Finite Group Databasebeta

← #10848849

Subgroups — #10848849

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic
2C2×#5093C211yesmaximal, characteristic, complemented
3C2×#5060C211yesmaximal, characteristic, complemented
4#82262C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#71725C211yesmaximal, characteristic, complemented
7#71700C212yesmaximal, complemented
8#71685C212yesmaximal
9#71632C212yesmaximal, complemented
10#71612C212yesmaximal, complemented
11#71582C211yesmaximal, characteristic, complemented
12#25613—31nomaximal, Sylow
13C3×#542—24no—
14C3×#542—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#124—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C22×#216C2212yes—
27C22×#214C2212yes—
28C2×#924C2212yes—
29C2×#869C2211yescharacteristic
30C2×#835C2212yes—
31C2×#832C2212yes—
32C2×#830C2212yes—
33#5093—24no—
34#5060—24no—
35#4981—21no—
36#4981—21no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39#4953—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42#4929C2212yes—
43C2×#803C2211yescharacteristic
44C2×C4×C3⋊C8C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#542D311yescharacteristic
49C2×#497—31no—
50C2×#487—31no—
51#2621—31no—
52#2001—32no—
53#2000—32no—
54#1993—31no—
55#1958—31no—
56#1878—32no—
57#1877—32no—
58#1750—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C22×Dic6C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yes—
68C3×#126—28no—
69C3×#124—28no—
70C3×#124—22no—
71C3×#124D414yes—
72C3×#123—28no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×#121—22no—
77C3×#121—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—22no—
102C2×#216—24no—
103C2×#216—24no—
104C2×#216—42no—
105C2×#214—24no—
106C2×#214—24no—
107C2×#214—42no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2312yes—
111#924—42no—
112#924—42no—
113C2×Dic3⋊C4—22no—
114C22×C3⋊C8C2312yes—
115C22×C3⋊C8C2312yes—
116#869—22no—
117#869—22no—
118#835—42no—
119#832—42no—
120C3⋊#95—22no—
121C3⋊#95—22no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124#830—24no—
125#810—24no—
126#810—24no—
127#803—22no—
128C4×C3⋊C8—22no—
129C22×Q16—32no—
130C22×SD16—32no—
131#542—24no—
132#542—24no—
133C2×C4⋊Q8—31no—
134C2×C4⋊Q8C2×D311yescharacteristic
135C2×#126—21no—
136C2×#124—21no—
137C2×#124C2×D311yescharacteristic
138C2×#123—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8—32no—
141C2×C22⋊Q8C2×D311yescharacteristic
142C2×#121C2×D311yescharacteristic
143C2×C4×Q8C2×D311yescharacteristic
144C2×C4×D4C2×D311yescharacteristic
145C2×#117C2×D311yescharacteristic
146#497—64no—
147#487—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×C4×C8—31no—
155#398—61no—
156#398—61no—
157#397—61no—
158#397—61no—
159#391—32no—
160#382—31no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—24no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—82no—
171C2×C3×Q8C2×D412yes—
172C2×C3×D4—24no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—82no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—82no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4C2×D412yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C42—24no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222#217—44no—
223#216—48no—
224#214—48no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3—42no—
228C4⋊Dic3—42no—
229Dic3⋊C4—44no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8#4114yes—
236C22×Q8—21no—
237C22×Q8—32no—
238C22×Q8C22×D311yescharacteristic
239C22×D4—21no—
240C22×D4C22×D311yescharacteristic
241C23×C4—21no—
242C23×C4C22×D311yescharacteristic
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—122no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C22×C8—32no—
250C22×C8—32no—
251C4⋊Q8—22no—
252C4⋊Q8—62no—
253C4⋊Q8—62no—
254C4⋊Q8#6714yes—
255#126—28no—
256#124—28no—
257#124—22no—
258#124#6714yes—
259#123—28no—
260C22⋊Q8—28no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—122no—
264C22⋊Q8—122no—
265#121—22no—
266#121—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×D4—22no—
270C4×D4—22no—
271#117—22no—
272#117—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4#6711yescharacteristic
281C2×C4⋊C4#6711yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4#6711yescharacteristic
291C2×C22⋊C4#6711yescharacteristic
292C2×C42—21no—
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#103—64no—
296#103—64no—
297#103—122no—
298#102—122no—
299C22⋊C8—64no—
300#97—62no—
301C4×C8—62no—
302#95—61no—
303#95—61no—
304#95—62no—
305#95—62no—
306C23×C3—21no—
307C23×C3—22no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C22×D411yescharacteristic
317C3×Q8—24no—
318C3×Q8—42no—
319C3×Q8—84no—
320C3×D4—24no—
321C3×D4—42no—
322C3×D4—84no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—84no—
342C2×C3×C4#12012yes—
343C2×C3×C4#12012yes—
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C2×#4112yes—
347C2×C3×C4H(5,𝔽2)12yes—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354Dic6—42no—
355Dic6—42no—
356Dic6—42no—
357Dic6—42no—
358C3⋊C8—24no—
359C3⋊C8—42no—
360C24—21no—
361C24C2×#6711yescharacteristic
362C2×Q8—24no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—82no—
373C2×Q8C2×#6712yes—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—82no—
379C2×D4C2×#6712yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—82no—
391C22×C4C2×#6711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C23×D311yescharacteristic
397Q16—128no—
398SD16—128no—
399C2×C8—34no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—124no—
415C4⋊C4—122no—
416C4⋊C4—122no—
417C4⋊C4C2×#6712yes—
418C4⋊C4C2×#6712yes—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—124no—
427C42—24no—
428C42—21no—
429C42—21no—
430C42C2×#6712yes—
431C42C2×#6712yes—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3C2×#12011yescharacteristic
443C22×C3#53211yescharacteristic
444C22×C3#53511yescharacteristic
445C22×C3#57712yes—
446C22×C3#57812yes—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—41no—
455C3×C4#54612yes—
456C3×C4#55012yes—
457Dic3—82no—
458Dic3—82no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23C22×#6711yescharacteristic
470Q8—24no—
471Q8—42no—
472Q8—84no—
473Q8—122no—
474Q8—122no—
475Q8—122no—
476Q8—122no—
477D4—24no—
478D4—42no—
479D4—84no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4#95312yes—
506C2×C4#95312yes—
507C2×C4D3×#4112yes—
508C2×C4#100912yes—
509C2×C4C22×#6711yescharacteristic
510C2×C4C22×#6711yescharacteristic
511C8—64no—
512C8—122no—
513C2×C3—81no—
514C2×C3—82no—
515C2×C3#243111yescharacteristic
516C2×C3#300911yescharacteristic
517C2×C3#301111yescharacteristic
518C2×C3#317812yes—
519C2×C3#322512yes—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—82no—
529C22—82no—
530C22#583011yescharacteristic
531C22#583211yescharacteristic
532C22#583612yes—
533C22#583812yes—
534C22C2×#95311yescharacteristic
535C4—22no—
536C4—22no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—42no—
541C4—42no—
542C4—41no—
543C4—242no—
544C4—242no—
545C4#582912yes—
546C4#583412yes—
547C3#2561311yescharacteristic, complemented, Sylow
548C2—81no—
549C2—82no—
550C2#8345812yes—
551C2#8346412yes—
552C2#8409211yescharacteristic
553C2#8410511yescharacteristic
554C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#542C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#542D3
hypocenterC3#25613
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
2. C2×#5093C2semidirect
3. C2×#5060C2semidirect
4. #82262C2semidirect
5. #71788C2semidirect
6. #71725C2semidirect
7. #71700C2semidirect
9. #71632C2semidirect
10. #71612C2semidirect
11. #71582C2semidirect
547. C3#25613semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×#50932
C2×#50602
#822622
#717882
#717252
#717002
#716852
#716322
#716122
#715822
#256133