Finite Group Databasebeta

← #10828824

Subgroups — #10828824

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×#5513C211yesmaximal, characteristic
2C22×C3×#107C211yesmaximal, characteristic, complemented
3#82196C211yesmaximal, characteristic
4C2×#4971C211yesmaximal, characteristic, complemented
5#71975C211yesmaximal, characteristic
6#71972C211yesmaximal, characteristic, complemented
7#71553C211yesmaximal, characteristic, complemented
8#70642C212yesmaximal, complemented
9#70635C212yesmaximal, complemented
10#70633C212yesmaximal, complemented
11#70618C212yesmaximal, complemented
12#24094—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#929—21no—
17C2×#929—21no—
18C2×C4×#67C2211yescharacteristic
19C2×#926—21no—
20C2×#924C2211yescharacteristic
21#5513—24no—
22#5513—24no—
23C2×#898—21no—
24C2×#897C2211yescharacteristic
25C2×#892C2211yescharacteristic
26C2×#890C2211yescharacteristic
27C2×#889—21no—
28C2×#887C2211yescharacteristic
29C2×C3×#107—24no—
30C2×C3×#107C2212yes—
31C2×C3×#107C2212yes—
32#5212C2211yescharacteristic
33#5211—21no—
34C2×#830C2211yescharacteristic
35C2×(C2×D3)⋊C8C2212yes—
36C2×#816C2212yes—
37#4992—21no—
38#4987C2212yes—
39#4983—21no—
40#4972C2212yes—
41#4971—24no—
42C2×#810C2212yes—
43C2×#808C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#552—31no—
49C2×#483—31no—
50C22×#107D311yescharacteristic, complemented
51#2486—31no—
52#2059—32no—
53#2036—32no—
54#1892—31no—
55#1869—31no—
56#1863—32no—
57#1854—32no—
58#1823—31no—
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67D411yescharacteristic
63C22×C4×D3C2311yescharacteristic
64C22×Dic6C2311yescharacteristic
65C22×C3×C8—22no—
66C22×C3×C8—24no—
67C22×C3×C8C2312yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D414yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×#217—21no—
78C2×#217—21no—
79C2×#217D411yescharacteristic
80#929—28no—
81#929—44no—
82C4×#67—22no—
83C4×#67—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4D411yescharacteristic
88#926—28no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2312yes—
92C2×C4⋊Dic3C2311yescharacteristic
93#924—22no—
94#924D414yes—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104#898—28no—
105#897—22no—
106#897—22no—
107#892—22no—
108#892—22no—
109#890—22no—
110#890—22no—
111#889—28no—
112#887—22no—
113#887—22no—
114C3×#107—24no—
115C3×#107—24no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#830—22no—
120(C2×D3)⋊C8—24no—
121#816—24no—
122#810—24no—
123#808—24no—
124C23×C8C2×D311yescharacteristic
125#552—64no—
126#552—64no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#125—61no—
130C2×#123—61no—
131C2×#123—61no—
132C2×C22⋊Q8—31no—
133C2×C22⋊Q8—31no—
134C2×C4×Q8—31no—
135C2×C4×D4—31no—
136C2×#117—31no—
137C2×#117—61no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C4⋊C4C2×D311yescharacteristic
140#483—64no—
141C2×#107—24no—
142C2×#107C2×D312yes—
143C2×#107C2×D312yes—
144C2×#106—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—32no—
149#398—61no—
150#397—61no—
151#389—32no—
152#381—61no—
153#380—31no—
154#380—32no—
155#340—32no—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C23×D3C2×D411yescharacteristic
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—22no—
177C2×#67—22no—
178C2×#67—22no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3C2×D411yescharacteristic
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—21no—
192C2×Dic6C2×D412yes—
193C2×C3×C8—28no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8#4114yes—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4C2×D412yes—
204C3×C4⋊C4C2×D412yes—
205#217—24no—
206#217—22no—
207#217—42no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—22no—
211(C2×D3)⋊C4—44no—
212C4⋊Dic3—24no—
213C4⋊Dic3—24no—
214C4⋊Dic3—21no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3C2×D412yes—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4—22no—
221Dic3⋊C4—24no—
222Dic3⋊C4—44no—
223C4×Dic3—24no—
224C4×Dic3—24no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×Q8—31no—
231C22×D4—31no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—31no—
235C23×C4C22×D311yescharacteristic
236C22×C8—22no—
237C22×C8—24no—
238C22×C8—32no—
239C22×C8C22×D312yes—
240#126—68no—
241#125—62no—
242#125—68no—
243#125—62no—
244#123—68no—
245#123—124no—
246C22⋊Q8—34no—
247C22⋊Q8—62no—
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250C4×Q8—62no—
251C4×Q8—62no—
252C4×D4—62no—
253C4×D4—62no—
254#117—62no—
255#117—68no—
256#117—62no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4#6714yes—
272C2×C4⋊C4C22×D312yes—
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—31no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—61no—
282C2×C42—31no—
283C2×C42—61no—
284C2×C42—61no—
285#107—24no—
286#107—24no—
287#106—64no—
288#103—64no—
289#103—64no—
290C22⋊C8—62no—
291C22⋊C8—64no—
292#95—62no—
293#95—62no—
294#95—61no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3#12012yes—
298C23×C3#13512yes—
299C23×C3AGL(1,ℤ/8)12yes—
300C23×C3C22×D411yescharacteristic
301C22×D3—21no—
302C22×D3—22no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—48no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—84no—
321C2×C3×C4#12012yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4112yes—
325C2×C3×C42−1+412yes—
326#67—44no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C4×D3—41no—
342C4×D3—41no—
343C4×D3—42no—
344Dic6—24no—
345Dic6—41no—
346Dic6—41no—
347C3×C8—24no—
348C3⋊C8—42no—
349C24—31no—
350C24D3×D411yescharacteristic
351C2×Q8—32no—
352C2×Q8—61no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—31no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4D3×D411yescharacteristic
380C22×C4D3×D412yes—
381C22×C4C2×#6712yes—
382C22×C4C23×D311yescharacteristic
383C2×C8—28no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8#23514yes—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—32no—
396C4⋊C4—42no—
397C4⋊C4—64no—
398C4⋊C4—64no—
399C4⋊C4—64no—
400C4⋊C4—64no—
401C4⋊C4—62no—
402C4⋊C4—64no—
403C4⋊C4—61no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—124no—
407C4⋊C4C2×#6712yes—
408C4⋊C4C2×#6712yes—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—62no—
412C22⋊C4—64no—
413C22⋊C4—62no—
414C22⋊C4—124no—
415C22⋊C4—122no—
416C42—64no—
417C42—64no—
418C42—61no—
419C42—61no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3#44914yes—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54712yes—
427C22×C3C2×#13511yescharacteristic
428C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
429C22×C3#57812yes—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—41no—
433C2×D3—41no—
434C2×D3—41no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—82no—
439C3×C4#52511yescharacteristic
440C3×C4#53611yescharacteristic
441C3×C4#54712yes—
442Dic3—42no—
443Dic3—41no—
444Dic3—41no—
445Dic3—41no—
446Dic3—42no—
447Dic3—82no—
448C23—22no—
449C23—22no—
450C23—61no—
451C23—61no—
452C23—62no—
453C23—62no—
454C23—61no—
455C23#91112yes—
456C23#91212yes—
457C23#93712yes—
458C23C2×D3×D411yescharacteristic
459Q8—64no—
460Q8—121no—
461Q8—121no—
462D4—124no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—48no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—61no—
486C2×C4—84no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4#93712yes—
496C2×C4C2×#23512yes—
497C2×C4C2×D3×D411yescharacteristic
498C2×C4#100712yes—
499C2×C4C22×#6711yescharacteristic
500C8—24no—
501C8—122no—
502C2×C3—24no—
503C2×C3#240611yescharacteristic
504C2×C3C2×#44911yescharacteristic
505C2×C3#301111yescharacteristic
506C2×C3#333612yes—
507C2×C3#334312yes—
508D3—81no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—22no—
513C22—121no—
514C22—122no—
515C22—122no—
516C22—121no—
517C22—121no—
518C22#515814yes—
519C22#550712yes—
520C22C2×#91111yescharacteristic
521C22C2×#91211yescharacteristic
522C22#576612yes—
523C22C2×#93711yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—121no—
529C4—122no—
530C4—121no—
531C4—121no—
532C4—122no—
533C4—242no—
534C4#550712yes—
535C4#579111yescharacteristic
536C4#581611yescharacteristic
537C3#2409411yescharacteristic, complemented, Sylow
538C2—24no—
539C2—241no—
540C2#7915512yes—
541C2#7915812yes—
542C2#8219211yescharacteristic
543C2C2×#515811yescharacteristic
544C2#8388411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#107C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenterC22×#107D3
hypocenterC3#24094
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C22×C3×#107C2semidirect
4. C2×#4971C2semidirect
6. #71972C2semidirect
7. #71553C2semidirect
8. #70642C2semidirect
9. #70635C2semidirect
10. #70633C2semidirect
11. #70618C2semidirect
50. C22×#107D3semidirect
537. C3#24094semidirect

Maximal subgroups

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

subgroupindex
C2×#55132
C22×C3×#1072
#821962
C2×#49712
#719752
#719722
#715532
#706422
#706352
#706332
#706182
#240943