Finite Group Databasebeta

← #11069426

Subgroups — #11069426

Order 768.

2154 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5729C212yesmaximal, complemented
2C3×#2551C211yesmaximal, characteristic, complemented
3#83259C212yesmaximal
4#83242C212yesmaximal, complemented
5#83233C212yesmaximal, complemented
6#77261C214yesmaximal, complemented
7#77221C214yesmaximal, complemented
8#76250C212yesmaximal, complemented
9#76224C212yesmaximal, complemented
10#76030C212yesmaximal, complemented
11#76010C212yesmaximal
12#74356C211yesmaximal, characteristic, complemented
13#74324C211yesmaximal, characteristic
14C4⋊C4×D12C211yesmaximal, characteristic
15#73716C211yesmaximal, characteristic, complemented
16C4×#890C211yesmaximal, characteristic, complemented
17#73617C211yesmaximal, characteristic
18#39369—31nomaximal, Sylow
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2212yes—
21C2×#946C2214yes—
22C2×#946C2214yes—
23C2×#944C2214yes—
24#5729—216no—
25C2×#896C2214yes—
26C2×#892C2212yes—
27C2×D3×C4⋊C4C2212yes—
28C2×#890C2211yescharacteristic
29C2×#889C2214yes—
30C2×#888C2212yes—
31C2×#888C2212yes—
32C2×#888C2212yes—
33C2×C4×D12C2212yes—
34C2×C4×D12C2211yescharacteristic
35C2×#872C2212yes—
36C2×#870C2212yes—
37C3×#395C2214yes—
38C3×#389C2212yes—
39C3×#389C2212yes—
40C3×#384C2211yescharacteristic
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×C4×C4⋊C4C2211yescharacteristic
43#5230C2214yes—
44#4993C2214yes—
45#4993C2214yes—
46#4990C2214yes—
47#4989C2214yes—
48#4988C2211yescharacteristic
49#4988C2212yes—
50#4988C2211yescharacteristic
51#4987C2212yes—
52#4987C2212yes—
53#4984C2211yescharacteristic
54#4982C2214yes—
55#4979C2214yes—
56#4977C2211yescharacteristic
57#4976C2214yes—
58#4975C2212yes—
59#4975C2212yes—
60#4975C2212yes—
61#4975C2212yes—
62Dic3×C4⋊C4C2212yes—
63#4972C2211yescharacteristic
64#4972C2211yescharacteristic
65#4972C2211yescharacteristic
66#4972C2212yes—
67C4×(C2×D3)⋊C4C2212yes—
68#4935C2211yescharacteristic
69C4×C4⋊Dic3C2212yes—
70C4×C4⋊Dic3C2211yescharacteristic
71C4×Dic3⋊C4C2212yes—
72C42×Dic3C2211yescharacteristic
73#4928C2212yes—
74#4928C2212yes—
75#4676C2214yes—
76#4673C2214yes—
77#4669C2212yes—
78#4669C2212yes—
79#4668C2214yes—
80#4658C2214yes—
81#4657C2214yes—
82#4655C2212yes—
83#4654C2212yes—
84#4652C2214yes—
85C2×#544—32no—
86#2868—31no—
87#2720—34no—
88#2677—34no—
89#2628—32no—
90#2552—31no—
91#2551—32no—
92#2551D311yescharacteristic, complemented
93#2542—32no—
94#2506—32no—
95#2452—32no—
96#2441—32no—
97#2440—32no—
98#2354—31no—
99D4×C4⋊C4—31no—
100C4×#125—31no—
101#2293—31no—
102C22×C3×Q8C2314yes—
103C22×D12C2311yescharacteristic
104C22×C4×D3C2312yes—
105C22×C4×D3C2314yes—
106C22×Dic6C2314yes—
107C3×C4⋊Q8—28no—
108C3×C4⋊Q8—44no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2314yes—
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C42C2312yes—
119C2×C3×C42C2312yes—
120C2×C3×C42C2311yescharacteristic
121#946—216no—
122#946—48no—
123#944—216no—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2311yescharacteristic
139C2×C4⋊Dic3C2311yescharacteristic
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2312yes—
151C2×C4×Dic3C2314yes—
152C2×C4×Dic3C2311yescharacteristic
153C2×C4×Dic3C2311yescharacteristic
154C2×C4×Dic3C2311yescharacteristic
155C2×C4×Dic3C2311yescharacteristic
156C2×C4×Dic3C2311yescharacteristic
157C2×C4×Dic3C2312yes—
158C2×C4×Dic3C2311yescharacteristic
159#896—216no—
160#892—28no—
161D3×C4⋊C4—28no—
162#890D418yescomplemented
163#889—216no—
164#888—28no—
165#888—44no—
166#888—44no—
167C4×D12—44no—
168C4×D12Q818yescomplemented
169#872—28no—
170#870—28no—
171C3×#95C2314yes—
172C3×#95C2312yes—
173C3×#95C2312yes—
174C3⋊#95C2312yes—
175C3⋊#95C2312yes—
176C3⋊#95C2314yes—
177C3⋊#95C2312yes—
178C3⋊#95C2312yes—
179C3⋊#95C2312yes—
180C3⋊#95C2312yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2312yes—
184#544—616no—
185C2×C4⋊Q8—34no—
186C2×C4⋊Q8—32no—
187C2×C4⋊Q8—32no—
188C2×C4⋊Q8—32no—
189C2×C4⋊Q8C2×D312yes—
190C2×C4⋊Q8C2×D312yes—
191C2×#125—31no—
192C2×#125—34no—
193C2×#125—32no—
194C2×C22⋊Q8—34no—
195C2×C22⋊Q8—34no—
196C2×C22⋊Q8—34no—
197C2×C4×D4—31no—
198C2×C4×D4—32no—
199C2×#117—32no—
200C2×#117—32no—
201C22×C4⋊C4—32no—
202#400—34no—
203#399—34no—
204#399—34no—
205#398—34no—
206#398—34no—
207#396—34no—
208#395—34no—
209#395—34no—
210#395C2×D314yes—
211#393—34no—
212#390—32no—
213#390—31no—
214#390—31no—
215#389—31no—
216#389—31no—
217#389—32no—
218#389—31no—
219#389—31no—
220#389—34no—
221#389—32no—
222#389—32no—
223#389C2×D312yes—
224#389C2×D312yes—
225#388—34no—
226#388—32no—
227#388—32no—
228#387—32no—
229#387—32no—
230#387—34no—
231#385—34no—
232#385—34no—
233#384—32no—
234#384—32no—
235#384—32no—
236#384—32no—
237#384—31no—
238#384C2×D311yescharacteristic
239#382—31no—
240#380—32no—
241#380—32no—
242C4×C4⋊C4—32no—
243C4×C4⋊C4—31no—
244C4×C4⋊C4—32no—
245C4×C4⋊C4—32no—
246C4×C4⋊C4C2×D311yescharacteristic
247C4×C4⋊C4C2×D311yescharacteristic
248C4×C22⋊C4—32no—
249C43—31no—
250C23×D3C2412yes—
251C2×C3×Q8—28no—
252C2×C3×Q8—44no—
253C2×C3×Q8—44no—
254C22×C3×C4C2412yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2414yes—
257C22×C3×C4C2412yes—
258C22×C3×C4C2412yes—
259C22×C3×C4C2411yescharacteristic
260C22×C3×C4C2411yescharacteristic
261C22×C3×C4C2411yescharacteristic
262C22×Dic3C2412yes—
263C22×Dic3C2414yes—
264C22×Dic3C2411yescharacteristic
265C22×Dic3C2412yes—
266C22×Dic3C2411yescharacteristic
267C22×Dic3C2412yes—
268C22×Dic3C2411yescharacteristic
269C22×Dic3C2411yescharacteristic
270C2×D12—22no—
271C2×D12—22no—
272C2×D12C2×Q814yes—
273C2×C4×D3—24no—
274C2×C4×D3—24no—
275C2×C4×D3—28no—
276C2×C4×D3—44no—
277C2×C4×D3—44no—
278C2×C4×D3C2×Q818yes—
279C2×Dic6—28no—
280C2×Dic6—44no—
281C2×Dic6—44no—
282C3×C4⋊C4—24no—
283C3×C4⋊C4—28no—
284C3×C4⋊C4—24no—
285C3×C4⋊C4—22no—
286C3×C4⋊C4—22no—
287C3×C4⋊C4—22no—
288C3×C4⋊C4—42no—
289C3×C4⋊C4—44no—
290C3×C4⋊C4C2×D414yes—
291C3×C42—24no—
292C3×C42—24no—
293C3×C42C2×Q814yes—
294(C2×D3)⋊C4—24no—
295(C2×D3)⋊C4—24no—
296(C2×D3)⋊C4—28no—
297(C2×D3)⋊C4—24no—
298(C2×D3)⋊C4—24no—
299(C2×D3)⋊C4—44no—
300(C2×D3)⋊C4—44no—
301(C2×D3)⋊C4C2×Q818yes—
302C4⋊Dic3—24no—
303C4⋊Dic3—24no—
304C4⋊Dic3—22no—
305C4⋊Dic3—24no—
306C4⋊Dic3—42no—
307C4⋊Dic3C2×D414yes—
308C4⋊Dic3C2×D418yes—
309C4⋊Dic3C2×Q814yes—
310Dic3⋊C4—28no—
311Dic3⋊C4—28no—
312Dic3⋊C4—28no—
313Dic3⋊C4—24no—
314Dic3⋊C4—24no—
315Dic3⋊C4—28no—
316Dic3⋊C4—28no—
317Dic3⋊C4—24no—
318Dic3⋊C4—44no—
319Dic3⋊C4—44no—
320Dic3⋊C4C2×D418yes—
321C4×Dic3—22no—
322C4×Dic3—22no—
323C4×Dic3—22no—
324C4×Dic3—24no—
325C4×Dic3—28no—
326C4×Dic3—22no—
327C4×Dic3—22no—
328C4×Dic3C2×D414yes—
329C22×Q8—34no—
330C22×Q8C22×D314yes—
331C22×D4—31no—
332C23×C4—32no—
333C23×C4—34no—
334C4⋊Q8—28no—
335C4⋊Q8—44no—
336C4⋊Q8—68no—
337C4⋊Q8—616no—
338C4⋊Q8—124no—
339C4⋊Q8—124no—
340#125—38no—
341#125—68no—
342#125—616no—
343C22⋊Q8—616no—
344C22⋊Q8—616no—
345C22⋊Q8—128no—
346C4×D4—38no—
347C4×D4—124no—
348#117—68no—
349#117—68no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—34no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—34no—
361C2×C4⋊C4—34no—
362C2×C4⋊C4—34no—
363C2×C4⋊C4—34no—
364C2×C4⋊C4—31no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—34no—
367C2×C4⋊C4—32no—
368C2×C4⋊C4—34no—
369C2×C4⋊C4—68no—
370C2×C4⋊C4C22×D312yes—
371C2×C4⋊C4C22×D314yes—
372C2×C4⋊C4C22×D312yes—
373C2×C4⋊C4C22×D311yescharacteristic
374C2×C4⋊C4C22×D311yescharacteristic
375C2×C4⋊C4C22×D311yescharacteristic
376C2×C4⋊C4C22×D312yes—
377C2×C4⋊C4C22×D314yes—
378C2×C4⋊C4C22×D311yescharacteristic
379C2×C22⋊C4—32no—
380C2×C22⋊C4—32no—
381C2×C22⋊C4—34no—
382C2×C22⋊C4—32no—
383C2×C22⋊C4—32no—
384C2×C22⋊C4—32no—
385C2×C22⋊C4—34no—
386C2×C22⋊C4—34no—
387C2×C42—31no—
388C2×C42—34no—
389C2×C42—31no—
390C2×C42—31no—
391C2×C42—31no—
392C2×C42—31no—
393C2×C42—31no—
394C2×C42—32no—
395C2×C42C22×D312yes—
396C2×C42C22×D312yes—
397C2×C42C22×D311yescharacteristic
398#95—32no—
399#95—32no—
400#95—34no—
401#95—32no—
402#95—32no—
403#95—32no—
404#95—32no—
405#95—32no—
406#95—34no—
407#95—34no—
408#95C22×D312yes—
409#95C22×D312yes—
410#95C22×D314yes—
411C23×C3C2511yescharacteristic
412C22×D3—22no—
413C22×D3—22no—
414C22×D3—22no—
415C22×D3—22no—
416C22×D3—22no—
417C22×D3—22no—
418C22×D3C22×Q814yes—
419C3×Q8—88no—
420C2×C3×C4—21no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—21no—
424C2×C3×C4—24no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—24no—
428C2×C3×C4—22no—
429C2×C3×C4—24no—
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×C4C22×D414yes—
437C2×C3×C4C22×D412yes—
438C2×C3×C4C22×Q812yes—
439C2×C3×C4C22×Q812yes—
440C2×C3×C4C22×Q812yes—
441C2×C3×C4H(5,𝔽2)12yes—
442C2×C3×C42−1+412yes—
443C2×C3×C42−1+412yes—
444C2×Dic3—22no—
445C2×Dic3—21no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—21no—
449C2×Dic3—22no—
450C2×Dic3—21no—
451C2×Dic3—21no—
452C2×Dic3—21no—
453C2×Dic3—21no—
454C2×Dic3—22no—
455C2×Dic3—22no—
456C2×Dic3—24no—
457C2×Dic3—24no—
458C2×Dic3—22no—
459C2×Dic3—24no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3C22×D414yes—
463C2×Dic3C22×D412yes—
464C2×Dic3C22×D412yes—
465C2×Dic3C22×Q814yes—
466C2×Dic3H(5,𝔽2)12yes—
467C2×Dic32−1+412yes—
468D12—28no—
469C4×D3—216no—
470C4×D3—88no—
471Dic6—88no—
472C24—32no—
473C2×Q8—28no—
474C2×Q8—44no—
475C2×Q8—44no—
476C2×Q8—68no—
477C2×Q8—124no—
478C2×Q8—124no—
479C2×D4—34no—
480C2×D4—62no—
481C2×D4—62no—
482C22×C4—32no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—32no—
487C22×C4—31no—
488C22×C4—38no—
489C22×C4—32no—
490C22×C4—34no—
491C22×C4—64no—
492C22×C4—64no—
493C22×C4—68no—
494C22×C4—124no—
495C22×C4—124no—
496C22×C4C23×D312yes—
497C22×C4C23×D312yes—
498C22×C4C23×D314yes—
499C22×C4C23×D312yes—
500C22×C4C23×D312yes—
501C22×C4C23×D311yescharacteristic
502C22×C4C23×D311yescharacteristic
503C22×C4C23×D311yescharacteristic
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—34no—
511C4⋊C4—38no—
512C4⋊C4—38no—
513C4⋊C4—34no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—68no—
519C4⋊C4—68no—
520C4⋊C4—68no—
521C4⋊C4—64no—
522C4⋊C4—68no—
523C4⋊C4—64no—
524C4⋊C4—68no—
525C4⋊C4—64no—
526C4⋊C4—64no—
527C4⋊C4—62no—
528C4⋊C4—124no—
529C4⋊C4—124no—
530C4⋊C4—122no—
531C4⋊C4D3×D414yes—
532C22⋊C4—38no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4—68no—
536C22⋊C4—64no—
537C22⋊C4—64no—
538C22⋊C4—124no—
539C22⋊C4—124no—
540C42—24no—
541C42—24no—
542C42—34no—
543C42—68no—
544C42—62no—
545C42—62no—
546C42—62no—
547C42—62no—
548C42—62no—
549C42—64no—
550C42D3×Q814yes—
551C22×C3C23×D411yescharacteristic
552C22×C3C23×Q811yescharacteristic
553C22×C3C2×H(5,𝔽2)11yescharacteristic
554C22×C3C2×H(5,𝔽2)11yescharacteristic
555C22×C3C2×2−1+411yescharacteristic
556C22×C3C2×2−1+411yescharacteristic
557C22×C3C2×2−1+411yescharacteristic
558C2×D3—24no—
559C2×D3—24no—
560C2×D3—24no—
561C2×D3—42no—
562C2×D3—42no—
563C2×D3—42no—
564C2×D3—42no—
565C3×C4—24no—
566C3×C4—22no—
567C3×C4—44no—
568C3×C4—42no—
569C3×C4—42no—
570C3×C4—42no—
571C3×C4#54414yescomplemented
572C3×C4D4×Q814yes—
573Dic3—24no—
574Dic3—22no—
575Dic3—22no—
576Dic3—24no—
577Dic3—22no—
578Dic3—22no—
579Dic3—44no—
580Dic3—42no—
581C23—34no—
582C23—62no—
583C23—62no—
584C23—62no—
585C23—62no—
586C23—62no—
587C23—62no—
588C23C24×D311yescharacteristic
589Q8—88no—
590Q8—248no—
591D4—68no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—32no—
609C2×C4—32no—
610C2×C4—34no—
611C2×C4—34no—
612C2×C4—32no—
613C2×C4—32no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—62no—
621C2×C4—61no—
622C2×C4—62no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—62no—
626C2×C4—616no—
627C2×C4—64no—
628C2×C4—64no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—64no—
633C2×C4—248no—
634C2×C4C2×D3×D414yes—
635C2×C4C2×D3×D412yes—
636C2×C4#100412yes—
637C2×C4C2×D3×Q812yes—
638C2×C4C2×D3×Q812yes—
639C2×C4C2×D3×Q812yes—
640C2×C4#100712yes—
641C2×C4#100712yes—
642C2×C3C2×#54411yescharacteristic
643C2×C3C2×D4×Q811yescharacteristic
644C2×C3#349811yescharacteristic
645C2×C3#349911yescharacteristic
646C2×C3#350311yescharacteristic
647C2×C3#354411yescharacteristic
648C2×C3#355211yescharacteristic
649D3—44no—
650C22—64no—
651C22—64no—
652C22—64no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22C22×D3×D411yescharacteristic
658C22C2×#100411yescharacteristic
659C22C22×D3×Q811yescharacteristic
660C22C2×#100711yescharacteristic
661C22C2×#100711yescharacteristic
662C22D3×H(5,𝔽2)11yescharacteristic
663C22D3×2−1+411yescharacteristic
664C4—22no—
665C4—24no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—62no—
671C4—62no—
672C4—64no—
673C4—62no—
674C4—64no—
675C4—62no—
676C4—122no—
677C4—124no—
678C4#572714yes—
679C4#572914yescomplemented
680C3#3936911yescharacteristic, complemented, Sylow
681C2—124no—
682C2C2×#572911yescharacteristic
683C2C2×#572711yescharacteristic
684C2#8658611yescharacteristic
685C2#8660811yescharacteristic
686C2#8674611yescharacteristic
687C2#8676411yescharacteristic
688C2#8704711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2551C2
prime radicalC22×D12C23
hypercenter#2551D3
hypocenterC3#39369
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2551

Complements

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

NH ≅ G/Ntype
1. C2×#5729C2semidirect
2. C3×#2551C2semidirect
4. #83242C2semidirect
5. #83233C2semidirect
6. #77261C2semidirect
7. #77221C2semidirect
8. #76250C2semidirect
9. #76224C2semidirect
10. #76030C2semidirect
12. #74356C2semidirect
15. #73716C2semidirect
16. C4×#890C2semidirect
92. #2551D3semidirect
162. #890D4semidirect
168. C4×D12Q8semidirect
571. C3×C4#544semidirect
679. C4#5729semidirect
680. C3#39369semidirect

Maximal subgroups

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

subgroupindex
C2×#57292
C3×#25512
#832592
#832422
#832332
#772612
#772212
#762502
#762242
#760302
#760102
#743562
#743242
C4⋊C4×D122
#737162
C4×#8902
#736172
#393693