Finite Group Databasebeta

← #11166378

Subgroups — #11166378

Order 768.

1980 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#552C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3#83093C214yesmaximal, complemented
4#83055C212yesmaximal, complemented
5#83044C212yesmaximal, complemented
6#83037C212yesmaximal, complemented
7#82871C214yesmaximal
8#82787C211yesmaximal, characteristic, complemented
9#82775C211yesmaximal, characteristic, complemented
10#82772C212yesmaximal, complemented
11#82470C212yesmaximal, complemented
12#82373C211yesmaximal, characteristic
13#82346C211yesmaximal, characteristic
14#82262C212yesmaximal, complemented
15#82261C212yesmaximal, complemented
16#82239C211yesmaximal, characteristic, complemented
17#82171C212yesmaximal, complemented
18#42964—31nomaximal, Sylow
19C3×#552—28no—
20C2×C3×#126C2212yes—
21C2×C3×#125C2211yescharacteristic
22C2×C3×#125C2212yes—
23C2×C3×#123C2212yes—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117C2212yes—
29C2×C3×#117C2211yescharacteristic
30C22×C3×C4⋊C4C2211yescharacteristic
31C2×#935C2212yes—
32C2×#926C2211yescharacteristic
33C22×C4⋊Dic3C2211yescharacteristic
34C2×#924C2214yes—
35#5519—28no—
36C2×#890C2212yes—
37C2×#888C2212yes—
38C2×#878C2214yes—
39C2×#870C2212yes—
40C2×#869C2212yes—
41#5229C2212yes—
42#5229C2212yes—
43#5222C2212yes—
44#5212C2211yescharacteristic
45#5212C2212yes—
46#5210C2214yes—
47C4×#217C2211yescharacteristic
48C4×#217C2212yes—
49#4986C2212yes—
50#4986C2211yescharacteristic
51#4985C2214yes—
52#4984C2212yes—
53#4983C2214yes—
54#4982C2214yes—
55#4982C2214yes—
56#4981C2212yes—
57#4981C2212yes—
58#4981C2214yes—
59#4981C2212yes—
60#4981C2211yescharacteristic
61#4981C2212yes—
62#4979C2214yes—
63#4979C2214yes—
64#4977C2214yes—
65#4976C2214yes—
66#4975C2212yes—
67#4975C2214yes—
68Dic3×C4⋊C4C2212yes—
69#4972C2212yes—
70#4953C2214yes—
71#4953C2212yes—
72#4953C2211yescharacteristic
73#4952C2214yes—
74#4952C2212yes—
75#4951C2214yes—
76#4949C2211yescharacteristic
77#4946C2214yes—
78#4945C2214yes—
79#4943C2212yes—
80#4936C2212yes—
81#4933C2214yes—
82#4932C2212yes—
83C4×Dic3⋊C4C2212yes—
84#4929C2214yes—
85#4928C2212yes—
86C2×#552D311yescharacteristic
87C2×#544—31no—
88#2822—32no—
89#2698—34no—
90#2680—34no—
91#2660—32no—
92#2625—31no—
93#2621—32no—
94#2611—32no—
95#2552—32no—
96#2502—31no—
97#2489—32no—
98#2486—31no—
99#2451—32no—
100#2440—32no—
101#2395—31no—
102#2346—31no—
103C22×C3×Q8C2311yescharacteristic
104C22×C3×D4C2311yescharacteristic
105C23×C3×C4C2311yescharacteristic
106C23×C3×C4C2312yes—
107C23×C3×C4C2311yescharacteristic
108C23×Dic3C2312yes—
109C22×Dic6C2314yes—
110C3×#126—28no—
111C3×#125—24no—
112C3×#125—28no—
113C3×#123—28no—
114C3×C22⋊Q8—24no—
115C3×C22⋊Q8—24no—
116C3×C4×Q8—24no—
117C3×C4×D4—24no—
118C3×#117—28no—
119C3×#117D418yes—
120C2×C3×C4⋊C4—24no—
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C4⋊C4C2312yes—
124C2×C3×C4⋊C4C2312yes—
125C2×C3×C4⋊C4C2312yes—
126C2×C3×C4⋊C4C2312yes—
127C2×C3×C4⋊C4C2312yes—
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C4⋊C4C2311yescharacteristic
130C2×C3×C4⋊C4C2311yescharacteristic
131C2×C3×C22⋊C4C2311yescharacteristic
132C2×C3×C22⋊C4C2312yes—
133C2×C3×C22⋊C4C2312yes—
134C2×C3×C22⋊C4C2311yescharacteristic
135C2×C3×C22⋊C4C2312yes—
136C2×C3×C42C2311yescharacteristic
137C2×C3×C42C2312yes—
138C2×C3×C42C2312yes—
139C2×#217C2312yes—
140C2×#217C2314yes—
141C2×#217C2312yes—
142C2×#217C2312yes—
143C2×#217C2312yes—
144C2×#217C2312yes—
145#935—44no—
146#926—24no—
147C2×C4⋊Dic3—24no—
148C2×C4⋊Dic3C2312yes—
149C2×C4⋊Dic3C2312yes—
150C2×C4⋊Dic3C2311yescharacteristic
151C2×C4⋊Dic3C2311yescharacteristic
152C2×C4⋊Dic3C2312yes—
153C2×C4⋊Dic3C2314yes—
154C2×C4⋊Dic3C2314yes—
155#924—48no—
156C2×Dic3⋊C4C2312yes—
157C2×Dic3⋊C4C2314yes—
158C2×Dic3⋊C4C2314yes—
159C2×Dic3⋊C4C2314yes—
160C2×Dic3⋊C4C2312yes—
161C2×Dic3⋊C4C2314yes—
162C2×Dic3⋊C4C2314yes—
163C2×Dic3⋊C4C2314yes—
164C2×C4×Dic3C2312yes—
165C2×C4×Dic3C2312yes—
166C2×C4×Dic3C2314yes—
167#890—28no—
168#888—28no—
169#878—216no—
170#870—28no—
171#869—28no—
172C3⋊#95C2314yes—
173C3⋊#95C2312yes—
174C3⋊#95C2314yes—
175C3⋊#95C2314yes—
176C3⋊#95C2312yes—
177C3⋊#95C2311yescharacteristic
178C3⋊#95C2314yes—
179C3⋊#95C2312yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2311yescharacteristic
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2312yes—
186C3⋊#95C2312yes—
187C3⋊#95C2314yes—
188#552—28no—
189#544—68no—
190C2×C4⋊Q8—32no—
191C2×C4⋊Q8—32no—
192C2×#126C2×D312yes—
193C2×#125—32no—
194C2×#125—32no—
195C2×#125C2×D312yes—
196C2×#125C2×D311yescharacteristic
197C2×#123—32no—
198C2×#123C2×D312yes—
199C2×C22⋊Q8—34no—
200C2×C22⋊Q8—34no—
201C2×C22⋊Q8C2×D311yescharacteristic
202C2×C22⋊Q8C2×D311yescharacteristic
203C2×C4×Q8C2×D311yescharacteristic
204C2×C4×D4C2×D311yescharacteristic
205C2×#117—31no—
206C2×#117C2×D312yes—
207C2×#117C2×D311yescharacteristic
208C22×C4⋊C4—31no—
209C22×C4⋊C4C2×D311yescharacteristic
210#400—31no—
211#400—32no—
212#400—34no—
213#399—31no—
214#398—32no—
215#398—32no—
216#398—34no—
217#398—34no—
218#398—34no—
219#398—32no—
220#398—32no—
221#398—34no—
222#398—31no—
223#397—32no—
224#397—31no—
225#397—34no—
226#397—34no—
227#397—32no—
228#395—34no—
229#395—34no—
230#394—32no—
231#391—32no—
232#391—32no—
233#391—34no—
234#389—32no—
235#389—34no—
236#389—32no—
237#389—32no—
238#388—34no—
239#388—34no—
240#387—34no—
241#385—34no—
242#385—34no—
243#384—34no—
244#384—32no—
245#382—34no—
246#382—32no—
247#381—32no—
248#380—31no—
249#380—32no—
250C4×C4⋊C4—32no—
251C4×C4⋊C4—32no—
252C4×C22⋊C4—31no—
253C4×C22⋊C4—32no—
254#376—32no—
255C24×C3C2411yescharacteristic
256C24×C3C2411yescharacteristic
257C2×C3×Q8—22no—
258C2×C3×Q8—24no—
259C2×C3×D4—22no—
260C2×C3×D4—22no—
261C2×C3×D4—41no—
262C22×C3×C4—24no—
263C22×C3×C4—24no—
264C22×C3×C4—24no—
265C22×C3×C4—22no—
266C22×C3×C4—24no—
267C22×C3×C4—42no—
268C22×C3×C4C2×D414yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2411yescharacteristic
271C22×C3×C4C2411yescharacteristic
272C22×C3×C4C2411yescharacteristic
273C22×C3×C4C2412yes—
274C22×C3×C4C2411yescharacteristic
275C22×C3×C4C2412yes—
276C22×C3×C4C2412yes—
277C22×C3×C4C2411yescharacteristic
278C22×Dic3—24no—
279C22×Dic3—42no—
280C22×Dic3—42no—
281C22×Dic3C2412yes—
282C22×Dic3C2414yes—
283C22×Dic3C2412yes—
284C22×Dic3C2412yes—
285C22×Dic3C2414yes—
286C22×Dic3C2412yes—
287C2×Dic6—28no—
288C2×Dic6—44no—
289C2×Dic6—44no—
290C3×C4⋊C4—22no—
291C3×C4⋊C4—24no—
292C3×C4⋊C4—24no—
293C3×C4⋊C4—24no—
294C3×C4⋊C4—22no—
295C3×C4⋊C4—24no—
296C3×C4⋊C4—24no—
297C3×C4⋊C4—24no—
298C3×C4⋊C4C2×D414yes—
299C3×C4⋊C4C2×D414yes—
300C3×C22⋊C4—22no—
301C3×C22⋊C4—24no—
302C3×C22⋊C4—22no—
303C3×C22⋊C4—24no—
304C3×C22⋊C4C2×D418yes—
305C3×C42—24no—
306C3×C42—22no—
307C3×C42C2×D418yes—
308#217—24no—
309#217—24no—
310#217—28no—
311#217—24no—
312#217—24no—
313#217—42no—
314C4⋊Dic3—22no—
315C4⋊Dic3—24no—
316C4⋊Dic3—28no—
317C4⋊Dic3—22no—
318C4⋊Dic3—24no—
319C4⋊Dic3—24no—
320C4⋊Dic3—44no—
321Dic3⋊C4—28no—
322Dic3⋊C4—28no—
323Dic3⋊C4—24no—
324Dic3⋊C4—42no—
325Dic3⋊C4—44no—
326Dic3⋊C4—44no—
327Dic3⋊C4—44no—
328Dic3⋊C4—44no—
329C4×Dic3—24no—
330C4×Dic3—24no—
331C4×Dic3—28no—
332C22×Q8—34no—
333C22×Q8C22×D311yescharacteristic
334C22×D4C22×D311yescharacteristic
335C23×C4—32no—
336C23×C4C22×D311yescharacteristic
337C23×C4C22×D312yes—
338C23×C4C22×D311yescharacteristic
339C4⋊Q8—68no—
340C4⋊Q8—68no—
341#126—28no—
342#125—24no—
343#125—28no—
344#125—68no—
345#125—68no—
346#123—28no—
347#123—124no—
348C22⋊Q8—24no—
349C22⋊Q8—24no—
350C22⋊Q8—616no—
351C22⋊Q8—128no—
352C4×Q8—24no—
353C4×D4—24no—
354#117—28no—
355#117—64no—
356#117#6718yes—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—31no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—34no—
363C2×C4⋊C4—34no—
364C2×C4⋊C4—34no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—34no—
368C2×C4⋊C4—34no—
369C2×C4⋊C4—34no—
370C2×C4⋊C4—34no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—64no—
374C2×C4⋊C4C22×D311yescharacteristic
375C2×C4⋊C4C22×D312yes—
376C2×C4⋊C4C22×D312yes—
377C2×C4⋊C4C22×D312yes—
378C2×C4⋊C4C22×D311yescharacteristic
379C2×C4⋊C4C22×D312yes—
380C2×C4⋊C4C22×D312yes—
381C2×C4⋊C4C22×D312yes—
382C2×C4⋊C4C22×D311yescharacteristic
383C2×C4⋊C4C22×D311yescharacteristic
384C2×C22⋊C4—32no—
385C2×C22⋊C4—32no—
386C2×C22⋊C4—34no—
387C2×C22⋊C4—32no—
388C2×C22⋊C4—32no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4C22×D311yescharacteristic
391C2×C22⋊C4C22×D312yes—
392C2×C22⋊C4C22×D311yescharacteristic
393C2×C22⋊C4C22×D312yes—
394C2×C22⋊C4C22×D312yes—
395C2×C42—32no—
396C2×C42—34no—
397C2×C42—32no—
398C2×C42C22×D312yes—
399C2×C42C22×D311yescharacteristic
400C2×C42C22×D312yes—
401#95—31no—
402#95—32no—
403#95—34no—
404#95—32no—
405#95—31no—
406#95—32no—
407#95—34no—
408#95—32no—
409#95—32no—
410#95—34no—
411#95—34no—
412#95—34no—
413#95—34no—
414#95—34no—
415#95—34no—
416#95—34no—
417C23×C3—21no—
418C23×C3—21no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—22no—
422C23×C3—22no—
423C23×C3—21no—
424C23×C3—21no—
425C23×C3C22×D412yes—
426C23×C3H(5,𝔽2)12yes—
427C23×C3C2511yescharacteristic
428C3×Q8—44no—
429C3×D4—44no—
430C2×C3×C4—22no—
431C2×C3×C4—24no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—28no—
435C2×C3×C4—21no—
436C2×C3×C4—22no—
437C2×C3×C4—21no—
438C2×C3×C4—21no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—21no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—24no—
445C2×C3×C4—22no—
446C2×C3×C4—21no—
447C2×C3×C4—48no—
448C2×C3×C4—44no—
449C2×C3×C4C22×D414yes—
450C2×C3×C4C22×D414yes—
451C2×C3×C4C22×D412yes—
452C2×C3×C4C22×D412yes—
453C2×C3×C42−1+414yes—
454C2×Dic3—22no—
455C2×Dic3—22no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3—24no—
459C2×Dic3—22no—
460C2×Dic3—24no—
461C2×Dic3—22no—
462C2×Dic3—24no—
463C2×Dic3—22no—
464C2×Dic3—24no—
465C2×Dic3—22no—
466C2×Dic3—24no—
467C2×Dic3—22no—
468C2×Dic3—22no—
469C2×Dic3—22no—
470C2×Dic3—24no—
471C2×Dic3—84no—
472C2×Dic32−1+414yes—
473Dic6—88no—
474C24C23×D311yescharacteristic
475C24C23×D311yescharacteristic
476C2×Q8—24no—
477C2×Q8—22no—
478C2×Q8—68no—
479C2×Q8—124no—
480C2×Q8—124no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—41no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—32no—
490C22×C4—34no—
491C22×C4—32no—
492C22×C4—32no—
493C22×C4—32no—
494C22×C4—34no—
495C22×C4—42no—
496C22×C4—64no—
497C22×C4—122no—
498C22×C4—122no—
499C22×C4C2×#6714yes—
500C22×C4C23×D312yes—
501C22×C4C23×D311yescharacteristic
502C22×C4C23×D311yescharacteristic
503C22×C4C23×D311yescharacteristic
504C22×C4C23×D312yes—
505C22×C4C23×D311yescharacteristic
506C22×C4C23×D312yes—
507C22×C4C23×D312yes—
508C22×C4C23×D311yescharacteristic
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—62no—
522C4⋊C4—62no—
523C4⋊C4—68no—
524C4⋊C4—68no—
525C4⋊C4—68no—
526C4⋊C4—124no—
527C4⋊C4—124no—
528C4⋊C4—124no—
529C4⋊C4—124no—
530C4⋊C4—124no—
531C4⋊C4—122no—
532C4⋊C4C2×#6714yes—
533C4⋊C4C2×#6714yes—
534C22⋊C4—22no—
535C22⋊C4—24no—
536C22⋊C4—22no—
537C22⋊C4—24no—
538C22⋊C4—64no—
539C22⋊C4—64no—
540C22⋊C4—64no—
541C22⋊C4—68no—
542C22⋊C4—64no—
543C22⋊C4—122no—
544C22⋊C4C2×#6718yes—
545C42—24no—
546C42—22no—
547C42—68no—
548C42—64no—
549C42—64no—
550C42C2×#6718yes—
551C22×C3—22no—
552C22×C3—22no—
553C22×C3—22no—
554C22×C3—22no—
555C22×C3—24no—
556C22×C3—41no—
557C22×C3—41no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3C23×D411yescharacteristic
562C22×C3C2×H(5,𝔽2)11yescharacteristic
563C22×C3C2×2−1+412yes—
564C22×C3C2×2−1+412yes—
565C22×C3#58511yescharacteristic
566C3×C4—22no—
567C3×C4—24no—
568C3×C4—24no—
569C3×C4—41no—
570C3×C4—42no—
571C3×C4—42no—
572C3×C4—41no—
573C3×C4—41no—
574C3×C4#53614yes—
575Dic3—24no—
576Dic3—42no—
577Dic3—44no—
578Dic3—42no—
579Dic3—42no—
580Dic3—44no—
581C23—21no—
582C23—22no—
583C23—22no—
584C23—21no—
585C23—21no—
586C23—22no—
587C23—22no—
588C23—21no—
589C23#100912yes—
590C23C22×#6712yes—
591C23C24×D311yescharacteristic
592Q8—44no—
593Q8—248no—
594D4—44no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—34no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—64no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—64no—
623C2×C4—64no—
624C2×C4—62no—
625C2×C4—64no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—64no—
632C2×C4—244no—
633C2×C4#101014yes—
634C2×C4C22×#6714yes—
635C2×C4C22×#6712yes—
636C2×C4C22×#6714yes—
637C2×C4C22×#6712yes—
638C2×C3—42no—
639C2×C3—42no—
640C2×C3C2×#53611yescharacteristic
641C2×C3#348811yescharacteristic
642C2×C3#349911yescharacteristic
643C2×C3#354412yes—
644C2×C3#356312yes—
645C22—24no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—41no—
655C22C2×#100911yescharacteristic
656C22C2×#101012yes—
657C22#596411yescharacteristic
658C22D3×2−1+412yes—
659C22C23×#6711yescharacteristic
660C4—22no—
661C4—24no—
662C4—24no—
663C4—41no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—41no—
668C4—64no—
669C4—124no—
670C4—122no—
671C4—124no—
672C4—122no—
673C4—122no—
674C4#582514yes—
675C3#4296411yescharacteristic, complemented, Sylow
676C2—42no—
677C2—42no—
678C2#8684712yes—
679C2#8686912yes—
680C2C2×#582511yescharacteristic
681C2#8727511yescharacteristic
682C2#8729211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#552C2
prime radicalC22×C3×D4C23
hypercenterC2×#552D3
hypocenterC3#42964
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
3. #83093C2semidirect
4. #83055C2semidirect
5. #83044C2semidirect
6. #83037C2semidirect
8. #82787C2semidirect
9. #82775C2semidirect
10. #82772C2semidirect
11. #82470C2semidirect
14. #82262C2semidirect
15. #82261C2semidirect
16. #82239C2semidirect
17. #82171C2semidirect
675. C3#42964semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×#55192
#830932
#830552
#830442
#830372
#828712
#827872
#827752
#827722
#824702
#823732
#823462
#822622
#822612
#822392
#821712
#429643