Finite Group Databasebeta

← #10829973

Subgroups — #10829973

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×#5090C211yesmaximal, characteristic, complemented
3#82482C211yesmaximal, characteristic, complemented
4C2×C4×#214C211yesmaximal, characteristic
5#71785C211yesmaximal, characteristic, complemented
6#71729C212yesmaximal
7#71722C212yesmaximal
8#71634C211yesmaximal, characteristic, complemented
9#71632C212yesmaximal, complemented
10#71629C212yesmaximal, complemented
11#71604C211yesmaximal, characteristic
12#24189—31nomaximal, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#124C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C22×#216C2212yes—
24C22×#214C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26#5229—21no—
27C2×#835C2212yes—
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#832C2211yescharacteristic
31C2×#832C2212yes—
32#5090—24no—
33C4×#214—24no—
34#4982—22no—
35#4972—21no—
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4935C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556D311yescharacteristic
47C2×#462—31no—
48C2×C4×SD16—31no—
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1956—31no—
54#1878—32no—
55#1865—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×Dic6C2311yescharacteristic
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D414yes—
64C3×#126—216no—
65C3×#124—24no—
66C3×#124—24no—
67C3×C22⋊Q8—28no—
68C3×C4×Q8—24no—
69C3×C4×Q8—24no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—22no—
73C3×C4×D4D414yes—
74C3×#117—28no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—22no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#216—24no—
88C2×#216—24no—
89C2×#216—42no—
90C2×#214—22no—
91C2×#214—22no—
92C2×#214—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2312yes—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—21no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C4×Dic6—22no—
104C4×Dic6—22no—
105#835—24no—
106#835—24no—
107#835—42no—
108#832—22no—
109#832—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#810—22no—
114#808—22no—
115C4⋊(C3⋊C8)—22no—
116C4×C3⋊C8—22no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—24no—
120#556—24no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#126—22no—
123C2×#124C2×D312yes—
124C2×C22⋊Q8—21no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D312yes—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—21no—
130#462—64no—
131C4×SD16—64no—
132C2×#106—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C4×C8—31no—
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8C2×D414yes—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4C2×D412yes—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6#4112yes—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—24no—
195C3×C42—28no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#216—48no—
201#214—28no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3—42no—
207C4⋊Dic3—81no—
208Dic3⋊C4—42no—
209Dic3⋊C4—82no—
210C4×Dic3—42no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—31no—
221C22×Q8#6711yescharacteristic
222C22×Q8C22×D312yes—
223C22×D4C22×D311yescharacteristic
224C23×C4—21no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—62no—
231C22×C8—31no—
232C22×C8—32no—
233C22×C8—31no—
234C4⋊Q8—22no—
235C4⋊Q8#6714yes—
236#126—216no—
237#124—24no—
238#124—24no—
239C22⋊Q8—28no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—62no—
245C4×Q8—62no—
246C4×D4—22no—
247C4×D4#6714yes—
248#117—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4#6711yescharacteristic
259C2×C4⋊C4#6712yes—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C42—22no—
265C2×C42—61no—
266C2×C42C22×D312yes—
267C2×C42C22×D311yescharacteristic
268#106—62no—
269C4⋊C8—62no—
270#103—64no—
271#103—64no—
272#103—62no—
273#103—122no—
274#102—62no—
275#102—122no—
276C4×C8—62no—
277#95—62no—
278#95—62no—
279#95—62no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3C22×D411yescharacteristic
288C3×Q8—24no—
289C3×Q8—24no—
290C3×Q8—42no—
291C3×Q8—42no—
292C3×Q8—44no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—41no—
296C3×D4—41no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—48no—
312C2×C3×C4#12012yes—
313C2×C3×C4#12012yes—
314C2×C3×C4#13512yes—
315C2×C3×C4#13712yes—
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×C3×C42−1+411yescharacteristic
322C2×Dic3—22no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331Dic6—22no—
332Dic6—22no—
333Dic6—41no—
334Dic6—41no—
335C3⋊C8—22no—
336C3⋊C8—41no—
337C3⋊C8—42no—
338C24—21no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—32no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8C2×#6714yes—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4C2×#6712yes—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—61no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4C2×#6712yes—
369C22×C4C2×#6712yes—
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C23×D311yescharacteristic
373Q16—128no—
374SD16—68no—
375C2×C8—32no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—62no—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—21no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—122no—
395C4⋊C4—122no—
396C4⋊C4—242no—
397C4⋊C4—241no—
398C4⋊C4C2×#6712yes—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—24no—
402C42—28no—
403C42—22no—
404C42—22no—
405C42—122no—
406C42C2×#6712yes—
407C42C2×#6712yes—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#54811yescharacteristic
417C22×C3#55011yescharacteristic
418C22×C3C2×#13511yescharacteristic
419C22×C3C2×#13711yescharacteristic
420C22×C3#57711yescharacteristic
421C22×C3#57811yescharacteristic
422C3×C4—22no—
423C3×C4—44no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4#45214yes—
428C3×C4#52511yescharacteristic
429C3×C4#53711yescharacteristic
430C3×C4#54611yescharacteristic
431C3×C4#54711yescharacteristic
432Dic3—42no—
433Dic3—82no—
434Dic3—81no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C22×#6711yescharacteristic
443Q8—24no—
444Q8—24no—
445Q8—44no—
446Q8—42no—
447Q8—42no—
448Q8—62no—
449Q8—62no—
450Q8—121no—
451Q8—121no—
452D4—22no—
453D4—22no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—62no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—121no—
477C2×C4—122no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4#95012yes—
481C2×C4#95112yes—
482C2×C4#95312yes—
483C2×C4#95312yes—
484C2×C4D3×#4111yescharacteristic
485C2×C4D3×#4111yescharacteristic
486C2×C4#100911yescharacteristic
487C2×C4#101011yescharacteristic
488C2×C4C22×#6711yescharacteristic
489C2×C4C22×#6711yescharacteristic
490C8—62no—
491C8—122no—
492C8—121no—
493C2×C3—82no—
494C2×C3#240711yescharacteristic
495C2×C3C2×#45211yescharacteristic
496C2×C3#301311yescharacteristic
497C2×C3#336411yescharacteristic
498C2×C3#336911yescharacteristic
499C2×C3#339411yescharacteristic
500C2×C3#339811yescharacteristic
501C22—42no—
502C22—42no—
503C22—42no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22C2×#95011yescharacteristic
509C22C2×#95111yescharacteristic
510C22#583111yescharacteristic
511C22#583411yescharacteristic
512C22#583711yescharacteristic
513C22#583811yescharacteristic
514C22C2×#95311yescharacteristic
515C4—22no—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—42no—
520C4—122no—
521C4—241no—
522C4—242no—
523C4#523914yes—
524C4#582411yescharacteristic
525C4#582811yescharacteristic
526C4#582911yescharacteristic
527C4#583311yescharacteristic
528C3#2418911yescharacteristic, complemented, Sylow
529C2—82no—
530C2#8367611yescharacteristic
531C2#8367711yescharacteristic
532C2#8367911yescharacteristic
533C2#8368011yescharacteristic
534C2C2×#523911yescharacteristic
535C2#8409711yescharacteristic
536C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#5090C2semidirect
3. #82482C2semidirect
5. #71785C2semidirect
8. #71634C2semidirect
9. #71632C2semidirect
10. #71629C2semidirect
528. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50902
#824822
C2×C4×#2142
#717852
#717292
#717222
#716342
#716322
#716292
#716042
#241893