Finite Group Databasebeta

← #10829274

Subgroups — #10829274

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×#5726C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3#82499C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70634C212yesmaximal, complemented
7#70632C212yesmaximal, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70471C211yesmaximal, characteristic, complemented
10#70272C212yesmaximal, complemented
11#70268C212yesmaximal, complemented
12#24132—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946—21no—
17C2×#946C2212yes—
18C22×#215C2211yescharacteristic
19#5726—24no—
20#5726—24no—
21C2×#895C2212yes—
22C2×#894—22no—
23C2×D3×C4⋊C4C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C3×#477—24no—
27C2×C3×#107C2212yes—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×#835C2211yescharacteristic
31C2×#821C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34#5026—24no—
35#4991—21no—
36#4987C2212yes—
37#4981—21no—
38C2×#809C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4940—22no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#477D311yescharacteristic, complemented
48C2×#440—31no—
49#2536—31no—
50#2033—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1872—32no—
55#1852—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C2×#239—28no—
60C2×#239—48no—
61C22×D12—22no—
62C22×D12D411yescharacteristic
63C22×D12C2311yescharacteristic
64C22×C4×D3—22no—
65C22×C4×D3—21no—
66C22×C4×D3C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yes—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C42—21no—
77C2×C3×C42C2311yescharacteristic
78#946—28no—
79#946—24no—
80#946—42no—
81C2×#215—22no—
82C2×#215—22no—
83C2×#215—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4D412yes—
89C2×C4⋊Dic3D411yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96#895—24no—
97#895—24no—
98#894—216no—
99D3×C4⋊C4—24no—
100D3×C4⋊C4—24no—
101C4×D12—22no—
102C4×D12—22no—
103C4×D12—22no—
104C4×D12D414yes—
105C3×#107—24no—
106C3×C4⋊C8—22no—
107C3×#103—24no—
108#835—22no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112#821—42no—
113(C2×D3)⋊C8—24no—
114(C2×D3)⋊C8—42no—
115#809—22no—
116#807—22no—
117#803—22no—
118C22×#41—61no—
119C22×SD16—31no—
120#547—64no—
121#547—64no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#123—62no—
124C2×C22⋊Q8—32no—
125C2×C22⋊Q8—61no—
126C2×#121—32no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×C4×D4—31no—
130C22×C4⋊C4—32no—
131#477—24no—
132#440—64no—
133C2×#107—31no—
134C2×#107C2×D312yes—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—31no—
137C2×#103C2×D312yes—
138C2×#102—31no—
139C2×#102—32no—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97—31no—
143#398—61no—
144#394—61no—
145#388—62no—
146#382—31no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C23×D3—21no—
152C23×D3C2×D412yes—
153C2×C3×Q8—24no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—44no—
160C2×C3×Q8#4112yescomplemented
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3C2×D412yes—
170#239—432no—
171C2×D12—24no—
172C2×D12—22no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—28no—
178C2×D12—21no—
179C2×D12—44no—
180C2×D12—44no—
181C2×D12C2×D412yes—
182C2×C4×D3—28no—
183C2×C4×D3—24no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—44no—
191C2×C4×D3—44no—
192C2×C4×D3—44no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4C2×D414yes—
202C3×C42—42no—
203C3×C42C2×D412yes—
204C3×C42C2×D412yes—
205#215—28no—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—44no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3C2×D412yes—
216Dic3⋊C4—28no—
217Dic3⋊C4—44no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8#4112yescomplemented
224C2×#41—68no—
225C2×#41—128no—
226C22×Q8—21no—
227C22×Q8C22×D311yescharacteristic
228C22×D4—31no—
229C22×D4—31no—
230C22×D4—62no—
231C23×C4—32no—
232C23×C4—62no—
233C23×C4—61no—
234C2×SD16—62no—
235C2×SD16—62no—
236C2×SD16—62no—
237C22×C8—31no—
238C22×C8—31no—
239C22×C8C22×D312yes—
240C4⋊Q8—22no—
241C4⋊Q8#6714yes—
242#123—616no—
243C22⋊Q8—68no—
244C22⋊Q8—64no—
245C22⋊Q8—122no—
246#121—64no—
247#121—64no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×D4—34no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4—62no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—21no—
271C2×C42C22×D311yescharacteristic
272#107—24no—
273#107—62no—
274C4⋊C8—22no—
275#103—24no—
276#103—62no—
277#102—62no—
278#102—122no—
279C22⋊C8—64no—
280C22⋊C8—122no—
281#97—62no—
282#95—62no—
283#95—61no—
284#95—61no—
285C23×C3C22×D411yescharacteristic
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3AGL(1,ℤ/8)14yes—
298C3×Q8—22no—
299C3×Q8—22no—
300C3×Q8—48no—
301C3×Q8—41no—
302C3×Q8—41no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4#12012yes—
314C2×C3×C4#12012yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C2×C3×C42−1+411yescharacteristic
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329D12—28no—
330D12—22no—
331D12—22no—
332D12—416no—
333D12—41no—
334D12—41no—
335C4×D3—48no—
336C4×D3—416no—
337C4×D3—42no—
338C4×D3—42no—
339C4×D3—42no—
340C4×D3—42no—
341C3×C8—42no—
342C3⋊C8—22no—
343C3⋊C8—41no—
344C24—32no—
345C24—61no—
346#41—1232no—
347C2×Q8—24no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—44no—
354C2×Q8#23912yescomplemented
355C2×D4—32no—
356C2×D4—64no—
357C2×D4—68no—
358C2×D4—62no—
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—21no—
369C22×C4—32no—
370C22×C4—68no—
371C22×C4—61no—
372C22×C4—64no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D311yescharacteristic
386SD16—68no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—32no—
391C2×C8—61no—
392C2×C8—61no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—61no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—32no—
401C4⋊C4—42no—
402C4⋊C4—68no—
403C4⋊C4—62no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4—124no—
408C4⋊C4C2×#6714yes—
409C22⋊C4—68no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—124no—
413C22⋊C4—124no—
414C42—42no—
415C42D3×D412yes—
416C42C2×#6712yes—
417C22×C3C2×#12011yescharacteristic
418C22×C3#54611yescharacteristic
419C22×C3#54711yescharacteristic
420C22×C3C2×AGL(1,ℤ/8)12yes—
421C22×C3#57812yes—
422C2×D3—24no—
423C2×D3—24no—
424C2×D3—24no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
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—44no—
438C3×C4—81no—
439C3×C4#52511yescharacteristic
440C3×C4#53711yescharacteristic
441C3×C4#54811yescharacteristic
442C3×C4#55011yescharacteristic
443Dic3—42no—
444Dic3—42no—
445Dic3—82no—
446C23—34no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—122no—
455C23—122no—
456C23—121no—
457C23—121no—
458C23C2×D3×D411yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—48no—
462Q8—41no—
463Q8—41no—
464D4—68no—
465D4—62no—
466D4—62no—
467D4—1216no—
468D4—121no—
469D4—121no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—128no—
485C2×C4—1216no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4#93712yes—
495C2×C4#93712yes—
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×#23911yescharacteristic
498C2×C4#100711yescharacteristic
499C2×C4D3×#4111yescharacteristic
500C2×C4#100911yescharacteristic
501C2×C4C22×#6711yescharacteristic
502C8—42no—
503C8—62no—
504C8—121no—
505C2×C3#240711yescharacteristic
506C2×C3#300311yescharacteristic
507C2×C3#301111yescharacteristic
508C2×C3#330812yes—
509C2×C3#334312yes—
510D3—44no—
511D3—82no—
512C22—64no—
513C22—64no—
514C22—64no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—242no—
523C22—242no—
524C22#563011yescharacteristic
525C22#563111yescharacteristic
526C22#576612yes—
527C22D3×AGL(1,ℤ/8)12yes—
528C22C2×#93711yescharacteristic
529C4—22no—
530C4—22no—
531C4—42no—
532C4—44no—
533C4—81no—
534C4—122no—
535C4—122no—
536C4—242no—
537C4#562711yescharacteristic
538C4#562911yescharacteristic
539C4#581711yescharacteristic
540C4#583411yescharacteristic
541C3#2413211yescharacteristic, complemented, Sylow
542C2—124no—
543C2—242no—
544C2#8112112yes—
545C2#8112712yes—
546C2#8281211yescharacteristic
547C2#8388411yescharacteristic
548C2#8394411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC2×#5726C2
hypercenterC2×#477D3
hypocenterC3#24132
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×C3×#477C2semidirect
3. #82499C2semidirect
4. C2×#5026C2semidirect
5. #71788C2semidirect
6. #70634C2semidirect
7. #70632C2semidirect
8. #70480C2semidirect
9. #70471C2semidirect
10. #70272C2semidirect
11. #70268C2semidirect
47. C2×#477D3semidirect
160. C2×C3×Q8#41semidirect
223. C2×C3⋊C8#41semidirect
354. C2×Q8#239semidirect
541. C3#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#57262
C2×C3×#4772
#824992
C2×#50262
#717882
#706342
#706322
#704802
#704712
#702722
#702682
#241323