Finite Group Databasebeta

← #10849323

Subgroups — #10849323

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×C3×#559C211yesmaximal, characteristic
2C2×#5055C211yesmaximal, characteristic, complemented
3#82475C211yesmaximal, characteristic, complemented
4C2×#5008C211yesmaximal, characteristic, complemented
5#71762C212yesmaximal
6#71722C212yesmaximal, complemented
7#71721C211yesmaximal, characteristic, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71629C212yesmaximal, complemented
10#71609C211yesmaximal, characteristic, complemented
11#71607C212yesmaximal, complemented
12#25659—31nomaximal, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×#123—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×#944—21no—
27C2×#869C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32C2×#832C2212yes—
33C2×#832C2212yes—
34#5055—24no—
35#5008—24no—
36#4981—21no—
37#4981—21no—
38#4973—21no—
39C2×#808C2212yes—
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2212yes—
42#4928C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#559D311yescharacteristic
49C2×#489—31no—
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×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4—21no—
67C22×Dic6—21no—
68C3×#127—22no—
69C3×#127—22no—
70C3×#124—28no—
71C3×#124—22no—
72C3×#124—22no—
73C3×#124—22no—
74C3×#124D414yes—
75C3×#123—28no—
76C3×#123—28no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×D4—28no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4D414yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101#944—44no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2312yes—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×C4×Dic3—21no—
108C22×C3⋊C8C2312yes—
109C22×C3⋊C8C2312yes—
110#869—22no—
111#869—22no—
112#835—24no—
113#835—24no—
114#832—24no—
115#832—24no—
116#832—24no—
117#832—24no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120C3⋊#95—21no—
121#808—24no—
122C4⋊(C3⋊C8)—22no—
123#803—22no—
124#559—24no—
125#559—24no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—61no—
128C2×#127C2×D311yescharacteristic
129C2×#124—21no—
130C2×#124C2×D311yescharacteristic
131C2×#124C2×D311yescharacteristic
132C2×#123—21no—
133C2×#123—21no—
134C2×C4×Q8C2×D311yescharacteristic
135C2×C4×Q8C2×D311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4C2×D311yescharacteristic
138C2×C4×D4C2×D311yescharacteristic
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—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—
160C24×C3—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—41no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—82no—
179C2×C3×D4C2×D412yes—
180C2×C3×D4C2×D412yes—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—82no—
186C22×C3×C4—82no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—22no—
195C22×Dic3—22no—
196C2×Dic6—42no—
197C2×Dic6—42no—
198C2×Dic6—42no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C42—24no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—42no—
222C4⋊Dic3#4114yes—
223Dic3⋊C4—44no—
224Dic3⋊C4—44no—
225C4×Dic3—42no—
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—61no—
233C22×Q8#6711yescharacteristic
234C22×Q8C22×D311yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C22×D311yescharacteristic
238C22×D4C22×D311yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C22×C8—32no—
242C22×C8—32no—
243C4⋊Q8—62no—
244C4⋊Q8—62no—
245C4⋊Q8—124no—
246#127—22no—
247#127—22no—
248#124—28no—
249#124—22no—
250#124—22no—
251#124—22no—
252#124#6714yes—
253#123—28no—
254#123—28no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—28no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4#6714yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4#6711yescharacteristic
271C2×C4⋊C4#6711yescharacteristic
272C2×C4⋊C4#6711yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
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—61no—
295#95—62no—
296#95—62no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—24no—
309C3×Q8—44no—
310C3×Q8—41no—
311C3×Q8—41no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—24no—
315C3×D4—48no—
316C3×D4—41no—
317C3×D4—41no—
318C3×D4—42no—
319C3×D4—84no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—84no—
337C2×C3×C4—84no—
338C2×C3×C4#12012yes—
339C2×C3×C4#12012yes—
340C2×C3×C4AGL(1,ℤ/8)12yes—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C2×#4112yes—
344C2×C3×C42−1+412yes—
345C2×Dic3—24no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351Dic6—84no—
352C3⋊C8—42no—
353C3⋊C8—42no—
354C24—21no—
355C24—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—122no—
363C2×Q8—122no—
364C2×Q8—122no—
365C2×Q8C2×#6712yes—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—82no—
377C2×D4C2×#6712yes—
378C2×D4C2×#6712yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—82no—
386C22×C4—82no—
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—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—34no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—124no—
410C4⋊C4—124no—
411C4⋊C4—122no—
412C4⋊C4C2×#6712yes—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C42—24no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—122no—
425C42C2×#6712yes—
426C42C2×#6712yes—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54812yes—
439C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
440C22×C3#57511yescharacteristic
441C22×C3#57712yes—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4#53211yescharacteristic
451C3×C4#53611yescharacteristic
452C3×C4#54712yes—
453Dic3—44no—
454Dic3—82no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C22×#6711yescharacteristic
466Q8—24no—
467Q8—44no—
468Q8—41no—
469Q8—41no—
470Q8—244no—
471D4—24no—
472D4—22no—
473D4—22no—
474D4—48no—
475D4—42no—
476D4—41no—
477D4—41no—
478D4—84no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—64no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4#94912yes—
504C2×C4#95312yes—
505C2×C4#95312yes—
506C2×C4D3×#4112yes—
507C2×C4#101012yes—
508C2×C4C22×#6711yescharacteristic
509C2×C4C22×#6711yescharacteristic
510C8—122no—
511C8—122no—
512C2×C3—82no—
513C2×C3—82no—
514C2×C3#243211yescharacteristic
515C2×C3#300611yescharacteristic
516C2×C3#300911yescharacteristic
517C2×C3#337112yes—
518C2×C3#337312yes—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×#94911yescharacteristic
530C22#582211yescharacteristic
531C22#583112yes—
532C22#583612yes—
533C22C2×#95311yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—41no—
541C4—42no—
542C4—124no—
543C4—242no—
544C4#582412yes—
545C4#582511yescharacteristic
546C4#583211yescharacteristic
547C3#2565911yescharacteristic, complemented, Sylow
548C2—82no—
549C2—82no—
550C2#8371512yes—
551C2#8371812yes—
552C2#8408211yescharacteristic
553C2#8409311yescharacteristic
554C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#559D3
hypocenterC3#25659
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×#5055C2semidirect
3. #82475C2semidirect
4. C2×#5008C2semidirect
6. #71722C2semidirect
7. #71721C2semidirect
8. #71635C2semidirect
9. #71629C2semidirect
10. #71609C2semidirect
11. #71607C2semidirect
547. C3#25659semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50552
#824752
C2×#50082
#717622
#717222
#717212
#716352
#716292
#716092
#716072
#256593