Finite Group Databasebeta

← #10828838

Subgroups — #10828838

Order 768.

1080 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5541C211yesmaximal, characteristic
2C3×#2487C211yesmaximal, characteristic
3C2×#4970C211yesmaximal, characteristic, complemented
4C22×#808C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic
6#69632C212yesmaximal, complemented
7#69606C212yesmaximal, complemented
8#69526C212yesmaximal, complemented
9#69514C211yesmaximal, characteristic, complemented
10#69506C212yesmaximal, complemented
11#69501C211yesmaximal, characteristic, complemented
12#24095—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#935—21no—
16C2×#926C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21#5541—24no—
22#5541—24no—
23C2×#889—21no—
24C2×#879—21no—
25C2×#877—21no—
26C2×#870C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#400—21no—
29C3×#399—21no—
30C3×#389C2212yes—
31C3×#386—21no—
32C3×#385—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C2×#830C2211yescharacteristic
36C2×#830C2212yes—
37#4970—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#808—24no—
41C2×#808C2212yes—
42C2×#808C2212yes—
43C2×#807C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#552—31no—
49C2×#484—31no—
50C22×#106—31no—
51#2487D311yescharacteristic
52#2059—32no—
53#2035—32no—
54#1892—31no—
55#1869—31no—
56#1865—32no—
57#1854—32no—
58#1823—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C22×Dic6C2311yescharacteristic
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4D411yescharacteristic
77C2×C3×C22⋊C4D411yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×#217—21no—
80C2×#217—21no—
81C2×#217—21no—
82#935—28no—
83#926—22no—
84#926—22no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3D414yes—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91#924—22no—
92#924—22no—
93#924—22no—
94#924D414yes—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2312yes—
106#889—28no—
107#879—28no—
108#877—28no—
109#870—22no—
110#870—22no—
111C4×Dic6—22no—
112C4×Dic6—22no—
113C3×#95—22no—
114C3×#95—21no—
115C3×#95—22no—
116#830—22no—
117#830—24no—
118#810—24no—
119#810—24no—
120#808—24no—
121#808—24no—
122#807—24no—
123C23×C8—31no—
124#552—64no—
125#552—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#125—61no—
129C2×#123—61no—
130C2×C22⋊Q8—31no—
131C2×C22⋊Q8—31no—
132C2×C4×Q8—31no—
133C2×C4×D4C2×D311yescharacteristic
134C2×#117—31no—
135C2×#117—61no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138#484—64no—
139C2×#107—32no—
140C2×#106—32no—
141C2×#106—32no—
142C2×#106—64no—
143C2×#103—32no—
144C2×#103—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—32no—
147#400—21no—
148#399—21no—
149#389C2×D312yes—
150#386—21no—
151#385—21no—
152#380C2×D311yescharacteristic
153#380C2×D312yes—
154#340—32no—
155#340—32no—
156#340—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C24×C3C2×D411yescharacteristic
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—42no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C22×Dic3—24no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—42no—
186C22×Dic3—42no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6C2×D412yes—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—82no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—82no—
202C3×C42—21no—
203C3×C42—21no—
204#217—24no—
205#217—24no—
206#217—24no—
207C4⋊Dic3—24no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3C2×D412yes—
212C4⋊Dic3C2×D412yes—
213C4⋊Dic3C2×D412yes—
214Dic3⋊C4—24no—
215Dic3⋊C4—24no—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219C4×Dic3—24no—
220C4×Dic3—24no—
221C2×C3⋊C8—28no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8#4114yes—
228C22×Q8—31no—
229C22×D4#6711yescharacteristic
230C23×C4—21no—
231C23×C4—61no—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—32no—
236C22×C8—64no—
237C22×C8—62no—
238#126—68no—
239#125—62no—
240#125—62no—
241#125—68no—
242#123—68no—
243C22⋊Q8—34no—
244C22⋊Q8—62no—
245C22⋊Q8—62no—
246C22⋊Q8—62no—
247C4×Q8—62no—
248C4×Q8—62no—
249C4×D4—22no—
250C4×D4—22no—
251#117—68no—
252#117—62no—
253#117—62no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4#6711yescharacteristic
270C2×C4⋊C4C22×D312yes—
271C2×C4⋊C4C22×D312yes—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4#6711yescharacteristic
278C2×C22⋊C4#6711yescharacteristic
279C2×C42—61no—
280C2×C42—61no—
281C2×C42C22×D311yescharacteristic
282#107—64no—
283#106—64no—
284#106—64no—
285#103—64no—
286#103—64no—
287C22⋊C8—64no—
288C22⋊C8—62no—
289#95—22no—
290#95—21no—
291#95—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3#12012yes—
300C23×C3#13512yes—
301C23×C3#13712yes—
302C23×C3C22×D411yescharacteristic
303C3×D4—44no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—84no—
325C2×C3×C4#12012yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4112yes—
329C2×C3×C42−1+412yes—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—42no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—48no—
341Dic6—24no—
342Dic6—41no—
343Dic6—41no—
344C3⋊C8—24no—
345C3⋊C8—42no—
346C24C2×#6711yescharacteristic
347C24C2×#6711yescharacteristic
348C2×Q8—32no—
349C2×Q8—61no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—42no—
364C22×C4—64no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—82no—
370C22×C4—122no—
371C22×C4—122no—
372C22×C4D3×D412yes—
373C22×C4D3×D412yes—
374C22×C4C2×#6711yescharacteristic
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6711yescharacteristic
377C22×C4C2×#6711yescharacteristic
378C22×C4C23×D311yescharacteristic
379C2×C8—34no—
380C2×C8—68no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—32no—
392C4⋊C4—32no—
393C4⋊C4—32no—
394C4⋊C4—42no—
395C4⋊C4—64no—
396C4⋊C4—64no—
397C4⋊C4—64no—
398C4⋊C4—64no—
399C4⋊C4—64no—
400C4⋊C4—64no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—82no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—42no—
408C22⋊C4—64no—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—82no—
412C42—21no—
413C42—21no—
414C42—64no—
415C42—64no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3#45214yes—
426C22×C3C2×#12011yescharacteristic
427C22×C3#54712yes—
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3#57712yes—
431C3×C4—22no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—82no—
436C3×C4—82no—
437C3×C4#52511yescharacteristic
438C3×C4#53611yescharacteristic
439C3×C4#54712yes—
440Dic3—42no—
441Dic3—42no—
442Dic3—42no—
443Dic3—42no—
444C23—22no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23#93712yes—
452C23#94212yes—
453C23#95012yes—
454C23C22×#6711yescharacteristic
455Q8—64no—
456Q8—121no—
457Q8—121no—
458D4—44no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—84no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—128no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4#93712yes—
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4D3×#4112yes—
495C2×C4#101012yes—
496C8—64no—
497C8—122no—
498C2×C3—24no—
499C2×C3—81no—
500C2×C3#240611yescharacteristic
501C2×C3C2×#45211yescharacteristic
502C2×C3#301011yescharacteristic
503C2×C3#333712yes—
504C2×C3#334412yes—
505C22—24no—
506C22—22no—
507C22—24no—
508C22—22no—
509C22—41no—
510C22—42no—
511C22—41no—
512C22—42no—
513C22—41no—
514C22#504714yes—
515C22C2×#93711yescharacteristic
516C22C2×#94211yescharacteristic
517C22C2×#95011yescharacteristic
518C22#582412yes—
519C22#583612yes—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—82no—
525C4—82no—
526C4—122no—
527C4—122no—
528C4—122no—
529C4—122no—
530C4#564811yescharacteristic
531C4#565512yes—
532C4#565611yescharacteristic
533C3#2409511yescharacteristic, complemented, Sylow
534C2—24no—
535C2—81no—
536C2C2×#504711yescharacteristic
537C2#8284111yescharacteristic
538C2#8339011yescharacteristic
539C2#8360612yes—
540C2#8362412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2487C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2487D3
hypocenterC3#24095
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. C2×#4970C2semidirect
4. C22×#808C2semidirect
6. #69632C2semidirect
7. #69606C2semidirect
8. #69526C2semidirect
9. #69514C2semidirect
10. #69506C2semidirect
11. #69501C2semidirect
533. C3#24095semidirect

Maximal subgroups

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

subgroupindex
C2×#55412
C3×#24872
C2×#49702
C22×#8082
#716102
#696322
#696062
#695262
#695142
#695062
#695012
#240953