Finite Group Databasebeta

← #10982990

Subgroups — #10982990

Order 768.

2314 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic, complemented
2C2×#5504C211yesmaximal, characteristic, complemented
3C3×#2487C211yesmaximal, characteristic, complemented
4#82400C211yesmaximal, characteristic
5#82179C211yesmaximal, characteristic, complemented
6#78191C212yesmaximal, complemented
7#78131C212yesmaximal, complemented
8#78053C212yesmaximal, complemented
9#78045C212yesmaximal, complemented
10#76223C212yesmaximal, complemented
11#76205C212yesmaximal, complemented
12#74717C214yesmaximal, complemented
13#74435C214yesmaximal, complemented
14#74020C212yesmaximal, complemented
15#74012C212yesmaximal, complemented
16#73950C211yesmaximal, characteristic, complemented
17#73881C211yesmaximal, characteristic, complemented
18#35759—31nomaximal, Sylow
19C22×#235C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C2×#936C2212yes—
23C2×C4×#67C2212yes—
24C2×C4×#67C2214yes—
25C2×#926C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C2×#924C2211yescharacteristic
28C2×#924C2211yescharacteristic
29#5542—28no—
30C2×#897C2212yes—
31#5504—28no—
32C2×#893C2212yes—
33C2×#892C2212yes—
34C2×D3×C4⋊C4C2212yes—
35C2×#887C2212yes—
36C2×#887C2212yes—
37C2×#879C2212yes—
38C2×#879C2212yes—
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#400C2212yes—
42C3×#399C2212yes—
43C3×#389C2212yes—
44C3×#386C2212yes—
45C3×#385C2212yes—
46C3×#380C2212yes—
47C3×#380C2211yescharacteristic
48#5223C2212yes—
49#5211C2214yes—
50C4×#217C2211yescharacteristic
51#4994C2214yes—
52#4991C2211yescharacteristic
53#4989C2214yes—
54#4987C2212yes—
55#4985C2212yes—
56#4985C2212yes—
57#4980C2214yes—
58#4979C2211yescharacteristic
59#4977C2212yes—
60Dic3×C4⋊C4C2212yes—
61#4973C2212yes—
62#4972C2212yes—
63#4972C2214yes—
64#4961C2214yes—
65#4960C2211yescharacteristic
66#4959C2214yes—
67#4959C2212yes—
68#4956C2214yes—
69#4951C2214yes—
70#4947C2211yescharacteristic
71#4945C2214yes—
72#4944C2214yes—
73C4×(C2×D3)⋊C4C2212yes—
74#4675C2214yes—
75#4673C2214yes—
76#4672C2212yes—
77#4666C2214yes—
78#4666C2214yes—
79#4656C2212yes—
80#4654C2214yes—
81#4652C2214yes—
82#4650C2212yes—
83#4648C2212yes—
84#4647C2214yes—
85#4646C2214yes—
86C2×#556—31no—
87C2×#520—31no—
88#2636—32no—
89#2616—31no—
90#2604—32no—
91#2581—32no—
92#2503—31no—
93#2487D311yescharacteristic, complemented
94#2466—34no—
95#2459—34no—
96#2439—32no—
97#2415—32no—
98#2398—32no—
99#2388—31no—
100#2360—32no—
101#2352—32no—
102#2346—31no—
103C22×C3×D4C2311yescharacteristic
104C23×C3×C4C2311yescharacteristic
105C23×C3×C4C2311yescharacteristic
106C23×C3×C4C2312yes—
107C22×#67C2312yes—
108C22×#67C2312yes—
109C2×#235—24no—
110C2×#235—28no—
111C22×D12C2311yescharacteristic
112C22×C4×D3C2312yes—
113C22×C4×D3C2314yes—
114C22×Dic6C2311yescharacteristic
115C22×Dic6C2311yescharacteristic
116C22×Dic6C2312yes—
117C3×C4×D4—24no—
118C2×C3×C4⋊C4—24no—
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C4⋊C4C2314yes—
124C2×C3×C22⋊C4C2311yescharacteristic
125C2×C3×C22⋊C4C2311yescharacteristic
126C2×C3×C22⋊C4C2314yes—
127C2×C3×C22⋊C4C2312yes—
128C2×C3×C42C2311yescharacteristic
129C2×#217C2312yes—
130C2×#217C2312yes—
131C2×#217C2312yes—
132C2×#217C2312yes—
133C2×#217C2312yes—
134C2×#217C2312yes—
135#936—28no—
136C4×#67—28no—
137C4×#67—216no—
138C2×(C2×D3)⋊C4C2312yes—
139C2×(C2×D3)⋊C4C2312yes—
140C2×(C2×D3)⋊C4C2314yes—
141C2×(C2×D3)⋊C4C2314yes—
142C2×(C2×D3)⋊C4C2314yes—
143C2×(C2×D3)⋊C4C2312yes—
144#926—24no—
145#926D418yescomplemented
146C2×C4⋊Dic3C2314yes—
147C2×C4⋊Dic3C2312yes—
148C2×C4⋊Dic3C2311yescharacteristic
149C2×C4⋊Dic3C2311yescharacteristic
150C2×C4⋊Dic3C2311yescharacteristic
151#924—24no—
152#924—24no—
153C2×Dic3⋊C4C2312yes—
154C2×Dic3⋊C4C2312yes—
155C2×Dic3⋊C4C2314yes—
156C2×Dic3⋊C4C2312yes—
157C2×Dic3⋊C4C2312yes—
158C2×Dic3⋊C4C2314yes—
159C2×C4×Dic3C2314yes—
160C2×C4×Dic3C2312yes—
161C2×C4×Dic3C2312yes—
162C2×C4×Dic3C2312yes—
163C2×C4×Dic3C2314yes—
164C2×C4×Dic3C2312yes—
165#897—28no—
166#893—28no—
167#892—28no—
168D3×C4⋊C4—28no—
169#887—28no—
170#887—28no—
171#879—28no—
172#879—28no—
173#869—24no—
174C4×Dic6—24no—
175C3×#95C2314yes—
176C3×#95C2314yes—
177C3×#95C2312yes—
178C3⋊#95C2312yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2311yescharacteristic
183C3⋊#95C2314yes—
184C3⋊#95C2311yescharacteristic
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2312yes—
189C22×#41—31no—
190#556—68no—
191C2×C4⋊Q8—31no—
192C2×#126—32no—
193C2×#126—32no—
194C2×#124—32no—
195C2×C22⋊Q8—31no—
196C2×C22⋊Q8—31no—
197C2×C22⋊Q8—32no—
198#520—68no—
199C2×C4×Q8—32no—
200C2×C4×Q8—32no—
201C2×C4×Q8—31no—
202C2×C4×D4—32no—
203C2×C4×D4—34no—
204C2×C4×D4—32no—
205C2×C4×D4C2×D311yescharacteristic
206C2×#117—31no—
207C2×#117—31no—
208C2×#117—32no—
209C22×C4⋊C4—32no—
210C22×C4⋊C4C2×D311yescharacteristic
211#400—32no—
212#400—32no—
213#400C2×D312yes—
214#399C2×D312yes—
215#397—34no—
216#396—34no—
217#395—31no—
218#395—32no—
219#394—31no—
220#394—34no—
221#394—34no—
222#394—32no—
223#394—32no—
224#393—31no—
225#393—34no—
226#392—32no—
227#392—31no—
228#391—32no—
229#391—34no—
230#389—32no—
231#389—34no—
232#389—34no—
233#389C2×D312yes—
234#388—34no—
235#388—34no—
236#388—34no—
237#387—34no—
238#387—34no—
239#387—34no—
240#386—34no—
241#386C2×D312yes—
242#385—34no—
243#385C2×D312yes—
244#382—32no—
245#382—32no—
246#381—34no—
247#380—32no—
248#380C2×D311yescharacteristic
249#380C2×D312yes—
250C4×C4⋊C4—32no—
251C4×C22⋊C4—31no—
252C4×C22⋊C4—32no—
253C4×C22⋊C4—34no—
254C4×C22⋊C4—34no—
255#376—32no—
256C24×C3C2411yescharacteristic
257C24×C3C2411yescharacteristic
258C23×D3C2412yes—
259C2×C3×D4—22no—
260C2×C3×D4—22no—
261C2×C3×D4—41no—
262C22×C3×C4—24no—
263C22×C3×C4—24no—
264C22×C3×C4—22no—
265C22×C3×C4—24no—
266C22×C3×C4—44no—
267C22×C3×C4C2×D414yes—
268C22×C3×C4C2414yes—
269C22×C3×C4C2414yes—
270C22×C3×C4C2411yescharacteristic
271C22×C3×C4C2411yescharacteristic
272C22×C3×C4C2411yescharacteristic
273C22×C3×C4C2411yescharacteristic
274C22×C3×C4C2411yescharacteristic
275C2×#67—24no—
276C2×#67—24no—
277C2×#67—24no—
278C2×#67—24no—
279C2×#67—42no—
280C2×#67—42no—
281C22×Dic3C2412yes—
282C22×Dic3C2412yes—
283C22×Dic3C2412yes—
284C22×Dic3C2412yes—
285C22×Dic3C2412yes—
286C22×Dic3C2412yes—
287C22×Dic3C2412yes—
288#235—232no—
289C2×D12—24no—
290C2×D12—22no—
291C2×C4×D3—24no—
292C2×C4×D3—24no—
293C2×C4×D3—28no—
294C2×C4×D3—44no—
295C2×C4×D3—44no—
296C2×C4×D3#4118yes—
297C2×Dic6—22no—
298C2×Dic6—24no—
299C2×Dic6—22no—
300C2×Dic6—24no—
301C2×Dic6—24no—
302C2×Dic6—24no—
303C2×Dic6—42no—
304C3×C4⋊C4—22no—
305C3×C4⋊C4—24no—
306C3×C4⋊C4—24no—
307C3×C4⋊C4—24no—
308C3×C4⋊C4—44no—
309C3×C22⋊C4—22no—
310C3×C22⋊C4—22no—
311C3×C22⋊C4—44no—
312C3×C22⋊C4—42no—
313C3×C42—22no—
314#217—24no—
315#217—24no—
316#217—24no—
317#217—24no—
318#217—42no—
319#217C2×D418yes—
320(C2×D3)⋊C4—28no—
321(C2×D3)⋊C4—24no—
322(C2×D3)⋊C4—24no—
323(C2×D3)⋊C4—28no—
324(C2×D3)⋊C4—44no—
325(C2×D3)⋊C4—42no—
326C4⋊Dic3—22no—
327C4⋊Dic3—24no—
328C4⋊Dic3—28no—
329C4⋊Dic3C2×D414yes—
330C4⋊Dic3C2×D414yes—
331Dic3⋊C4—24no—
332Dic3⋊C4—24no—
333Dic3⋊C4—24no—
334Dic3⋊C4—28no—
335Dic3⋊C4—24no—
336Dic3⋊C4—44no—
337C4×Dic3—24no—
338C4×Dic3—24no—
339C4×Dic3—24no—
340C4×Dic3—28no—
341C4×Dic3—44no—
342C4×Dic3C2×D418yes—
343C2×#41—68no—
344C2×#41—64no—
345C22×Q8—31no—
346C22×Q8—31no—
347C22×Q8—32no—
348C22×D4—32no—
349C22×D4—32no—
350C22×D4—31no—
351C22×D4C22×D311yescharacteristic
352C23×C4—32no—
353C23×C4—34no—
354C23×C4C22×D311yescharacteristic
355C23×C4C22×D311yescharacteristic
356C23×C4C22×D312yes—
357C4⋊Q8—64no—
358#126—68no—
359#126—68no—
360#124—68no—
361C22⋊Q8—68no—
362C22⋊Q8—64no—
363C22⋊Q8—64no—
364C4×Q8—64no—
365C4×Q8—68no—
366C4×Q8—68no—
367C4×D4—24no—
368C4×D4—68no—
369C4×D4—616no—
370C4×D4—68no—
371#117—38no—
372#117—68no—
373#117—64no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—31no—
376C2×C4⋊C4—31no—
377C2×C4⋊C4—34no—
378C2×C4⋊C4—31no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—32no—
381C2×C4⋊C4—32no—
382C2×C4⋊C4—32no—
383C2×C4⋊C4—34no—
384C2×C4⋊C4—32no—
385C2×C4⋊C4—34no—
386C2×C4⋊C4—68no—
387C2×C4⋊C4C22×D312yes—
388C2×C4⋊C4C22×D312yes—
389C2×C4⋊C4C22×D311yescharacteristic
390C2×C4⋊C4C22×D312yes—
391C2×C4⋊C4C22×D314yes—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—32no—
394C2×C22⋊C4—32no—
395C2×C22⋊C4—32no—
396C2×C22⋊C4—32no—
397C2×C22⋊C4—34no—
398C2×C22⋊C4—32no—
399C2×C22⋊C4—34no—
400C2×C22⋊C4—32no—
401C2×C22⋊C4—32no—
402C2×C22⋊C4—34no—
403C2×C22⋊C4—32no—
404C2×C22⋊C4C22×D311yescharacteristic
405C2×C22⋊C4C22×D311yescharacteristic
406C2×C22⋊C4C22×D312yes—
407C2×C22⋊C4C22×D314yes—
408C2×C42—32no—
409C2×C42—34no—
410C2×C42—32no—
411C2×C42—32no—
412C2×C42—32no—
413C2×C42—34no—
414C2×C42C22×D311yescharacteristic
415#95—31no—
416#95—34no—
417#95—34no—
418#95—34no—
419#95—31no—
420#95—32no—
421#95—32no—
422#95—34no—
423#95—34no—
424#95—34no—
425#95—34no—
426#95C22×D312yes—
427#95C22×D314yes—
428#95C22×D314yes—
429C23×C3—22no—
430C23×C3—21no—
431C23×C3—22no—
432C23×C3—21no—
433C23×C3—21no—
434C23×C3—22no—
435C23×C3—21no—
436C23×C3—22no—
437C23×C3—21no—
438C23×C3C22×D412yes—
439C23×C3C2511yescharacteristic
440C22×D3—22no—
441C22×D3—22no—
442C22×D3—22no—
443C22×D3—22no—
444C22×D3—22no—
445C22×D3—22no—
446C22×D3C2×#4114yes—
447C3×D4—44no—
448C2×C3×C4—21no—
449C2×C3×C4—22no—
450C2×C3×C4—21no—
451C2×C3×C4—28no—
452C2×C3×C4—21no—
453C2×C3×C4—24no—
454C2×C3×C4—21no—
455C2×C3×C4—22no—
456C2×C3×C4—24no—
457C2×C3×C4—24no—
458C2×C3×C4—24no—
459C2×C3×C4—22no—
460C2×C3×C4—24no—
461C2×C3×C4—24no—
462C2×C3×C4—44no—
463C2×C3×C4—84no—
464C2×C3×C4C22×D412yes—
465C2×C3×C4C22×D412yes—
466C2×C3×C4C2×#4114yes—
467C2×C3×C42−1+412yes—
468#67—216no—
469#67—48no—
470C2×Dic3—22no—
471C2×Dic3—22no—
472C2×Dic3—22no—
473C2×Dic3—24no—
474C2×Dic3—22no—
475C2×Dic3—22no—
476C2×Dic3—22no—
477C2×Dic3—22no—
478C2×Dic3—22no—
479C2×Dic3—24no—
480C2×Dic3—22no—
481C2×Dic3—22no—
482C2×Dic3—22no—
483C2×Dic3—22no—
484C2×Dic3—22no—
485C2×Dic3—22no—
486C2×Dic3C22×D414yes—
487C2×Dic3C22×D414yes—
488C2×Dic3C2×#4114yes—
489D12—28no—
490C4×D3—216no—
491C4×D3—88no—
492Dic6—28no—
493Dic6—48no—
494Dic6—44no—
495C24—32no—
496C24C23×D311yescharacteristic
497C24C23×D311yescharacteristic
498#41—632no—
499C2×Q8—64no—
500C2×Q8—64no—
501C2×Q8—62no—
502C2×Q8—62no—
503C2×Q8—64no—
504C2×Q8—64no—
505C2×Q8—122no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—41no—
509C2×D4—64no—
510C2×D4—64no—
511C2×D4—64no—
512C2×D4—64no—
513C2×D4—64no—
514C2×D4—62no—
515C2×D4—122no—
516C2×D4—122no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—22no—
520C22×C4—24no—
521C22×C4—32no—
522C22×C4—32no—
523C22×C4—32no—
524C22×C4—38no—
525C22×C4—32no—
526C22×C4—32no—
527C22×C4—32no—
528C22×C4—32no—
529C22×C4—44no—
530C22×C4—68no—
531C22×C4—64no—
532C22×C4—64no—
533C22×C4—124no—
534C22×C4—124no—
535C22×C4D3×D414yes—
536C22×C4C23×D314yes—
537C22×C4C23×D314yes—
538C22×C4C23×D311yescharacteristic
539C22×C4C23×D311yescharacteristic
540C22×C4C23×D311yescharacteristic
541C22×C4C23×D311yescharacteristic
542C22×C4C23×D311yescharacteristic
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—24no—
547C4⋊C4—34no—
548C4⋊C4—34no—
549C4⋊C4—44no—
550C4⋊C4—68no—
551C4⋊C4—64no—
552C4⋊C4—64no—
553C4⋊C4—64no—
554C4⋊C4—68no—
555C4⋊C4—64no—
556C4⋊C4—62no—
557C4⋊C4—64no—
558C4⋊C4—124no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—38no—
562C22⋊C4—42no—
563C22⋊C4—44no—
564C22⋊C4—68no—
565C22⋊C4—64no—
566C22⋊C4—68no—
567C22⋊C4—64no—
568C22⋊C4—64no—
569C22⋊C4—64no—
570C22⋊C4—64no—
571C22⋊C4—64no—
572C22⋊C4—124no—
573C22⋊C4—122no—
574C22⋊C4—122no—
575C42—22no—
576C42—38no—
577C42—68no—
578C42—64no—
579C42—64no—
580C42—64no—
581C42—124no—
582C22×C3—22no—
583C22×C3—22no—
584C22×C3—22no—
585C22×C3—42no—
586C22×C3—42no—
587C22×C3—41no—
588C22×C3—41no—
589C22×C3—42no—
590C22×C3—41no—
591C22×C3—42no—
592C22×C3C23×D411yescharacteristic
593C22×C3C22×#4112yes—
594C22×C3C2×2−1+411yescharacteristic
595C22×C3#58511yescharacteristic
596C22×C3#58512yes—
597C2×D3—24no—
598C2×D3—24no—
599C2×D3—24no—
600C2×D3—42no—
601C2×D3—42no—
602C2×D3—42no—
603C2×D3—42no—
604C3×C4—22no—
605C3×C4—22no—
606C3×C4—41no—
607C3×C4—44no—
608C3×C4—44no—
609C3×C4—41no—
610C3×C4#52514yes—
611Dic3—24no—
612Dic3—24no—
613Dic3—24no—
614Dic3—42no—
615Dic3—42no—
616Dic3—42no—
617Dic3—42no—
618C23—21no—
619C23—22no—
620C23—21no—
621C23—22no—
622C23—21no—
623C23—21no—
624C23—22no—
625C23—21no—
626C23—22no—
627C23—34no—
628C23—62no—
629C23—62no—
630C23—62no—
631C23—62no—
632C23—62no—
633C23—62no—
634C23C2×D3×D412yes—
635C23C24×D311yescharacteristic
636Q8—68no—
637Q8—128no—
638Q8—124no—
639D4—44no—
640D4—68no—
641D4—616no—
642D4—128no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—28no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—34no—
658C2×C4—34no—
659C2×C4—34no—
660C2×C4—44no—
661C2×C4—62no—
662C2×C4—62no—
663C2×C4—62no—
664C2×C4—62no—
665C2×C4—64no—
666C2×C4—616no—
667C2×C4—62no—
668C2×C4—62no—
669C2×C4—62no—
670C2×C4—62no—
671C2×C4—62no—
672C2×C4—62no—
673C2×C4—64no—
674C2×C4—62no—
675C2×C4—62no—
676C2×C4—62no—
677C2×C4—62no—
678C2×C4—84no—
679C2×C4—248no—
680C2×C4C2×D3×D412yes—
681C2×C4C2×D3×D412yes—
682C2×C4#100712yes—
683C2×C4D3×#4114yes—
684C2×C3—42no—
685C2×C3—81no—
686C2×C3C2×#52511yescharacteristic
687C2×C3#348911yescharacteristic
688C2×C3#349312yes—
689C2×C3#350011yescharacteristic
690C2×C3#350712yes—
691D3—44no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—42no—
696C22—42no—
697C22—41no—
698C22—42no—
699C22—42no—
700C22—41no—
701C22—41no—
702C22—64no—
703C22—64no—
704C22—64no—
705C22—122no—
706C22—122no—
707C22—122no—
708C22—122no—
709C22C22×D3×D411yescharacteristic
710C22C2×#100711yescharacteristic
711C22C2×D3×#4112yes—
712C22#596411yescharacteristic
713C22#596612yes—
714C4—22no—
715C4—22no—
716C4—44no—
717C4—41no—
718C4—44no—
719C4—41no—
720C4—64no—
721C4—64no—
722C4—64no—
723C4—122no—
724C4—122no—
725C4—122no—
726C4—122no—
727C4#565614yes—
728C3#3575911yescharacteristic, complemented, Sylow
729C2—42no—
730C2—81no—
731C2—124no—
732C2#8598511yescharacteristic
733C2C2×#565611yescharacteristic
734C2#8631711yescharacteristic
735C2#8691012yes—
736C2#8695012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2487C2
prime radical#82400C2
hypercenter#2487D3
hypocenterC3#35759
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2487

Complements

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

NH ≅ G/Ntype
1. C2×#5542C2semidirect
2. C2×#5504C2semidirect
3. C3×#2487C2semidirect
5. #82179C2semidirect
6. #78191C2semidirect
7. #78131C2semidirect
8. #78053C2semidirect
9. #78045C2semidirect
10. #76223C2semidirect
11. #76205C2semidirect
12. #74717C2semidirect
13. #74435C2semidirect
14. #74020C2semidirect
15. #74012C2semidirect
16. #73950C2semidirect
17. #73881C2semidirect
93. #2487D3semidirect
145. #926D4semidirect
728. C3#35759semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×#55042
C3×#24872
#824002
#821792
#781912
#781312
#780532
#780452
#762232
#762052
#747172
#744352
#740202
#740122
#739502
#738812
#357593