Finite Group Databasebeta

← #10829268

Subgroups — #10829268

Order 768.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5555C211yesmaximal, characteristic
2C3×#2536C211yesmaximal, characteristic, complemented
3C2×#5026C211yesmaximal, characteristic, complemented
4C2×#5021C211yesmaximal, characteristic, complemented
5#71699C212yesmaximal, complemented
6#71692C212yesmaximal, complemented
7#71615C211yesmaximal, characteristic, complemented
8#69556C211yesmaximal, characteristic, complemented
9#69555C211yesmaximal, characteristic, complemented
10#69523C212yesmaximal, complemented
11#69504C212yesmaximal, complemented
12#24132—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C22×#215C2211yescharacteristic
18C2×#930C2212yes—
19C22×C4⋊Dic3C2212yes—
20#5555—24no—
21#5555—24no—
22C2×#897—21no—
23C2×#886—22no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#398—21no—
27C3×#394—21no—
28C3×#388—22no—
29C3×#382C2211yescharacteristic
30C3×#380C2212yes—
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5026—24no—
36#5021—24no—
37C2×#809C2211yescharacteristic
38C2×#809C2212yes—
39C2×#807C2211yescharacteristic
40C2×#807C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#477—31no—
48C2×#440—31no—
49#2536D311yescharacteristic, complemented
50#2033—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1872—32no—
55#1852—32no—
56#1804—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4D411yescharacteristic
59C23×C3×C4C2312yes—
60C22×#67—22no—
61C23×Dic3—22no—
62C2×#237—28no—
63C2×#237—48no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×Dic6—21no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—42no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4D412yes—
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83C2×#215—22no—
84C2×#215—22no—
85C2×#215—22no—
86#930—24no—
87#930—24no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98C22×C3⋊C8C2311yescharacteristic
99#897—28no—
100#886—216no—
101C4×D12—22no—
102C4×D12D414yes—
103#869—22no—
104#869D414yes—
105C3×#95—22no—
106C3×#95—21no—
107C3×#95—21no—
108#835—24no—
109#835—22no—
110#830—24no—
111#830—42no—
112#809—22no—
113#809—42no—
114#807—24no—
115#807—22no—
116C4⋊(C3⋊C8)—22no—
117#803—22no—
118C22×#41—61no—
119C22×SD16—31no—
120#547—64no—
121#547—64no—
122C2×C4⋊Q8—31no—
123C2×#123—62no—
124C2×C22⋊Q8—61no—
125C2×C22⋊Q8C2×D312yes—
126C2×#121—32no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4—32no—
131#477—64no—
132#440—64no—
133C2×#107—32no—
134C2×#107—31no—
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97—31no—
143#398—21no—
144#394—21no—
145#388—22no—
146#382C2×D311yescharacteristic
147#380C2×D312yes—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3C2×D412yes—
152C23×D3—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8#4112yescomplemented
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—28no—
175C2×#67—44no—
176C2×#67—44no—
177C22×Dic3—28no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—44no—
181C22×Dic3—44no—
182#237—432no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12C2×D412yes—
189C2×C4×D3—24no—
190C2×C4×D3—44no—
191C2×Dic6—24no—
192C2×Dic6—44no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—44no—
199C3×C4⋊C4—42no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—44no—
203C3×C42—42no—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206#215—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—24no—
209C4⋊Dic3—28no—
210C4⋊Dic3—24no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3C2×D414yes—
214C4⋊Dic3C2×D412yes—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8#4112yescomplemented
224C2×#41—68no—
225C2×#41—128no—
226C22×Q8—61no—
227C22×Q8C22×D311yescharacteristic
228C22×D4—31no—
229C22×D4—62no—
230C22×D4#6711yescharacteristic
231C23×C4—62no—
232C23×C4—61no—
233C23×C4C22×D312yes—
234C2×SD16—62no—
235C2×SD16—62no—
236C2×SD16—62no—
237C22×C8—31no—
238C22×C8—31no—
239C22×C8—32no—
240C4⋊Q8—34no—
241C4⋊Q8—62no—
242#123—616no—
243C22⋊Q8—24no—
244C22⋊Q8—42no—
245C22⋊Q8—68no—
246#121—64no—
247#121—64no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—34no—
253C4×D4—62no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4#6711yescharacteristic
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4#6712yes—
270C2×C42—21no—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273#107—64no—
274C4⋊C8—62no—
275#103—62no—
276#103—64no—
277#102—62no—
278#102—122no—
279C22⋊C8—64no—
280C22⋊C8—122no—
281#97—62no—
282#95—22no—
283#95—21no—
284#95—21no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3AGL(1,ℤ/8)14yes—
292C23×C3C22×D411yescharacteristic
293C22×D3—22no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—42no—
297C22×D3—42no—
298C3×Q8—22no—
299C3×Q8—22no—
300C3×Q8—41no—
301C3×Q8—41no—
302C3×D4—28no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12012yes—
321C2×C3×C4#12012yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×C3×C42−1+411yescharacteristic
328#67—416no—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—416no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335D12—22no—
336D12—22no—
337D12—41no—
338D12—41no—
339C4×D3—48no—
340Dic6—48no—
341C3⋊C8—22no—
342C3⋊C8—41no—
343C3⋊C8—42no—
344C24—61no—
345C24C2×#6712yes—
346#41—1232no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—64no—
353C2×Q8—124no—
354C2×Q8#23712yescomplemented
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—32no—
358C2×D4—68no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—124no—
365C2×D4—124no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—68no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—64no—
379C22×C4—124no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4C2×#6712yes—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C23×D311yescharacteristic
386SD16—68no—
387C2×C8—32no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—62no—
392C2×C8—61no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—61no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—32no—
402C4⋊C4—34no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—68no—
406C4⋊C4—64no—
407C4⋊C4—61no—
408C4⋊C4—62no—
409C22⋊C4—24no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—68no—
413C22⋊C4—64no—
414C42—42no—
415C42D3×D412yes—
416C42D3×D412yes—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C2×#12011yescharacteristic
425C22×C3#54611yescharacteristic
426C22×C3#54711yescharacteristic
427C22×C3C2×AGL(1,ℤ/8)12yes—
428C22×C3#57812yes—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—82no—
433C2×D3—82no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—81no—
439C3×C4—82no—
440C3×C4#52511yescharacteristic
441C3×C4#53711yescharacteristic
442C3×C4#54811yescharacteristic
443C3×C4#55011yescharacteristic
444Dic3—44no—
445Dic3—42no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—62no—
453C23—121no—
454C23—122no—
455C23—121no—
456C23—122no—
457C23#94914yes—
458C23C22×#6711yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—41no—
462Q8—41no—
463Q8—128no—
464D4—28no—
465D4—62no—
466D4—62no—
467D4—1216no—
468D4—121no—
469D4—121no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—62no—
488C2×C4—64no—
489C2×C4—128no—
490C2×C4—1216no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4#93712yes—
495C2×C4#93712yes—
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C2×C4C2×#23711yescharacteristic
499C2×C4#100411yescharacteristic
500C2×C4D3×#4111yescharacteristic
501C2×C4#101011yescharacteristic
502C8—62no—
503C8—122no—
504C8—121no—
505C2×C3—44no—
506C2×C3#240711yescharacteristic
507C2×C3#300311yescharacteristic
508C2×C3#301111yescharacteristic
509C2×C3#330812yes—
510C2×C3#334312yes—
511D3—82no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—242no—
523C22—242no—
524C22C2×#93711yescharacteristic
525C22#580611yescharacteristic
526C22C2×#94912yes—
527C22#582411yescharacteristic
528C22#583812yes—
529C4—22no—
530C4—22no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—81no—
535C4—122no—
536C4—124no—
537C4#559811yescharacteristic
538C4#559911yescharacteristic
539C4#560611yescharacteristic
540C4#560811yescharacteristic
541C3#2413211yescharacteristic, complemented, Sylow
542C2—44no—
543C2—242no—
544C2#8270111yescharacteristic
545C2#8292711yescharacteristic
546C2#8341111yescharacteristic
547C2#8353612yes—
548C2#8359212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2536C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenter#2536D3
hypocenterC3#24132
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. C3×#2536C2semidirect
3. C2×#5026C2semidirect
4. C2×#5021C2semidirect
5. #71699C2semidirect
6. #71692C2semidirect
7. #71615C2semidirect
8. #69556C2semidirect
9. #69555C2semidirect
10. #69523C2semidirect
11. #69504C2semidirect
49. #2536D3semidirect
158. C2×C3×Q8#41semidirect
223. C2×C3⋊C8#41semidirect
354. C2×Q8#237semidirect
541. C3#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
C3×#25362
C2×#50262
C2×#50212
#716992
#716922
#716152
#695562
#695552
#695232
#695042
#241323