow
661C2—22no—
662C2—22no—
663C2—22no—
664C2—22no—
665C2—22no—
666C2—41no—
667C2—41no—
668C2#8212111yescharacteristic
669C2C2×#513211yescharacteristic
670C2C22×#82411yescharacteristic
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#407C2
socleC22×C3C22×C22⋊C4
prime radicalC25×C3C2×C4
cosocleC22×C3×C4C24
hypercenterC2×#407D3
hypocenterC3#21753
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
4. #72003C2semidirect
5. #69990C2semidirect
6. #69986C2semidirect
7. #69973C2semidirect
40. C2×#824C22semidirect
41. #4554C22semidirect
345. C23×C3#101semidirect
553. C23#824semidirect
660. C3#21753semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C2×C3×#4072
C2×#52072
#720032
#699902
#699862
#699732
#217533