Finite Group Databasebeta

← #10981907

Subgroups — #10981907

Order 768.

2190 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×#5572C211yesmaximal, characteristic
2C2×Q8×Dic6C211yesmaximal, characteristic, complemented
3C3×#2581C211yesmaximal, characteristic, complemented
4#82491C211yesmaximal, characteristic
5#82479C211yesmaximal, characteristic, complemented
6#78209C212yesmaximal
7#78181C212yesmaximal, complemented
8#77191C212yesmaximal, complemented
9#77170C212yesmaximal
10#76614C212yesmaximal
11#76508C212yesmaximal, complemented
12#76223C212yesmaximal
13#75982C212yesmaximal, complemented
14#74987C214yesmaximal, complemented
15#74983C214yesmaximal
16#74194C211yesmaximal, characteristic
17#74157C211yesmaximal, characteristic, complemented
18#35711—31nomaximal, Sylow
19C22×D3×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×Q8×Dic3C2212yes—
23C2×Q8×Dic3C2211yescharacteristic
24#5572—28no—
25Q8×Dic6—28no—
26C2×#892C2212yes—
27C2×#892C2211yescharacteristic
28C2×#888C2212yes—
29C2×#888C2214yes—
30C2×#887C2212yes—
31C2×#887C2211yescharacteristic
32C2×#887C2212yes—
33C2×#872C2211yescharacteristic
34C2×#872C2212yes—
35C2×#869C2211yescharacteristic
36C2×#869C2212yes—
37C2×C4×Dic6C2212yes—
38C2×C4×Dic6C2212yes—
39C2×C4×Dic6C2211yescharacteristic
40C3×#400C2212yes—
41C3×#395C2212yes—
42C3×#391C2212yes—
43C3×#389C2212yes—
44C3×#387C2214yes—
45C3×#382C2211yescharacteristic
46#5231C2211yescharacteristic
47#5230C2211yescharacteristic
48#5229C2212yes—
49#4994C2212yes—
50#4994C2214yes—
51#4993C2211yescharacteristic
52#4987C2212yes—
53#4986C2212yes—
54#4986C2212yes—
55#4985C2211yescharacteristic
56#4984C2212yes—
57#4982C2212yes—
58#4980C2214yes—
59#4977C2212yes—
60#4976C2214yes—
61Dic3×C4⋊C4C2214yes—
62#4972C2212yes—
63C4×(C2×D3)⋊C4C2212yes—
64C4×C4⋊Dic3C2211yescharacteristic
65C4×Dic3⋊C4C2212yes—
66#4929C2214yes—
67#4677C2212yes—
68#4673C2214yes—
69#4673C2212yes—
70#4667C2214yes—
71#4666C2214yes—
72#4661C2214yes—
73#4659C2212yes—
74#4656C2214yes—
75#4656C2212yes—
76#4655C2214yes—
77#4650C2214yes—
78#4650C2212yes—
79#4649C2212yes—
80#4647C2214yes—
81#4647C2214yes—
82#4646C2214yes—
83#4646C2214yes—
84#4646C2214yes—
85#4646C2214yes—
86C2×Q82—31no—
87C2×#519—31no—
88#2628—32no—
89#2628—31no—
90#2581—32no—
91#2581D311yescharacteristic, complemented
92#2568—32no—
93#2568—31no—
94#2451—32no—
95#2451—34no—
96#2439—32no—
97#2439—34no—
98#2396—32no—
99#2396—31no—
100#2351—32no—
101#2351—31no—
102Q8×C22⋊C4—32no—
103C22×C3×Q8C2311yescharacteristic
104C22×C3×Q8C2311yescharacteristic
105C2×D3×Q8—24no—
106C2×D3×Q8—24no—
107C2×D3×Q8Q818yescomplemented
108C22×C4×D3C2312yes—
109C22×C4×D3C2311yescharacteristic
110C22×Dic6C2312yes—
111C22×Dic6C2314yes—
112C22×Dic6C2312yes—
113C22×Dic6C2311yescharacteristic
114C3×C4×Q8—24no—
115C3×C4×Q8—24no—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C4⋊C4C2311yescharacteristic
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C42C2312yes—
123C2×C3×C42C2312yes—
124C2×C3×C42C2311yescharacteristic
125Q8×Dic3—24no—
126Q8×Dic3—28no—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2311yescharacteristic
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2311yescharacteristic
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2311yescharacteristic
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2311yescharacteristic
145C2×Dic3⋊C4C2311yescharacteristic
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2312yes—
151C2×C4×Dic3C2314yes—
152C2×C4×Dic3C2314yes—
153C2×C4×Dic3C2312yes—
154C2×C4×Dic3C2311yescharacteristic
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2312yes—
157#892—24no—
158#892—28no—
159#888—216no—
160#888—28no—
161#887—24no—
162#887—28no—
163#887—28no—
164#872—24no—
165#872—28no—
166#869—24no—
167#869—28no—
168C4×Dic6—28no—
169C4×Dic6—28no—
170C4×Dic6D418yes—
171C3×#95C2314yes—
172C3×#95C2314yes—
173C3×#95C2314yes—
174C3×#95C2312yes—
175C3×#95C2312yes—
176C3⋊#95C2312yes—
177C3⋊#95C2314yes—
178C3⋊#95C2312yes—
179C3⋊#95C2314yes—
180C3⋊#95C2312yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2314yes—
186C3⋊#95C2311yescharacteristic
187C3⋊#95C2311yescharacteristic
188C3⋊#95C2312yes—
189C23×Q8—31no—
190Q82—68no—
191C2×C4⋊Q8—32no—
192C2×C4⋊Q8—34no—
193C2×C4⋊Q8—31no—
194C2×C4⋊Q8—32no—
195#519—68no—
196C2×C4×Q8—32no—
197C2×C4×Q8—31no—
198C2×C4×Q8—32no—
199C2×C4×Q8—32no—
200C2×C4×Q8—31no—
201C2×C4×Q8—31no—
202C2×C4×Q8—32no—
203C2×C4×Q8—32no—
204C2×C4×Q8C2×D311yescharacteristic
205C2×C4×Q8C2×D311yescharacteristic
206C2×#117—32no—
207C2×#117—31no—
208C2×#117—31no—
209C2×#117—32no—
210#400—31no—
211#400—32no—
212#400—32no—
213#400—32no—
214#400C2×D312yes—
215#399—31no—
216#399—32no—
217#398—32no—
218#398—34no—
219#396—34no—
220#396—32no—
221#395—31no—
222#395—32no—
223#395—34no—
224#395C2×D312yes—
225#393—31no—
226#393—34no—
227#393—32no—
228#391—32no—
229#391—34no—
230#391—34no—
231#391—34no—
232#391—34no—
233#391—34no—
234#391C2×D312yes—
235#389—34no—
236#389—32no—
237#389—32no—
238#389—34no—
239#389—34no—
240#389C2×D312yes—
241#387—34no—
242#387—34no—
243#387C2×D314yes—
244#382—32no—
245#382—34no—
246#382—32no—
247#382C2×D311yescharacteristic
248#380—32no—
249#380—34no—
250C4×C4⋊C4—34no—
251C4×C4⋊C4—31no—
252C4×C4⋊C4—32no—
253C4×C4⋊C4—32no—
254C4×C22⋊C4—32no—
255C4×C22⋊C4—34no—
256C23×D3C2411yescharacteristic
257C2×C3×Q8—22no—
258C2×C3×Q8—22no—
259C2×C3×Q8—22no—
260C2×C3×Q8—24no—
261C2×C3×Q8C2×Q814yes—
262C22×C3×C4C2412yes—
263C22×C3×C4C2414yes—
264C22×C3×C4C2412yes—
265C22×C3×C4C2411yescharacteristic
266C22×C3×C4C2411yescharacteristic
267C22×C3×C4C2412yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2411yescharacteristic
270C22×Dic3C2412yes—
271C22×Dic3C2412yes—
272C22×Dic3C2411yescharacteristic
273C22×Dic3C2414yes—
274C22×Dic3C2411yescharacteristic
275C22×Dic3C2412yes—
276C22×Dic3C2411yescharacteristic
277C22×Dic3C2412yes—
278D3×Q8—232no—
279C2×C4×D3—22no—
280C2×C4×D3—22no—
281C2×C4×D3—24no—
282C2×C4×D3—24no—
283C2×C4×D3C2×Q814yes—
284C2×C4×D3C2×Q818yes—
285C2×Dic6—24no—
286C2×Dic6—24no—
287C2×Dic6—24no—
288C2×Dic6—28no—
289C2×Dic6—44no—
290C2×Dic6—44no—
291C2×Dic6—44no—
292C2×Dic6C2×D414yes—
293C2×Dic6C2×Q818yes—
294C2×Dic6C2×Q814yes—
295C2×Dic6#4114yes—
296C3×C4⋊C4—24no—
297C3×C4⋊C4—22no—
298C3×C4⋊C4—24no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4—24no—
301C3×C4⋊C4—28no—
302C3×C42—24no—
303C3×C42—24no—
304C3×C42C2×D414yes—
305(C2×D3)⋊C4—22no—
306(C2×D3)⋊C4—22no—
307(C2×D3)⋊C4—24no—
308(C2×D3)⋊C4—24no—
309(C2×D3)⋊C4—42no—
310(C2×D3)⋊C4—44no—
311C4⋊Dic3—24no—
312C4⋊Dic3—24no—
313C4⋊Dic3—24no—
314C4⋊Dic3—28no—
315C4⋊Dic3—42no—
316C4⋊Dic3—44no—
317C4⋊Dic3C2×D414yes—
318Dic3⋊C4—22no—
319Dic3⋊C4—22no—
320Dic3⋊C4—24no—
321Dic3⋊C4—28no—
322Dic3⋊C4—24no—
323Dic3⋊C4—28no—
324Dic3⋊C4—44no—
325Dic3⋊C4—42no—
326Dic3⋊C4C2×D418yes—
327C4×Dic3—22no—
328C4×Dic3—24no—
329C4×Dic3—24no—
330C4×Dic3—24no—
331C4×Dic3—28no—
332C4×Dic3—24no—
333C4×Dic3—28no—
334C4×Dic3C2×D418yes—
335C22×Q8—38no—
336C22×Q8—31no—
337C22×Q8—32no—
338C22×Q8—34no—
339C22×Q8—32no—
340C22×Q8—64no—
341C22×Q8—64no—
342C22×Q8C22×D311yescharacteristic
343C22×Q8C22×D311yescharacteristic
344C23×C4—32no—
345C23×C4—31no—
346C4⋊Q8—68no—
347C4⋊Q8—68no—
348C4⋊Q8—616no—
349C4⋊Q8—64no—
350C4×Q8—24no—
351C4×Q8—24no—
352C4×Q8—38no—
353C4×Q8—68no—
354C4×Q8—68no—
355C4×Q8—68no—
356C4×Q8—68no—
357C4×Q8—64no—
358C4×Q8—68no—
359C4×Q8—64no—
360#117—68no—
361#117—68no—
362#117—64no—
363#117—64no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—32no—
369C2×C4⋊C4—31no—
370C2×C4⋊C4—34no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—34no—
375C2×C4⋊C4—32no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—32no—
379C2×C4⋊C4—34no—
380C2×C4⋊C4C22×D311yescharacteristic
381C2×C4⋊C4C22×D312yes—
382C2×C4⋊C4C22×D312yes—
383C2×C4⋊C4C22×D312yes—
384C2×C4⋊C4C22×D311yescharacteristic
385C2×C4⋊C4C22×D314yes—
386C2×C22⋊C4—32no—
387C2×C22⋊C4—31no—
388C2×C22⋊C4—31no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—32no—
391C2×C22⋊C4—34no—
392C2×C42—32no—
393C2×C42—32no—
394C2×C42—31no—
395C2×C42—32no—
396C2×C42—32no—
397C2×C42—34no—
398C2×C42—34no—
399C2×C42—32no—
400C2×C42C22×D312yes—
401C2×C42C22×D312yes—
402C2×C42C22×D311yescharacteristic
403#95—32no—
404#95—32no—
405#95—34no—
406#95—32no—
407#95—31no—
408#95—32no—
409#95—34no—
410#95—31no—
411#95—34no—
412#95—34no—
413#95—34no—
414#95—34no—
415#95—34no—
416#95C22×D314yes—
417#95C22×D312yes—
418#95C22×D314yes—
419#95C22×D314yes—
420#95C22×D312yes—
421C23×C3C2511yescharacteristic
422C22×D3—21no—
423C22×D3—21no—
424C22×D3—22no—
425C22×D3—22no—
426C22×D3C22×Q812yes—
427C3×Q8—28no—
428C3×Q8—28no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—24no—
438C2×C3×C4—24no—
439C2×C3×C4—24no—
440C2×C3×C4—24no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4C22×D412yes—
444C2×C3×C4C22×D412yes—
445C2×C3×C4C22×D412yes—
446C2×C3×C4C22×Q814yes—
447C2×C3×C4C22×Q812yes—
448C2×C3×C4C2×#4112yes—
449C2×C3×C42−1+414yes—
450C2×Dic3—21no—
451C2×Dic3—21no—
452C2×Dic3—24no—
453C2×Dic3—21no—
454C2×Dic3—21no—
455C2×Dic3—24no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3—22no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3—22no—
463C2×Dic3—22no—
464C2×Dic3—24no—
465C2×Dic3—24no—
466C2×Dic3C22×D414yes—
467C2×Dic3C22×D414yes—
468C2×Dic3C22×Q812yes—
469C2×Dic3C22×Q814yes—
470C2×Dic3C2×#4114yes—
471C4×D3—28no—
472C4×D3—216no—
473Dic6—216no—
474Dic6—88no—
475Dic6—84no—
476Dic6C22⋊Q8116yes—
477C24—31no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—24no—
482C2×Q8—34no—
483C2×Q8—34no—
484C2×Q8—38no—
485C2×Q8—34no—
486C2×Q8—64no—
487C2×Q8—64no—
488C2×Q8—68no—
489C2×Q8—632no—
490C2×Q8—64no—
491C2×Q8—124no—
492C2×Q8—124no—
493C2×Q8—124no—
494C2×Q8D3×Q814yes—
495C22×C4—32no—
496C22×C4—31no—
497C22×C4—32no—
498C22×C4—38no—
499C22×C4—34no—
500C22×C4—32no—
501C22×C4—32no—
502C22×C4—31no—
503C22×C4—34no—
504C22×C4—31no—
505C22×C4—64no—
506C22×C4—64no—
507C22×C4—62no—
508C22×C4—62no—
509C22×C4C23×D312yes—
510C22×C4C23×D314yes—
511C22×C4C23×D312yes—
512C22×C4C23×D311yescharacteristic
513C22×C4C23×D311yescharacteristic
514C22×C4C23×D312yes—
515C22×C4C23×D312yes—
516C22×C4C23×D311yescharacteristic
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—28no—
523C4⋊C4—38no—
524C4⋊C4—34no—
525C4⋊C4—64no—
526C4⋊C4—64no—
527C4⋊C4—68no—
528C4⋊C4—68no—
529C4⋊C4—68no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—64no—
533C4⋊C4—64no—
534C4⋊C4—64no—
535C4⋊C4—124no—
536C4⋊C4—122no—
537C4⋊C4—122no—
538C4⋊C4—124no—
539C22⋊C4—64no—
540C22⋊C4—62no—
541C22⋊C4—62no—
542C22⋊C4—64no—
543C22⋊C4—124no—
544C22⋊C4—122no—
545C42—24no—
546C42—24no—
547C42—38no—
548C42—64no—
549C42—64no—
550C42—64no—
551C42—68no—
552C42—68no—
553C42—62no—
554C42—64no—
555C42D3×D414yes—
556C22×C3C23×D411yescharacteristic
557C22×C3C23×Q811yescharacteristic
558C22×C3C22×#4111yescharacteristic
559C22×C3C2×2−1+412yes—
560C22×C3#58512yes—
561C2×D3—22no—
562C2×D3—24no—
563C2×D3—41no—
564C2×D3—41no—
565C2×D3—42no—
566C3×C4—24no—
567C3×C4—22no—
568C3×C4—24no—
569C3×C4—22no—
570C3×C4—42no—
571C3×C4—42no—
572C3×C4—44no—
573C3×C4C2×C22⋊Q814yes—
574Dic3—22no—
575Dic3—24no—
576Dic3—41no—
577Dic3—44no—
578Dic3—41no—
579Dic3—42no—
580Dic3—42no—
581Dic3C2×C22⋊Q818yes—
582C23—32no—
583C23—61no—
584C23—61no—
585C23—62no—
586C23—62no—
587C23C24×D311yescharacteristic
588Q8—28no—
589Q8—28no—
590Q8—316no—
591Q8—616no—
592Q8—244no—
593Q8—248no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—34no—
609C2×C4—34no—
610C2×C4—34no—
611C2×C4—34no—
612C2×C4—32no—
613C2×C4—616no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—64no—
617C2×C4—68no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—64no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—64no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—64no—
631C2×C4C2×D3×D412yes—
632C2×C4C2×D3×D412yes—
633C2×C4C2×D3×D412yes—
634C2×C4C2×D3×Q814yes—
635C2×C4C2×D3×Q812yes—
636C2×C4D3×#4112yes—
637C2×C4#101014yes—
638C2×C3C22×C22⋊Q811yescharacteristic
639C2×C3#348611yescharacteristic
640C2×C3#350011yescharacteristic
641C2×C3#350212yes—
642C2×C3#351812yes—
643D3—42no—
644C22—62no—
645C22—64no—
646C22—121no—
647C22—121no—
648C22—122no—
649C22C22×D3×D411yescharacteristic
650C22C22×D3×Q811yescharacteristic
651C22C2×D3×#4111yescharacteristic
652C22C2×#101012yes—
653C22#596812yes—
654C4—22no—
655C4—24no—
656C4—22no—
657C4—24no—
658C4—38no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—64no—
663C4—62no—
664C4—121no—
665C4—124no—
666C4—121no—
667C4—122no—
668C4—122no—
669C4D3×C22⋊Q814yes—
670C3#3571111yescharacteristic, complemented, Sylow
671C2—122no—
672C2C2×D3×C22⋊Q811yescharacteristic
673C2#8602211yescharacteristic
674C2#8634111yescharacteristic
675C2#8698912yes—
676C2#8703112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2581C2
prime radicalC23×D3C24
hypercenter#2581D3
hypocenterC3#35711
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic6C2semidirect
3. C3×#2581C2semidirect
5. #82479C2semidirect
7. #78181C2semidirect
8. #77191C2semidirect
11. #76508C2semidirect
13. #75982C2semidirect
14. #74987C2semidirect
17. #74157C2semidirect
91. #2581D3semidirect
107. C2×D3×Q8Q8semidirect
670. C3#35711semidirect

Maximal subgroups

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

subgroupindex
C2×#55722
C2×Q8×Dic62
C3×#25812
#824912
#824792
#782092
#781812
#771912
#771702
#766142
#765082
#762232
#759822
#749872
#749832
#741942
#741572
#357113