Finite Group Databasebeta

← #10845939

Subgroups — #10845939

Order 768.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4864C212yesmaximal, complemented
2C2×#4848C211yesmaximal, characteristic, complemented
3C2×#4834C211yesmaximal, characteristic, complemented
4#74369C211yesmaximal, characteristic, complemented
5#73799C211yesmaximal, characteristic
6C3×#1948C211yesmaximal, characteristic, complemented
7#70581C212yesmaximal, complemented
8#69580C211yesmaximal, characteristic, complemented
9C4×#810C211yesmaximal, characteristic, complemented
10#68887C212yesmaximal, complemented
11#68883C212yesmaximal, complemented
12#25387—31nomaximal, Sylow
13C22×#204C2212yes—
14C22×D3×C8C2212yes—
15C2×#897C2212yes—
16C2×#895C2212yes—
17C2×#895C2212yes—
18C2×#874—21no—
19C2×#874C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C3×#384C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2212yes—
27C2×(C2×D3)⋊C8C2212yes—
28#4994—22no—
29#4988—21no—
30#4987C2212yes—
31#4984—21no—
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#809C2212yes—
35C2×#809C2212yes—
36#4939—21no—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4864—28no—
39#4848—24no—
40#4834—24no—
41#4671—22no—
42C3×#340C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#465—32no—
46C2×#460—31no—
47C2×C8×D4—31no—
48#2541—31no—
49#2538—31no—
50#2034—32no—
51#2000—32no—
52#1957—31no—
53#1948D311yescharacteristic, complemented
54#1893—32no—
55C4×#103—31no—
56C22×D12—21no—
57C22×D12—22no—
58C22×D12—22no—
59C22×D12—21no—
60C22×D12D411yescharacteristic
61C22×D12C2312yes—
62C22×C4×D3C2312yes—
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2312yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4D412yes—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×C4×Dic3—21no—
82C22×C3⋊C8C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84C2×#204—24no—
85C2×#204—24no—
86C2×#204—42no—
87C2×D3×C8—24no—
88C2×D3×C8—24no—
89C2×D3×C8—42no—
90#897—24no—
91#897—42no—
92#895—24no—
93#895—24no—
94#895—42no—
95#895—42no—
96#874—22no—
97#874—22no—
98#874—44no—
99C4×D12—22no—
100C4×D12—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×#106—24no—
104C3×C4⋊C8D414yescomplemented
105C3×#95—21no—
106C3⋊#95—22no—
107#821—24no—
108(C2×D3)⋊C8—24no—
109#810—22no—
110#810D414yescomplemented
111#809—24no—
112#809—42no—
113C4×C3⋊C8—22no—
114C22×SD16—32no—
115C23×C8—32no—
116C2×#127—31no—
117C2×#127—61no—
118C2×C22⋊Q8—32no—
119C2×#121—32no—
120C2×#121—32no—
121C2×C4×Q8—31no—
122C2×C4×D4—31no—
123#465—68no—
124#460—64no—
125C8×D4—64no—
126C2×#106C2×D312yes—
127C2×C4⋊C8C2×D311yescharacteristic
128C2×#103—31no—
129C2×#103—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×C22⋊C8—32no—
134C2×C4×C8—31no—
135#396—62no—
136#392—62no—
137#390—61no—
138#390—61no—
139#389—61no—
140#384C2×D311yescharacteristic
141#380—32no—
142C4×C4⋊C4C2×D311yescharacteristic
143#340—31no—
144#340—31no—
145#340C2×D312yes—
146C23×D3—22no—
147C23×D3C2×D412yes—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3C2×D412yes—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—44no—
166C2×D12—44no—
167C2×D12—44no—
168C2×D12—44no—
169C2×D12—42no—
170C2×D12—42no—
171C2×D12—42no—
172C2×D12—42no—
173C2×D12—81no—
174C2×D12—81no—
175C2×D12—82no—
176C2×D12—82no—
177C2×D12SD1614yescomplemented
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C2×D412yes—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8C2×D414yes—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—81no—
199C3×C4⋊C4C2×D412yes—
200C3×C4⋊C4#4112yes—
201C3×C42—42no—
202C3×C42C2×D412yes—
203C3×C42#4112yes—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—82no—
209C4⋊Dic3—22no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—44no—
213Dic3⋊C4—42no—
214C4×Dic3—42no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8C2×D412yes—
221#204—48no—
222D3×C8—48no—
223C22×Q8—31no—
224C22×D4—31no—
225C22×D4—32no—
226C22×D4—62no—
227C22×D4—62no—
228C22×D4—61no—
229C22×D4—61no—
230C23×C4—32no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C22×C8—31no—
235C22×C8—31no—
236C22×C8—64no—
237C22×C8—64no—
238C22×C8—122no—
239C22×C8C22×D312yes—
240#127—62no—
241#127—62no—
242#127—124no—
243C22⋊Q8—64no—
244C22⋊Q8—122no—
245#121—64no—
246#121—64no—
247#121—122no—
248#121—122no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4C22×D312yes—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—32no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—21no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#106—24no—
271C4⋊C8D1214yescomplemented
272#103—34no—
273#103—62no—
274#102—64no—
275#102—64no—
276#102—122no—
277C22⋊C8—64no—
278C4×C8—62no—
279#95—21no—
280#95—62no—
281C23×C3C22×D411yescharacteristic
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3C2×SD1614yes—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4#12112yes—
308C2×C3×C4#12112yes—
309C2×C3×C4C2×SD1612yes—
310C2×C3×C4#13512yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326D12—28no—
327D12—42no—
328D12—42no—
329D12—42no—
330D12—42no—
331D12—84no—
332D12—88no—
333D12—88no—
334D12—84no—
335C4×D3—42no—
336C4×D3—42no—
337C4×D3—42no—
338C4×D3—42no—
339Dic6—22no—
340Dic6—22no—
341Dic6—41no—
342Dic6—41no—
343C3×C8—24no—
344C3⋊C8—22no—
345C3⋊C8—41no—
346C24—32no—
347C24—62no—
348C2×Q8—32no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×D4—34no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—124no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4—241no—
372C2×D4—241no—
373C2×D4—242no—
374C2×D4—242no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—32no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390SD16—128no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—32no—
394C2×C8—61no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—128no—
400C2×C8C2×D1214yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—62no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—81no—
410C4⋊C4—124no—
411C4⋊C4—122no—
412C4⋊C4D3×D412yes—
413C4⋊C4#23912yes—
414C22⋊C4—64no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—124no—
418C22⋊C4—242no—
419C42—42no—
420C42—122no—
421C42C2×D1212yes—
422C42#23912yes—
423C22×C3C2×#12111yescharacteristic
424C22×C3D4211yescharacteristic
425C22×C3#54611yescharacteristic
426C22×C3C22×SD1611yescharacteristic
427C22×C3C2×#13511yescharacteristic
428C22×C3#57611yescharacteristic
429C22×C3#57711yescharacteristic
430C2×D3—24no—
431C2×D3—24no—
432C2×D3—24no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—82no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C2×D3—82no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—82no—
448C3×C4#46514yescomplemented
449C3×C4#53211yescharacteristic
450C3×C4#54011yescharacteristic
451C3×C4D4211yescharacteristic
452C3×C4#55011yescharacteristic
453Dic3—42no—
454Dic3—42no—
455Dic3—81no—
456C23—34no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23C2×D3×D411yescharacteristic
471Q8—62no—
472Q8—62no—
473Q8—121no—
474Q8—121no—
475D4—68no—
476D4—122no—
477D4—122no—
478D4—122no—
479D4—122no—
480D4—244no—
481D4—248no—
482D4—248no—
483D4—244no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4#89512yes—
509C2×C4#89512yes—
510C2×C4D3×SD1612yes—
511C2×C4#91612yes—
512C2×C4C22×D1211yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×#23911yescharacteristic
515C2×C4C2×#23911yescharacteristic
516C2×C4#100911yescharacteristic
517C2×C4#100911yescharacteristic
518C8—24no—
519C8—62no—
520C8—121no—
521C2×C3#242811yescharacteristic
522C2×C3C2×#46511yescharacteristic
523C2×C3#306211yescharacteristic
524C2×C3D4×SD1611yescharacteristic
525C2×C3#328711yescharacteristic
526C2×C3#329511yescharacteristic
527C2×C3#330411yescharacteristic
528D3—44no—
529D3—162no—
530C22—64no—
531C22—64no—
532C22—64no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—122no—
537C22—242no—
538C22—242no—
539C22—242no—
540C22—242no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22C2×#89511yescharacteristic
545C22#560911yescharacteristic
546C22#563011yescharacteristic
547C22C2×D3×SD1611yescharacteristic
548C22C2×#91611yescharacteristic
549C22#577411yescharacteristic
550C22#577811yescharacteristic
551C4—22no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—122no—
556C4—122no—
557C4—241no—
558C4#486414yescomplemented
559C4D4×D1211yescharacteristic
560C4#557711yescharacteristic
561C4#569711yescharacteristic
562C4#572511yescharacteristic
563C3#2538711yescharacteristic, complemented, Sylow
564C2—124no—
565C2—482no—
566C2#7622011yescharacteristic
567C2C2×#486411yescharacteristic
568C2#7921311yescharacteristic
569C2#8039411yescharacteristic
570C2#8044811yescharacteristic
571C2#8102311yescharacteristic
572C2#8107511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radical#73799C2
hypercenter#1948D3
hypocenterC3#25387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. C2×#4864C2semidirect
2. C2×#4848C2semidirect
3. C2×#4834C2semidirect
4. #74369C2semidirect
6. C3×#1948C2semidirect
7. #70581C2semidirect
8. #69580C2semidirect
9. C4×#810C2semidirect
10. #68887C2semidirect
11. #68883C2semidirect
53. #1948D3semidirect
104. C3×C4⋊C8D4semidirect
110. #810D4semidirect
177. C2×D12SD16semidirect
271. C4⋊C8D12semidirect
448. C3×C4#465semidirect
558. C4#4864semidirect
563. C3#25387semidirect

Maximal subgroups

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

subgroupindex
C2×#48642
C2×#48482
C2×#48342
#743692
#737992
C3×#19482
#705812
#695802
C4×#8102
#688872
#688832
#253873