Finite Group Databasebeta

← #10828827

Subgroups — #10828827

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5541C211yesmaximal, characteristic
2C3×#2486C211yesmaximal, characteristic
3C2×#4971C211yesmaximal, characteristic, complemented
4C22×#807C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic
6#69631C212yesmaximal, complemented
7#69606C212yesmaximal, complemented
8#69527C212yesmaximal, complemented
9#69514C211yesmaximal, characteristic, complemented
10#69506C212yesmaximal, complemented
11#69501C211yesmaximal, characteristic, complemented
12#24094—31nomaximal, Sylow
13C2×C3×#123—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#935—21no—
17C2×#926C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22#5541—24no—
23#5541—24no—
24C2×#889—21no—
25C2×#879—21no—
26C2×#877—21no—
27C2×#870C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C3×#398—21no—
30C3×#397—21no—
31C3×#389C2212yes—
32C3×#381—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C2×#830C2211yescharacteristic
36C2×#830C2212yes—
37#4971—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#808C2212yes—
41C2×#807—24no—
42C2×#807C2212yes—
43C2×#807C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#552—31no—
49C2×#483—31no—
50C22×#107—31no—
51#2486D311yescharacteristic
52#2059—32no—
53#2036—32no—
54#1892—31no—
55#1869—31no—
56#1863—32no—
57#1854—32no—
58#1823—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C22×Dic6C2311yescharacteristic
65C3×#123—44no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4D411yescharacteristic
78C2×C3×C22⋊C4D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×#217—21no—
81C2×#217—21no—
82C2×#217—21no—
83#935—28no—
84#926—22no—
85#926—22no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3D414yes—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92#924—22no—
93#924—22no—
94#924—22no—
95#924D414yes—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C22×C3⋊C8—22no—
104C22×C3⋊C8—24no—
105C22×C3⋊C8C2312yes—
106C22×C3⋊C8C2312yes—
107#889—28no—
108#879—28no—
109#877—28no—
110#870—22no—
111#870—22no—
112C4×Dic6—22no—
113C4×Dic6—22no—
114C3×#95—22no—
115C3×#95—21no—
116C3×#95—22no—
117#830—22no—
118#830—24no—
119#810—24no—
120#810—24no—
121#808—24no—
122#807—24no—
123#807—24no—
124C23×C8—31no—
125#552—64no—
126#552—64no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#125—61no—
130C2×#123—21no—
131C2×#123—61no—
132C2×C22⋊Q8—31no—
133C2×C22⋊Q8—31no—
134C2×C4×Q8—31no—
135C2×C4×D4C2×D311yescharacteristic
136C2×#117—31no—
137C2×#117—61no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4C2×D311yescharacteristic
140#483—64no—
141C2×#107—32no—
142C2×#107—32no—
143C2×#107—64no—
144C2×#106—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—32no—
149#398—21no—
150#397—21no—
151#389C2×D312yes—
152#381—21no—
153#380C2×D311yescharacteristic
154#380C2×D312yes—
155#340—32no—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—24no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—42no—
188C22×Dic3—42no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—21no—
192C2×Dic6C2×D412yes—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—44no—
199C3×C4⋊C4—42no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—42no—
204C3×C42—21no—
205C3×C42—21no—
206#217—24no—
207#217—24no—
208#217—24no—
209C4⋊Dic3—24no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3C2×D412yes—
214C4⋊Dic3C2×D412yes—
215C4⋊Dic3C2×D412yes—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4—24no—
221C4×Dic3—24no—
222C4×Dic3—24no—
223C2×C3⋊C8—28no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8#4114yes—
230C22×Q8—31no—
231C22×D4#6711yescharacteristic
232C23×C4—21no—
233C23×C4—61no—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236C22×C8—32no—
237C22×C8—32no—
238C22×C8—64no—
239C22×C8—62no—
240#126—68no—
241#125—62no—
242#125—62no—
243#125—68no—
244#123—44no—
245#123—68no—
246C22⋊Q8—34no—
247C22⋊Q8—62no—
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250C4×Q8—62no—
251C4×Q8—62no—
252C4×D4—22no—
253C4×D4—22no—
254#117—68no—
255#117—62no—
256#117—62no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4#6711yescharacteristic
273C2×C4⋊C4C22×D312yes—
274C2×C4⋊C4C22×D312yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4#6711yescharacteristic
281C2×C22⋊C4#6711yescharacteristic
282C2×C42—61no—
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#107—64no—
287#106—64no—
288#103—64no—
289#103—64no—
290C22⋊C8—64no—
291C22⋊C8—62no—
292#95—22no—
293#95—21no—
294#95—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3#12012yes—
303C23×C3#13512yes—
304C23×C3AGL(1,ℤ/8)12yes—
305C23×C3C22×D411yescharacteristic
306C3×D4—44no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—84no—
328C2×C3×C4#12012yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×C3×C42−1+412yes—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—48no—
344Dic6—24no—
345Dic6—41no—
346Dic6—41no—
347C3⋊C8—24no—
348C3⋊C8—42no—
349C24C2×#6711yescharacteristic
350C24C2×#6711yescharacteristic
351C2×Q8—32no—
352C2×Q8—61no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—64no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—122no—
375C22×C4—122no—
376C22×C4D3×D412yes—
377C22×C4D3×D412yes—
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381C22×C4C2×#6711yescharacteristic
382C22×C4C23×D311yescharacteristic
383C2×C8—34no—
384C2×C8—68no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—32no—
396C4⋊C4—32no—
397C4⋊C4—32no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—64no—
401C4⋊C4—64no—
402C4⋊C4—64no—
403C4⋊C4—64no—
404C4⋊C4—64no—
405C4⋊C4—64no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C42—21no—
417C42—21no—
418C42—64no—
419C42—64no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3#44914yes—
430C22×C3C2×#12011yescharacteristic
431C22×C3#54712yes—
432C22×C3C2×#13511yescharacteristic
433C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
434C22×C3#57812yes—
435C3×C4—22no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—82no—
440C3×C4—82no—
441C3×C4#52511yescharacteristic
442C3×C4#53611yescharacteristic
443C3×C4#54712yes—
444Dic3—42no—
445Dic3—42no—
446Dic3—42no—
447Dic3—42no—
448C23—22no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23#93212yes—
456C23#93712yes—
457C23#95012yes—
458C23C22×#6711yescharacteristic
459Q8—64no—
460Q8—121no—
461Q8—121no—
462D4—44no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—84no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—128no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4#93712yes—
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C2×C4D3×#4112yes—
499C2×C4#101012yes—
500C8—64no—
501C8—122no—
502C2×C3—24no—
503C2×C3—81no—
504C2×C3#240611yescharacteristic
505C2×C3C2×#44911yescharacteristic
506C2×C3#301111yescharacteristic
507C2×C3#333612yes—
508C2×C3#334312yes—
509C22—24no—
510C22—24no—
511C22—22no—
512C22—22no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22#503714yes—
519C22C2×#93211yescharacteristic
520C22C2×#93711yescharacteristic
521C22C2×#95011yescharacteristic
522C22#582412yes—
523C22#583812yes—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—82no—
529C4—82no—
530C4—122no—
531C4—122no—
532C4—122no—
533C4—122no—
534C4#564811yescharacteristic
535C4#565512yes—
536C4#565611yescharacteristic
537C3#2409411yescharacteristic, complemented, Sylow
538C2—24no—
539C2—81no—
540C2C2×#503711yescharacteristic
541C2#8284111yescharacteristic
542C2#8341111yescharacteristic
543C2#8351812yes—
544C2#8359012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2486C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2486D3
hypocenterC3#24094
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. C2×#4971C2semidirect
4. C22×#807C2semidirect
6. #69631C2semidirect
7. #69606C2semidirect
8. #69527C2semidirect
9. #69514C2semidirect
10. #69506C2semidirect
11. #69501C2semidirect
537. C3#24094semidirect

Maximal subgroups

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

subgroupindex
C2×#55412
C3×#24862
C2×#49712
C22×#8072
#716102
#696312
#696062
#695272
#695142
#695062
#695012
#240943