Finite Group Databasebeta

← #10827628

Subgroups — #10827628

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5541C211yesmaximal, characteristic
2C2×#5120C211yesmaximal, characteristic, complemented
3#82349C211yesmaximal, characteristic
4C2×#4739C211yesmaximal, characteristic, complemented
5C3×#1870C211yesmaximal, characteristic
6#71989C211yesmaximal, characteristic, complemented
7#69964C212yesmaximal, complemented
8#69930C212yesmaximal, complemented
9#69926C211yesmaximal, characteristic, complemented
10#68690C212yesmaximal, complemented
11#68685C212yesmaximal, complemented
12#23983—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#935—21no—
17C2×#926C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C2×#924C2212yes—
21C2×#924C2211yescharacteristic
22C22×Dic12C2212yes—
23#5541—24no—
24#5541—24no—
25C2×#889—21no—
26C2×#879—21no—
27C2×#877—21no—
28C2×#870C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2212yes—
32#5220—21no—
33#5212C2211yescharacteristic
34#5120—24no—
35C2×#816C2212yes—
36C2×#816C2212yes—
37C2×#816C2212yes—
38C2×#816C2212yes—
39#4979—21no—
40#4947—21no—
41#4946—21no—
42#4929C2212yes—
43#4739—24no—
44C3×#336C2212yes—
45#4551C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#552—31no—
49C2×#451—31no—
50C2×#419—31no—
51#2409—31no—
52#2026—32no—
53#2001—32no—
54#1894—31no—
55#1878—32no—
56#1872—32no—
57#1870D311yescharacteristic
58#1825—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—21no—
63C23×Dic3—21no—
64C22×Dic6—22no—
65C22×Dic6—24no—
66C22×Dic6—22no—
67C22×Dic6—21no—
68C22×Dic6—42no—
69C22×Dic6C2311yescharacteristic
70C22×Dic6C2312yes—
71C2×C3×M16—22no—
72C2×C3×M16—42no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C22⋊C4D411yescharacteristic
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#217—21no—
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—21no—
85C2×#217—22no—
86#935—28no—
87#926—22no—
88#926D414yes—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—22no—
97#924—22no—
98#924—22no—
99#924—24no—
100#924—42no—
101#924D414yes—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—22no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×Dic12—24no—
111C2×Dic12—24no—
112C2×Dic12—42no—
113#889—28no—
114#879—28no—
115#877—28no—
116#870—22no—
117#870—22no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×C22⋊C8—22no—
121C3×C22⋊C8—42no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124C3⋊#95—21no—
125#816—24no—
126#816—24no—
127#816—24no—
128#816—42no—
129C23×Q8—31no—
130C22×Q16—32no—
131C22×M16C2×D311yescharacteristic
132#552—64no—
133#552—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#125—61no—
137C2×#123—61no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—32no—
141C2×C4×Q8—31no—
142C2×C4×D4C2×D311yescharacteristic
143C2×#117—31no—
144C2×#117—61no—
145C22×C4⋊C4—31no—
146#451—64no—
147#419—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×C22⋊C8C2×D311yescharacteristic
153C2×C22⋊C8C2×D312yes—
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336C2×D312yes—
164C24×C3C2×D411yescharacteristic
165C24×C3C2×D411yescharacteristic
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—41no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—44no—
190C22×Dic3—42no—
191C22×Dic3—42no—
192C22×Dic3—42no—
193C2×Dic6—28no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—21no—
201C2×Dic6—22no—
202C2×Dic6—48no—
203C2×Dic6—44no—
204C2×Dic6—44no—
205C2×Dic6—42no—
206C2×Dic6C2×D412yes—
207C2×Dic6#4114yes—
208C3×M16—44no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C3×C4⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C42—21no—
219C3×C42—21no—
220#217—24no—
221#217—24no—
222#217—24no—
223#217—44no—
224#217—42no—
225C4⋊Dic3—24no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3C2×D412yes—
232C4⋊Dic3C2×D412yes—
233C4⋊Dic3C2×D412yes—
234Dic3⋊C4—24no—
235Dic3⋊C4—24no—
236Dic3⋊C4—24no—
237Dic3⋊C4—24no—
238Dic3⋊C4—24no—
239Dic3⋊C4—44no—
240C4×Dic3—24no—
241C4×Dic3—24no—
242Dic12—48no—
243C22×Q8—31no—
244C22×Q8—32no—
245C22×Q8—64no—
246C22×Q8—61no—
247C22×Q8—62no—
248C22×Q8—62no—
249C22×Q8—122no—
250C22×D4D1211yescharacteristic
251C23×C4—61no—
252C23×C4—61no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255C2×Q16—64no—
256C2×Q16—64no—
257C2×Q16—122no—
258C2×M16—22no—
259C2×M16—42no—
260C22×C8C22×D312yes—
261C22×C8C22×D312yes—
262#126—68no—
263#125—62no—
264#125—62no—
265#125—68no—
266#123—68no—
267C22⋊Q8—34no—
268C22⋊Q8—62no—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271C22⋊Q8—64no—
272C22⋊Q8—122no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—22no—
276C4×D4—22no—
277#117—34no—
278#117—68no—
279#117—62no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4D1211yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4D1211yescharacteristic
300C2×C22⋊C4D1211yescharacteristic
301C2×C42—61no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304#103—64no—
305#103—64no—
306#103—64no—
307#103—122no—
308C22⋊C8—22no—
309C22⋊C8—42no—
310#95—61no—
311#95—62no—
312#95—62no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3#12012yes—
322C23×C3#13712yes—
323C23×C3C22×D411yescharacteristic
324C3×D4—44no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4#12012yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4112yes—
343C2×C3×C4H(5,𝔽2)12yes—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—24no—
349C2×Dic3—48no—
350C2×Dic3—48no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362Dic6—24no—
363Dic6—24no—
364Dic6—24no—
365Dic6—48no—
366Dic6—41no—
367Dic6—41no—
368Dic6—42no—
369Dic6—42no—
370C3×C8—42no—
371C3×C8—42no—
372C24C2×D1211yescharacteristic
373C24C2×D1211yescharacteristic
374C2×Q8—32no—
375C2×Q8—34no—
376C2×Q8—68no—
377C2×Q8—61no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—62no—
382C2×Q8—62no—
383C2×Q8—62no—
384C2×Q8—62no—
385C2×Q8—124no—
386C2×Q8—128no—
387C2×Q8—124no—
388C2×Q8—122no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—41no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—64no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—62no—
405C22×C4—124no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4C2×D1211yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4C2×D1211yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4D3×D412yes—
414C22×C4D3×D412yes—
415C22×C4C23×D311yescharacteristic
416Q16—128no—
417M16—44no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—22no—
425C4⋊C4—32no—
426C4⋊C4—32no—
427C4⋊C4—32no—
428C4⋊C4—64no—
429C4⋊C4—64no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—124no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—64no—
443C22⋊C4—64no—
444C22⋊C4—64no—
445C22⋊C4—124no—
446C22⋊C4—122no—
447C42—21no—
448C42—21no—
449C42—64no—
450C42—64no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#54612yes—
463C22×C3C2×#13711yescharacteristic
464C22×C3#57511yescharacteristic
465C22×C3#57812yes—
466C3×C4—22no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4#52511yescharacteristic
471C3×C4#53511yescharacteristic
472C3×C4#54612yes—
473Dic3—44no—
474Dic3—42no—
475Dic3—42no—
476Dic3—42no—
477Dic3—42no—
478Dic3—82no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23#88212yes—
488C23#90912yes—
489C23C22×D1211yescharacteristic
490Q8—64no—
491Q8—64no—
492Q8—64no—
493Q8—128no—
494Q8—121no—
495Q8—122no—
496Q8—121no—
497Q8—122no—
498D4—44no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—128no—
519C2×C4—128no—
520C2×C4—122no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4#88212yes—
532C2×C4C2×D3×D411yescharacteristic
533C2×C4C2×D3×D411yescharacteristic
534C2×C4#100412yes—
535C2×C4D3×#4112yes—
536C8—42no—
537C8—42no—
538C2×C3—42no—
539C2×C3—42no—
540C2×C3#240411yescharacteristic
541C2×C3#300711yescharacteristic
542C2×C3#301111yescharacteristic
543C2×C3#331212yes—
544C2×C3#332712yes—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—41no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22C2×#88211yescharacteristic
556C22#555412yes—
557C22C2×#90911yescharacteristic
558C22#574811yescharacteristic
559C22#575312yes—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—122no—
565C4—122no—
566C4—122no—
567C4—124no—
568C4—122no—
569C4—242no—
570C4#559211yescharacteristic
571C4#561012yes—
572C4#565611yescharacteristic
573C3#2398311yescharacteristic, complemented, Sylow
574C2—42no—
575C2—42no—
576C2#7488511yescharacteristic
577C2#7864011yescharacteristic
578C2#7867911yescharacteristic
579C2#7951112yes—
580C2#7956212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1870C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#1870D3
hypocenterC3#23983
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. C2×#5120C2semidirect
4. C2×#4739C2semidirect
6. #71989C2semidirect
7. #69964C2semidirect
8. #69930C2semidirect
9. #69926C2semidirect
10. #68690C2semidirect
11. #68685C2semidirect
573. C3#23983semidirect

Maximal subgroups

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

subgroupindex
C2×#55412
C2×#51202
#823492
C2×#47392
C3×#18702
#719892
#699642
#699302
#699262
#686902
#686852
#239833