Finite Group Databasebeta

← #10849334

Subgroups — #10849334

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 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#82478C211yesmaximal, characteristic, complemented
4C2×#5007C211yesmaximal, characteristic, complemented
5#71762C212yesmaximal
6#71722C212yesmaximal, complemented
7#71721C211yesmaximal, characteristic, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71628C212yesmaximal, complemented
10#71609C211yesmaximal, characteristic, complemented
11#71606C212yesmaximal, complemented
12#25660—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×#869C2211yescharacteristic
27#5229—21no—
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32C2×#832C2212yes—
33C2×#832C2212yes—
34#5055—24no—
35#5007—24no—
36#4979—21no—
37#4979—21no—
38#4975—21no—
39C2×#807C2212yes—
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×#474—31no—
51#2567—31no—
52#2042—32no—
53#2041—32no—
54#1985—31no—
55#1983—31no—
56#1863—32no—
57#1862—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
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2312yes—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8C2312yes—
108C22×C3⋊C8C2312yes—
109#869—22no—
110#869—22no—
111#835—24no—
112#835—24no—
113#832—24no—
114#832—24no—
115#832—24no—
116#832—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119C3⋊#95—21no—
120#807—24no—
121C4⋊(C3⋊C8)—22no—
122#803—22no—
123#559—24no—
124#559—24no—
125C2×C4⋊Q8—31no—
126C2×#127C2×D311yescharacteristic
127C2×#124—21no—
128C2×#124C2×D311yescharacteristic
129C2×#124C2×D311yescharacteristic
130C2×#123—21no—
131C2×#123—21no—
132C2×C4×Q8C2×D311yescharacteristic
133C2×C4×Q8C2×D311yescharacteristic
134C2×C4×D4—21no—
135C2×C4×D4C2×D311yescharacteristic
136C2×C4×D4C2×D311yescharacteristic
137#489—64no—
138#474—64no—
139C2×#107—32no—
140C2×C4⋊C8—31no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#97—31no—
148#395—61no—
149#395—61no—
150#391—61no—
151#389—32no—
152#389—32no—
153#384—61no—
154C4×C4⋊C4—31no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3—21no—
159C24×C3—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—41no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—82no—
178C2×C3×D4C2×D412yes—
179C2×C3×D4C2×D412yes—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—82no—
185C22×C3×C4—82no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—22no—
195C2×Dic6—42no—
196C2×Dic6—82no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4C2×D412yes—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C42—24no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42C2×D412yes—
215C3×C42C2×D412yes—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—42no—
220C4⋊Dic3#4114yes—
221Dic3⋊C4—82no—
222Dic3⋊C4—82no—
223C4×Dic3—42no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×Q8—61no—
231C22×Q8#6711yescharacteristic
232C22×Q8C22×D311yescharacteristic
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C22×D311yescharacteristic
236C22×D4C22×D311yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C22×C8—32no—
240C22×C8—32no—
241C4⋊Q8—62no—
242C4⋊Q8—62no—
243#127—22no—
244#127—22no—
245#124—28no—
246#124—22no—
247#124—22no—
248#124—22no—
249#124#6714yes—
250#123—28no—
251#123—28no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×D4—28no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4#6714yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4#6711yescharacteristic
268C2×C4⋊C4#6711yescharacteristic
269C2×C4⋊C4#6711yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—21no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280C2×C42C22×D311yescharacteristic
281C2×C42C22×D311yescharacteristic
282#107—64no—
283C4⋊C8—62no—
284#103—64no—
285#103—64no—
286#102—64no—
287#102—64no—
288#102—64no—
289#102—64no—
290#97—62no—
291#95—61no—
292#95—62no—
293#95—62no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3C22×D411yescharacteristic
305C3×Q8—24no—
306C3×Q8—44no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—24no—
312C3×D4—48no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4—42no—
316C3×D4—84no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—84no—
334C2×C3×C4—84no—
335C2×C3×C4#12012yes—
336C2×C3×C4#12012yes—
337C2×C3×C4#13712yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C2×#4112yes—
341C2×C3×C42−1+412yes—
342C2×Dic3—24no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348Dic6—84no—
349C3⋊C8—42no—
350C3⋊C8—42no—
351C24—21no—
352C24—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—122no—
360C2×Q8—242no—
361C2×Q8C2×#6712yes—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—82no—
373C2×D4C2×#6712yes—
374C2×D4C2×#6712yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C23×D311yescharacteristic
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—34no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—122no—
406C4⋊C4—242no—
407C4⋊C4—242no—
408C4⋊C4C2×#6712yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C42—24no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—122no—
421C42C2×#6712yes—
422C42C2×#6712yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#54812yes—
435C22×C3C2×#13711yescharacteristic
436C22×C3#57511yescharacteristic
437C22×C3#57612yes—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—42no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—42no—
446C3×C4#53211yescharacteristic
447C3×C4#53611yescharacteristic
448C3×C4#54712yes—
449Dic3—44no—
450Dic3—82no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23C22×#6711yescharacteristic
462Q8—24no—
463Q8—44no—
464Q8—41no—
465Q8—41no—
466Q8—244no—
467D4—24no—
468D4—22no—
469D4—22no—
470D4—48no—
471D4—42no—
472D4—41no—
473D4—41no—
474D4—84no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—64no—
492C2×C4—84no—
493C2×C4—84no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4#95112yes—
500C2×C4#95312yes—
501C2×C4#95312yes—
502C2×C4D3×#4112yes—
503C2×C4#101012yes—
504C2×C4C22×#6711yescharacteristic
505C2×C4C22×#6711yescharacteristic
506C8—122no—
507C8—122no—
508C2×C3—82no—
509C2×C3—82no—
510C2×C3#243211yescharacteristic
511C2×C3#300711yescharacteristic
512C2×C3#300811yescharacteristic
513C2×C3#337012yes—
514C2×C3#337512yes—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22#582211yescharacteristic
526C22C2×#95111yescharacteristic
527C22#583112yes—
528C22#583512yes—
529C22C2×#95311yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—42no—
538C4—124no—
539C4—242no—
540C4#582412yes—
541C4#582511yescharacteristic
542C4#583211yescharacteristic
543C3#2566011yescharacteristic, complemented, Sylow
544C2—82no—
545C2—82no—
546C2#8371412yes—
547C2#8371912yes—
548C2#8408411yescharacteristic
549C2#8409311yescharacteristic
550C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#559D3
hypocenterC3#25660
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. #82478C2semidirect
4. C2×#5007C2semidirect
6. #71722C2semidirect
7. #71721C2semidirect
8. #71635C2semidirect
9. #71628C2semidirect
10. #71609C2semidirect
11. #71606C2semidirect
543. C3#25660semidirect

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
#824782
C2×#50072
#717622
#717222
#717212
#716352
#716282
#716092
#716062
#256603