Finite Group Databasebeta

← #10831788

Subgroups — #10831788

Order 768.

1064 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×C3×#557C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic
4C2×#5026C211yesmaximal, characteristic
5#71792C211yesmaximal, characteristic
6#71782C211yesmaximal, characteristic, complemented
7#71743C211yesmaximal, characteristic
8#71741C211yesmaximal, characteristic, complemented
9#71739C211yesmaximal, characteristic, complemented
10#71635C211yesmaximal, characteristic, complemented
11#71633C211yesmaximal, characteristic
12#71633C211yesmaximal, characteristic
13#71629C211yesmaximal, characteristic, complemented
14#71628C211yesmaximal, characteristic, complemented
15#71626C211yesmaximal, characteristic, complemented
16#24346—31nomaximal, Sylow
17C3×#557—24no—
18C3×#557—24no—
19C2×C3×C4⋊Q8—21no—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#125—21no—
22C2×C3×#125—21no—
23C2×C3×#125C2211yescharacteristic
24C2×C3×#125C2211yescharacteristic
25C2×C3×C4×Q8—21no—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×Q8C2211yescharacteristic
29C2×C3×C4×Q8C2211yescharacteristic
30C2×#946—21no—
31C22×#215C2211yescharacteristic
32C2×C4×D12C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5066—24no—
37#5026—24no—
38#4994—21no—
39#4994—21no—
40#4987—21no—
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4935C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557D311yescharacteristic, complemented
65C2×#478—31no—
66C2×#440—31no—
67#2538—31no—
68#2059—31no—
69#2057—31no—
70#2025—31no—
71#2023—31no—
72#1992—31no—
73#1985—31no—
74#1880—31no—
75#1880—31no—
76#1865—31no—
77#1863—31no—
78#1803—31no—
79C22×C3×Q8—21no—
80C22×C3×Q8D411yescharacteristic
81C22×C3×Q8C2311yescharacteristic
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C3×C4⋊Q8—28no—
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—22no—
87C3×#125—28no—
88C3×#125—28no—
89C3×#125—22no—
90C3×#125—22no—
91C3×#125—22no—
92C3×#125D414yes—
93C3×C4×Q8—28no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×Q8—22no—
101C3×C4×Q8D414yes—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4D411yescharacteristic
111C2×C3×C4⋊C4D411yescharacteristic
112C2×C3×C4⋊C4D411yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122#946—44no—
123C2×#215—22no—
124C2×#215—22no—
125C2×#215Q814yescomplemented
126C2×(C2×D3)⋊C4—21no—
127C2×(C2×D3)⋊C4—21no—
128C2×(C2×D3)⋊C4—21no—
129C2×(C2×D3)⋊C4—21no—
130C2×(C2×D3)⋊C4—21no—
131C2×C4⋊Dic3—21no—
132C2×C4⋊Dic3—21no—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2311yescharacteristic
136C2×Dic3⋊C4—21no—
137C2×Dic3⋊C4—21no—
138C22×C3⋊C8C2311yescharacteristic
139C22×C3⋊C8C2311yescharacteristic
140C22×C3⋊C8C2311yescharacteristic
141C22×C3⋊C8C2311yescharacteristic
142C4×D12—22no—
143C4×D12—22no—
144#835—22no—
145#835—22no—
146#835—22no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#809—22no—
150#809—22no—
151#809—22no—
152#808—22no—
153#807—22no—
154#807—22no—
155C4⋊(C3⋊C8)—22no—
156#803—22no—
157C22×SD16—31no—
158#557—24no—
159#557—24no—
160C2×C4⋊Q8—21no—
161C2×C4⋊Q8C2×D311yescharacteristic
162C2×#125—21no—
163C2×#125—21no—
164C2×#125C2×D311yescharacteristic
165C2×#125C2×D311yescharacteristic
166C2×C22⋊Q8—61no—
167C2×C4×Q8—21no—
168C2×C4×Q8C2×D311yescharacteristic
169C2×C4×Q8C2×D311yescharacteristic
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4—31no—
173#478—64no—
174#440—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#97—31no—
186#396—61no—
187#396—61no—
188#390—31no—
189#390—31no—
190#389—31no—
191#389—31no—
192#384—31no—
193#380—61no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199#340—31no—
200#336—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204C23×D3—21no—
205C2×C3×Q8—24no—
206C2×C3×Q8—22no—
207C2×C3×Q8—22no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—41no—
212C2×C3×Q8—42no—
213C2×C3×Q8—42no—
214C2×C3×Q8C2×D412yes—
215C2×C3×Q8C2×Q812yes—
216C2×C3×Q8#4112yes—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12C2×Q812yes—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4C2×D412yes—
255C3×C4⋊C4C2×D412yes—
256C3×C4⋊C4C2×D412yes—
257C3×C42—24no—
258C3×C42—24no—
259C3×C42—24no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42C2×D412yes—
265C3×C42C2×D412yes—
266#215—28no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269(C2×D3)⋊C4—42no—
270(C2×D3)⋊C4—81no—
271(C2×D3)⋊C4—81no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—42no—
279C4⋊Dic3—81no—
280Dic3⋊C4—42no—
281Dic3⋊C4—42no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8C2×Q812yes—
294C22×Q8—21no—
295C22×Q8#6711yescharacteristic
296C22×Q8C22×D311yescharacteristic
297C22×D4—31no—
298C23×C4—61no—
299C2×SD16—34no—
300C2×SD16—62no—
301C2×SD16—62no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—31no—
306C4⋊Q8—28no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309#125—28no—
310#125—28no—
311#125—22no—
312#125—22no—
313#125—22no—
314#125#6714yes—
315C22⋊Q8—124no—
316C4×Q8—28no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8#6714yes—
325C4×D4—62no—
326C4×D4—62no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4#6711yescharacteristic
343C2×C4⋊C4#6711yescharacteristic
344C2×C4⋊C4#6711yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42C22×D311yescharacteristic
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—62no—
362C4⋊C8—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#97—62no—
370#95—61no—
371#95—61no—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8—44no—
384C3×Q8—48no—
385C3×Q8C22⋊Q814yescomplemented
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4#12012yes—
414C2×C3×C4#12012yes—
415C2×C3×C4AGL(1,ℤ/8)12yes—
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×Q811yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C42−1+411yescharacteristic
421C2×C3×C42−1+411yescharacteristic
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431D12—22no—
432D12—22no—
433D12—41no—
434D12—41no—
435C4×D3—84no—
436C3⋊C8—22no—
437C3⋊C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—61no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—41no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8D3×Q812yes—
451C2×Q8#23912yes—
452C2×Q8C2×#6712yes—
453C2×D4—32no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477C2×C8—32no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—122no—
510C4⋊C4—122no—
511C4⋊C4—122no—
512C4⋊C4—241no—
513C4⋊C4C2×#6712yes—
514C4⋊C4C2×#6712yes—
515C4⋊C4C2×#6712yes—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C42—24no—
522C42—24no—
523C42—24no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42C2×#6712yes—
529C42C2×#6712yes—
530C22×C3C2×#12011yescharacteristic
531C22×C3#54811yescharacteristic
532C22×C3D4×Q811yescharacteristic
533C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57711yescharacteristic
536C22×C3#57811yescharacteristic
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—24no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—42no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—42no—
554C3×C4C2×C22⋊Q811yescharacteristic
555C3×C4#53611yescharacteristic
556C3×C4#54811yescharacteristic
557C3×C4D4×Q811yescharacteristic
558Dic3—81no—
559Dic3—81no—
560Dic3—81no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C22×#6711yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—44no—
573Q8—48no—
574Q8#94614yescomplemented
575D4—62no—
576D4—62no—
577D4—121no—
578D4—121no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—244no—
616C2×C4#94912yes—
617C2×C4#95312yes—
618C2×C4#95312yes—
619C2×C4C2×D3×Q811yescharacteristic
620C2×C4C2×#23911yescharacteristic
621C2×C4#100711yescharacteristic
622C2×C4#100711yescharacteristic
623C2×C4C22×#6711yescharacteristic
624C2×C4C22×#6711yescharacteristic
625C8—62no—
626C8—121no—
627C8—121no—
628C8—121no—
629C2×C3#240911yescharacteristic
630C2×C3#300611yescharacteristic
631C2×C3#301311yescharacteristic
632C2×C3#336811yescharacteristic
633C2×C3#337711yescharacteristic
634C2×C3#338611yescharacteristic
635C2×C3#338811yescharacteristic
636D3—82no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22Q8×#6711yescharacteristic
645C22#581711yescharacteristic
646C22C2×#94911yescharacteristic
647C22#582211yescharacteristic
648C22#583711yescharacteristic
649C22#583811yescharacteristic
650C22C2×#95311yescharacteristic
651C4—24no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—241no—
662C4—241no—
663C4—241no—
664C4C2×#94611yescharacteristic
665C4Q8×#6711yescharacteristic
666C4#581611yescharacteristic
667C4#581711yescharacteristic
668C3#2434611yescharacteristic, complemented, Sylow
669C2—242no—
670C2#8369011yescharacteristic
671C2#8369411yescharacteristic
672C2#8369511yescharacteristic
673C2#8370011yescharacteristic
674C2#8406811yescharacteristic
675C2#8408211yescharacteristic
676C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#557C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#557D3
hypocenterC3#24346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
6. #71782C2semidirect
8. #71741C2semidirect
9. #71739C2semidirect
10. #71635C2semidirect
13. #71629C2semidirect
14. #71628C2semidirect
15. #71626C2semidirect
64. C2×#557D3semidirect
125. C2×#215Q8semidirect
385. C3×Q8C22⋊Q8semidirect
574. Q8#946semidirect
668. C3#24346semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#50662
#825002
C2×#50262
#717922
#717822
#717432
#717412
#717392
#716352
#716332
#716292
#716282
#716262
#243463