Finite Group Databasebeta

← #10828592

Subgroups — #10828592

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic, complemented
3C2×#5034C211yesmaximal, characteristic
4#82234C211yesmaximal, characteristic, complemented
5#71783C212yesmaximal, complemented
6#71722C212yesmaximal, complemented
7#71693C211yesmaximal, characteristic, complemented
8#71678C211yesmaximal, characteristic
9#71629C212yesmaximal, complemented
10#71607C212yesmaximal, complemented
11#71584C211yesmaximal, characteristic, complemented
12#24074—31nomaximal, Sylow
13C3×#542—24no—
14C3×#542—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#124—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C2×#926C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32C2×#830C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#5043—24no—
35#5034—24no—
36#4977—21no—
37#4975—21no—
38C2×#808C2212yes—
39C2×#808C2212yes—
40#4946—21no—
41#4946—21no—
42#4928C2212yes—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#542D311yescharacteristic
49C2×#481—31no—
50C2×#483—31no—
51#2317—31no—
52#2058—32no—
53#2042—32no—
54#1875—31no—
55#1873—31no—
56#1865—32no—
57#1864—32no—
58#1818—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C23×Dic3—21no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×#126—28no—
69C3×#124—28no—
70C3×#124—22no—
71C3×#124—22no—
72C3×#123—28no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8D414yes—
76C3×#121—22no—
77C3×#121D414yes—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102#926—24no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2312yes—
108C2×Dic3⋊C4—22no—
109C2×Dic3⋊C4—22no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#835—24no—
114#835—24no—
115#832—24no—
116#832—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#830—22no—
120#830—22no—
121#808—24no—
122#808—24no—
123#542—24no—
124#542—24no—
125C2×C4⋊Q8C2×D311yescharacteristic
126C2×#126—21no—
127C2×#124—21no—
128C2×#124C2×D311yescharacteristic
129C2×#123—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×#121C2×D311yescharacteristic
133C2×C4×Q8C2×D311yescharacteristic
134C2×C4×D4C2×D311yescharacteristic
135C2×#117—31no—
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138#483—64no—
139#481—64no—
140C2×#106—32no—
141C2×#106—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×C22⋊C8—31no—
147C2×C22⋊C8—31no—
148#389—32no—
149#389—32no—
150#385—61no—
151#385—61no—
152#384—61no—
153#382—61no—
154#340—32no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—24no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—82no—
165C2×C3×Q8C2×D412yes—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—82no—
171C2×C3×D4C2×D412yes—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—24no—
178C22×C3×C4—82no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—42no—
190C22×Dic3—82no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4C2×D412yes—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C42—24no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—21no—
212#217—24no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4⋊Dic3#4114yes—
218Dic3⋊C4—82no—
219Dic3⋊C4—82no—
220C4×Dic3—24no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C22×Q8—21no—
228C22×Q8C22×D311yescharacteristic
229C22×D4—21no—
230C22×D4C22×D311yescharacteristic
231C23×C4—21no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—32no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238#126—28no—
239#124—28no—
240#124—22no—
241#124—22no—
242#123—28no—
243C22⋊Q8—28no—
244C22⋊Q8—22no—
245C22⋊Q8#6714yes—
246#121—22no—
247#121#6714yes—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×D4—22no—
251C4×D4—22no—
252#117—22no—
253#117—22no—
254#117—64no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4#6711yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C4⋊C4C22×D311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4#6711yescharacteristic
276C2×C22⋊C4#6711yescharacteristic
277C2×C42—21no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280C2×C42C22×D311yescharacteristic
281#106—64no—
282#106—64no—
283#103—64no—
284#103—64no—
285#102—64no—
286#102—64no—
287C22⋊C8—62no—
288C22⋊C8—62no—
289#95—62no—
290#95—62no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3#12012yes—
301C23×C3C22×D411yescharacteristic
302C3×Q8—24no—
303C3×Q8—42no—
304C3×Q8—84no—
305C3×D4—24no—
306C3×D4—42no—
307C3×D4—84no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—84no—
328C2×C3×C4#12012yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×C3×C42−1+412yes—
333C2×Dic3—24no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—84no—
340C3⋊C8—42no—
341C3⋊C8—42no—
342C24—21no—
343C24C2×#6711yescharacteristic
344C2×Q8—24no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—82no—
349C2×Q8C2×#6712yes—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—82no—
355C2×D4C2×#6712yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—82no—
365C22×C4—122no—
366C22×C4—242no—
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C2×#6712yes—
373C22×C4C2×#6712yes—
374C22×C4C23×D311yescharacteristic
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8—62no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—34no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—122no—
393C4⋊C4—242no—
394C4⋊C4—242no—
395C4⋊C4C2×#6712yes—
396C4⋊C4C2×#6712yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—64no—
405C42—24no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—64no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#52511yescharacteristic
423C22×C3#53611yescharacteristic
424C22×C3#57712yes—
425C22×C3#57712yes—
426C3×C4—22no—
427C3×C4—42no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4#54712yes—
435C3×C4#54712yes—
436Dic3—44no—
437Dic3—82no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23#95312yes—
448C23C22×#6711yescharacteristic
449Q8—24no—
450Q8—42no—
451Q8—84no—
452D4—24no—
453D4—42no—
454D4—84no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—64no—
475C2×C4—84no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—244no—
482C2×C4#95312yes—
483C2×C4D3×#4112yes—
484C2×C4#101012yes—
485C2×C4C22×#6711yescharacteristic
486C2×C4C22×#6711yescharacteristic
487C8—122no—
488C8—122no—
489C2×C3—42no—
490C2×C3—82no—
491C2×C3#240611yescharacteristic
492C2×C3#300911yescharacteristic
493C2×C3#301011yescharacteristic
494C2×C3#312512yes—
495C2×C3#324412yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—82no—
505C22—82no—
506C22#582511yescharacteristic
507C22#582811yescharacteristic
508C22#583612yes—
509C22#583712yes—
510C22C2×#95311yescharacteristic
511C4—22no—
512C4—42no—
513C4—41no—
514C4—41no—
515C4—42no—
516C4—42no—
517C4—41no—
518C4—41no—
519C4—124no—
520C4—242no—
521C4#582412yes—
522C4#582412yes—
523C3#2407411yescharacteristic, complemented, Sylow
524C2—42no—
525C2—82no—
526C2#8345512yes—
527C2#8346212yes—
528C2#8409011yescharacteristic
529C2#8410511yescharacteristic
530C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#542C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#542D3
hypocenterC3#24074
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
2. C2×#5043C2semidirect
4. #82234C2semidirect
5. #71783C2semidirect
6. #71722C2semidirect
7. #71693C2semidirect
9. #71629C2semidirect
10. #71607C2semidirect
11. #71584C2semidirect
523. C3#24074semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×#50432
C2×#50342
#822342
#717832
#717222
#716932
#716782
#716292
#716072
#715842
#240743