Finite Group Databasebeta

← #10849324

Subgroups — #10849324

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×#5585C211yesmaximal, characteristic
2C3×#2546C211yesmaximal, characteristic, complemented
3C2×#5084C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69632C212yesmaximal, complemented
7#69628C212yesmaximal, complemented
8#69627C212yesmaximal
9#69626C212yesmaximal, complemented
10#69488C211yesmaximal, characteristic, complemented
11#69474C211yesmaximal, characteristic, complemented
12#25659—31nomaximal, Sylow
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#947—21no—
17#5585—24no—
18#5585—24no—
19C2×#894—21no—
20C2×#894—21no—
21C2×#893—21no—
22C2×#875C2211yescharacteristic
23C2×#875C2211yescharacteristic
24C2×#874C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#398—21no—
29C3×#398—21no—
30C3×#391—21no—
31C3×#389C2212yes—
32C3×#389C2212yes—
33C3×C4×C4⋊C4C2211yescharacteristic
34#5089—24no—
35#5084—24no—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#559—31no—
49C2×#489—31no—
50C2×#476—31no—
51#2546D311yescharacteristic, complemented
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×Q8D411yescharacteristic
61C22×D12—21no—
62C22×D12—21no—
63C22×D12C2311yescharacteristic
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×C4×D3—21no—
67C22×Dic6C2311yescharacteristic
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—44no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85#947—28no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—21no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2312yes—
96C22×C3⋊C8C2312yes—
97#894—28no—
98#894—28no—
99#893—28no—
100#875—22no—
101#875—22no—
102#875—22no—
103#875D414yes—
104#874—22no—
105#874—22no—
106C4×D12—22no—
107C4×D12—22no—
108C4×D12—22no—
109C4×D12D414yes—
110C4×Dic6—22no—
111C4×Dic6—22no—
112C3×#95—21no—
113C3×#95—22no—
114C3×#95—22no—
115#810—24no—
116#810—24no—
117#809—24no—
118#809—24no—
119#809—24no—
120#809—24no—
121#808—24no—
122C4⋊(C3⋊C8)—22no—
123#803—22no—
124#559—64no—
125#559—64no—
126C2×C4⋊Q8—21no—
127C2×C4⋊Q8C2×D311yescharacteristic
128C2×#127—31no—
129C2×#124—31no—
130C2×#124—31no—
131C2×#124—61no—
132C2×#123—61no—
133C2×#123—61no—
134C2×C4×Q8—31no—
135C2×C4×Q8C2×D311yescharacteristic
136C2×C4×D4—31no—
137C2×C4×D4—31no—
138C2×C4×D4—61no—
139#489—64no—
140#476—64no—
141C2×#106—32no—
142C2×C4⋊C8—31no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×#97—31no—
150#398—21no—
151#398—21no—
152#391—21no—
153#389C2×D312yes—
154#389C2×D312yes—
155C4×C4⋊C4C2×D311yescharacteristic
156#340—32no—
157#336—32no—
158#336—32no—
159C23×D3—21no—
160C23×D3—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—41no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—21no—
177C22×Dic3—21no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—24no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—24no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12—82no—
189C2×D12C2×D412yes—
190C2×D12C2×D412yes—
191C2×C4×D3—24no—
192C2×C4×D3—24no—
193C2×C4×D3—82no—
194C2×C4×D3—82no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—21no—
198C2×Dic6C2×D412yes—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—44no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—44no—
208C3×C4⋊C4#4114yes—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—42no—
214C3×C42C2×D412yes—
215C3×C42C2×D412yes—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—24no—
222C4⋊Dic3—21no—
223C4⋊Dic3C2×D412yes—
224Dic3⋊C4—24no—
225C4×Dic3—24no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C22×Q8—21no—
233C22×Q8—31no—
234C22×Q8#6711yescharacteristic
235C22×D4—31no—
236C22×D4—31no—
237C22×D4—61no—
238C22×D4—61no—
239C23×C4—61no—
240C23×C4—61no—
241C22×C8—32no—
242C22×C8—32no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—44no—
246#127—62no—
247#127—62no—
248#124—34no—
249#124—62no—
250#124—62no—
251#124—68no—
252#124—62no—
253#123—68no—
254#123—68no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—34no—
260C4×D4—62no—
261C4×D4—62no—
262C4×D4—68no—
263C4×D4—62no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4#6711yescharacteristic
270C2×C4⋊C4#6711yescharacteristic
271C2×C4⋊C4#6711yescharacteristic
272C2×C4⋊C4C22×D312yes—
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—21no—
281C2×C42—61no—
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284C2×C42C22×D311yescharacteristic
285#106—64no—
286C4⋊C8—62no—
287#103—64no—
288#103—64no—
289#102—64no—
290#102—64no—
291#102—64no—
292#102—64no—
293#97—62no—
294#95—21no—
295#95—22no—
296#95—22no—
297C23×C3C22×D411yescharacteristic
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—42no—
308C3×Q8—44no—
309C3×Q8—84no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—24no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4#12012yes—
329C2×C3×C4#12012yes—
330C2×C3×C4AGL(1,ℤ/8)12yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C42−1+412yes—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339D12—22no—
340D12—22no—
341D12—24no—
342D12—48no—
343D12—41no—
344D12—41no—
345D12—42no—
346D12—84no—
347C4×D3—84no—
348C4×D3—84no—
349Dic6—24no—
350Dic6—41no—
351Dic6—41no—
352C3⋊C8—42no—
353C3⋊C8—42no—
354C24—61no—
355C24—61no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—32no—
359C2×Q8—41no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—61no—
364C2×Q8—62no—
365C2×Q8—62no—
366C2×D4—32no—
367C2×D4—32no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—122no—
377C2×D4—122no—
378C2×D4—242no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—64no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—64no—
385C22×C4—242no—
386C22×C4—242no—
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C2×#6711yescharacteristic
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C23×D311yescharacteristic
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—62no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—32no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—64no—
411C4⋊C4—61no—
412C4⋊C4#23914yes—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—42no—
424C42—64no—
425C42D3×D412yes—
426C42D3×D412yes—
427C22×C3C2×#12011yescharacteristic
428C22×C3#54812yes—
429C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
430C22×C3#57511yescharacteristic
431C22×C3#57712yes—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—44no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—82no—
450C3×C4#53211yescharacteristic
451C3×C4#53611yescharacteristic
452C3×C4#54712yes—
453Dic3—42no—
454Dic3—42no—
455C23—62no—
456C23—62no—
457C23—121no—
458C23—121no—
459C23—122no—
460C23—122no—
461C23—121no—
462C23—121no—
463C23—122no—
464C23—122no—
465C23C22×#6711yescharacteristic
466Q8—44no—
467Q8—64no—
468Q8—84no—
469Q8—121no—
470Q8—121no—
471D4—62no—
472D4—62no—
473D4—64no—
474D4—128no—
475D4—121no—
476D4—121no—
477D4—122no—
478D4—244no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—244no—
502C2×C4—244no—
503C2×C4#93212yes—
504C2×C4#93712yes—
505C2×C4#93712yes—
506C2×C4C2×D3×D411yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C2×#23912yes—
509C2×C4#100712yes—
510C8—122no—
511C8—122no—
512C2×C3#243211yescharacteristic
513C2×C3#300611yescharacteristic
514C2×C3#300911yescharacteristic
515C2×C3#337112yes—
516C2×C3#337312yes—
517D3—82no—
518D3—82no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22—242no—
526C22—242no—
527C22—242no—
528C22—242no—
529C22C2×#93211yescharacteristic
530C22C2×#93711yescharacteristic
531C22#581712yes—
532C22#582211yescharacteristic
533C22#583712yes—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—44no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—82no—
542C4—122no—
543C4—122no—
544C4#572411yescharacteristic
545C4#572511yescharacteristic
546C4#572612yes—
547C3#2565911yescharacteristic, complemented, Sylow
548C2—242no—
549C2—242no—
550C2#8262411yescharacteristic
551C2#8325511yescharacteristic
552C2#8340411yescharacteristic
553C2#8356912yes—
554C2#8369112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2546C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2546D3
hypocenterC3#25659
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C3×#2546C2semidirect
3. C2×#5084C2semidirect
4. C2×#5089C2semidirect
5. #71635C2semidirect
6. #69632C2semidirect
7. #69628C2semidirect
9. #69626C2semidirect
10. #69488C2semidirect
11. #69474C2semidirect
51. #2546D3semidirect
547. C3#25659semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C3×#25462
C2×#50842
C2×#50892
#716352
#696322
#696282
#696272
#696262
#694882
#694742
#256593