Finite Group Databasebeta

← #10849332

Subgroups — #10849332

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5681C211yesmaximal, characteristic
2C2×C3×#489C211yesmaximal, characteristic, complemented
3#83239C211yesmaximal, characteristic, complemented
4C2×#5084C211yesmaximal, characteristic, complemented
5#71721C211yesmaximal, characteristic, complemented
6#70279C212yesmaximal, complemented
7#70237C212yesmaximal, complemented
8#70211C212yesmaximal
9#70169C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69812C211yesmaximal, characteristic, complemented
12#25659—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C2×#947C2211yescharacteristic
16C2×Q8×Dic3C2211yescharacteristic
17C2×#944—21no—
18C2×#940C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20#5681—24no—
21#5681—24no—
22C2×#886—21no—
23C2×#885—21no—
24C2×#883—21no—
25C2×#67⋊C4—21no—
26C2×C4×D12C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#489—24no—
29C2×C3×#103C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#97C2211yescharacteristic
32#5229—21no—
33C2×#835C2212yes—
34C2×#832C2212yes—
35C2×#821C2212yes—
36#5084—24no—
37#4983—21no—
38#4983—21no—
39#4972C2212yes—
40#4972C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yes—
43C4×C4⋊Dic3C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4551C2212yes—
46#4550C2212yes—
47#4550C2212yes—
48C2×#559—31no—
49C2×#489D311yescharacteristic, complemented
50C2×#476—31no—
51#2546—31no—
52#2042—32no—
53#2041—32no—
54#1985—31no—
55#1983—31no—
56#1865—32no—
57#1864—32no—
58#1762—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6D411yescharacteristic
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#124—22no—
72C3×#124D414yes—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42C2311yescharacteristic
78C2×#217—21no—
79#947—22no—
80#947—22no—
81Q8×Dic3—22no—
82Q8×Dic3—22no—
83#944—44no—
84#940—22no—
85#940—22no—
86D4×Dic3—22no—
87D4×Dic3—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4D411yescharacteristic
98C2×Dic3⋊C4D411yescharacteristic
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104#886—28no—
105#885—28no—
106#883—28no—
107#67⋊C4—28no—
108C4×D12—22no—
109C4×D12D414yes—
110C4×Dic6—22no—
111C4×Dic6—22no—
112C3×#103—24no—
113C3×#102—24no—
114C3×#97—22no—
115#835—24no—
116#832—24no—
117C3⋊#95—21no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#821—24no—
121#809—24no—
122#808—24no—
123C4⋊(C3⋊C8)—22no—
124#559—64no—
125#559—64no—
126C2×C4⋊Q8—61no—
127C2×C4⋊Q8C2×D311yescharacteristic
128C2×#127—31no—
129C2×#124—31no—
130C2×#124—61no—
131C2×#124C2×D311yescharacteristic
132C2×#123—61no—
133C2×#123—61no—
134C2×C4×Q8—31no—
135C2×C4×Q8—31no—
136C2×C4×D4—31no—
137C2×C4×D4—31no—
138C2×C4×D4—61no—
139#489—24no—
140#476—64no—
141C2×#106—32no—
142C2×C4⋊C8—31no—
143C2×#103—32no—
144C2×#103C2×D312yes—
145C2×#102—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102C2×D312yes—
149C2×#97C2×D311yescharacteristic
150#398—61no—
151#398—61no—
152#391—61no—
153#389—32no—
154#389—32no—
155C4×C4⋊C4—31no—
156#340—32no—
157#336—32no—
158#336—32no—
159C24×C3—21no—
160C23×D3—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4C2×D412yes—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×#67—24no—
178C2×#67—24no—
179C2×#67—42no—
180C2×#67—42no—
181C2×#67—82no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—24no—
185C22×Dic3—82no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—21no—
193C2×D12C2×D412yes—
194C2×C4×D3—24no—
195C2×C4×D3—82no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—41no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—42no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209#217—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213C4⋊Dic3—24no—
214C4⋊Dic3—22no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217C4⋊Dic3C2×D412yes—
218C4⋊Dic3#4114yes—
219Dic3⋊C4—22no—
220Dic3⋊C4—22no—
221Dic3⋊C4—44no—
222Dic3⋊C4—44no—
223C4×Dic3—24no—
224C4×Dic3—21no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C4×Dic3—42no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C22×Q8—21no—
233C22×Q8—31no—
234C22×Q8C22×D311yescharacteristic
235C22×D4—31no—
236C22×D4—61no—
237C22×D4—61no—
238C22×D4C22×D311yescharacteristic
239C23×C4—61no—
240C23×C4—61no—
241C22×C8—32no—
242C22×C8C22×D312yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—124no—
246#127—62no—
247#127—62no—
248#124—22no—
249#124—68no—
250#124—62no—
251#124—62no—
252#124#6714yes—
253#123—68no—
254#123—68no—
255C4×Q8—62no—
256C4×Q8—62no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—34no—
260C4×D4—68no—
261C4×D4—62no—
262C4×D4—62no—
263C4×D4—62no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#106—64no—
286C4⋊C8—62no—
287#103—24no—
288#103—64no—
289#102—24no—
290#102—64no—
291#102—64no—
292#102—64no—
293#97—22no—
294#95—62no—
295#95—62no—
296#95—61no—
297C23×C3—22no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3C22×D411yescharacteristic
303C22×D3—22no—
304C22×D3—42no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—42no—
308C3×Q8—24no—
309C3×Q8—41no—
310C3×Q8—41no—
311C3×Q8—84no—
312C3×D4—24no—
313C3×D4—41no—
314C3×D4—41no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12012yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4112yes—
328C2×C3×C42−1+412yes—
329#67—48no—
330#67—84no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—84no—
346C2×Dic3AGL(1,ℤ/8)12yes—
347D12—22no—
348D12—22no—
349D12—42no—
350C4×D3—84no—
351Dic6—44no—
352C3×C8—42no—
353C3⋊C8—42no—
354C24—21no—
355C24—61no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—62no—
363C2×Q8—62no—
364C2×Q8—121no—
365C2×Q8C2×#6712yes—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—32no—
370C2×D4—64no—
371C2×D4—64no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—61no—
375C2×D4—122no—
376C2×D4—122no—
377C2×D4—242no—
378C2×D4C2×#6712yes—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4—242no—
390C22×C4—242no—
391C22×C4D3×D411yescharacteristic
392C22×C4D3×D411yescharacteristic
393C22×C4C23×D311yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—62no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—34no—
404C4⋊C4—42no—
405C4⋊C4—64no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—61no—
411C4⋊C4—124no—
412C4⋊C4—124no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C42—64no—
420C42—61no—
421C42—61no—
422C42—61no—
423C42—61no—
424C42—122no—
425C42D3×D412yes—
426C42C2×#6712yes—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54812yes—
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3#57511yescharacteristic
436C22×C3#57712yes—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—82no—
441C2×D3—82no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—42no—
445C3×C4—82no—
446C3×C4#53211yescharacteristic
447C3×C4#53611yescharacteristic
448C3×C4#54712yes—
449Dic3—22no—
450Dic3—44no—
451Dic3—41no—
452Dic3—42no—
453Dic3—41no—
454Dic3—41no—
455C23—22no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—62no—
461C23—122no—
462C23—121no—
463C23—121no—
464C23—122no—
465C23C2×D3×D411yescharacteristic
466Q8—24no—
467Q8—41no—
468Q8—41no—
469Q8—84no—
470Q8—124no—
471D4—24no—
472D4—41no—
473D4—41no—
474D4—62no—
475D4—62no—
476D4—128no—
477D4—122no—
478D4—244no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—32no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—64no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—61no—
496C2×C4—62no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—244no—
503C2×C4—244no—
504C2×C4#93712yes—
505C2×C4#93712yes—
506C2×C4C2×D3×D411yescharacteristic
507C2×C4D3×#4112yes—
508C2×C4#101012yes—
509C2×C4C22×#6711yescharacteristic
510C8—42no—
511C8—122no—
512C2×C3—82no—
513C2×C3#243211yescharacteristic
514C2×C3#300611yescharacteristic
515C2×C3#300911yescharacteristic
516C2×C3#337112yes—
517C2×C3#337312yes—
518D3—82no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—122no—
525C22—122no—
526C22—122no—
527C22—242no—
528C22—242no—
529C22#567712yes—
530C22D3×AGL(1,ℤ/8)11yescharacteristic
531C22#577311yescharacteristic
532C22#577812yes—
533C22C2×#93711yescharacteristic
534C4—22no—
535C4—22no—
536C4—42no—
537C4—62no—
538C4—82no—
539C4—124no—
540C4—121no—
541C4—121no—
542C4—122no—
543C4—121no—
544C4#566811yescharacteristic
545C4#567511yescharacteristic
546C4#582412yes—
547C3#2565911yescharacteristic, complemented, Sylow
548C2—82no—
549C2—242no—
550C2#8167312yes—
551C2#8167412yes—
552C2#8295911yescharacteristic
553C2#8394611yescharacteristic
554C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenterC2×#489D3
hypocenterC3#25659
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. C2×C3×#489C2semidirect
3. #83239C2semidirect
4. C2×#5084C2semidirect
5. #71721C2semidirect
6. #70279C2semidirect
7. #70237C2semidirect
9. #70169C2semidirect
10. #69838C2semidirect
11. #69812C2semidirect
49. C2×#489D3semidirect
547. C3#25659semidirect

Maximal subgroups

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

subgroupindex
C2×#56812
C2×C3×#4892
#832392
C2×#50842
#717212
#702792
#702372
#702112
#701692
#698382
#698122
#256593