Finite Group Databasebeta

← #10900879

Subgroups — #10900879

Order 768.

1260 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×#536C211yesmaximal, characteristic
2C2×#5239C211yesmaximal, characteristic
3C2×#5228C211yesmaximal, characteristic, complemented
4#82263C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic
6#72132C211yesmaximal, characteristic, complemented
7#71793C212yesmaximal, complemented
8#71781C212yesmaximal, complemented
9#71729C212yesmaximal, complemented
10#71722C212yesmaximal, complemented
11#71584C211yesmaximal, characteristic, complemented
12#29633—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C3×#536—24no—
16C3×#536—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×#124C2212yes—
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×#117C2211yescharacteristic
23C22×#216C2212yes—
24C22×#216C2211yescharacteristic
25C22×#214C2211yescharacteristic
26C2×#924C2211yescharacteristic
27#5239—24no—
28#5228—24no—
29C2×#835C2212yes—
30C2×#835C2211yescharacteristic
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×#835C2212yes—
34C2×#832C2211yescharacteristic
35C2×#832C2212yes—
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#4985—21no—
40#4983—21no—
41#4952—22no—
42#4929C2212yes—
43#4928C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#536D311yescharacteristic
47C2×#452—31no—
48C2×#451—31no—
49#2625—31no—
50#2043—32no—
51#2042—32no—
52#2016—32no—
53#2015—32no—
54#1818—31no—
55#1788—31no—
56#1787—31no—
57C2×C3×#41—22no—
58C2×C3×#41—42no—
59C2×C3×#41—42no—
60C2×C3×#41D414yes—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8—42no—
65C22×C3×Q8D414yes—
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C22×Dic6C2311yescharacteristic
74C3×C4⋊Q8—24no—
75C3×C4⋊Q8—24no—
76C3×#124—24no—
77C3×#124—24no—
78C3×#123—216no—
79C3×C22⋊Q8—28no—
80C3×C22⋊Q8—28no—
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42C2312yes—
91C2×#217—21no—
92C2×#217—21no—
93C2×#216—24no—
94C2×#216—22no—
95C2×#216—42no—
96C2×#216—42no—
97C2×#216—42no—
98C2×#214—22no—
99C2×#214—42no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2312yes—
102#924—22no—
103#924—22no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—22no—
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C22×C3⋊C8C2311yescharacteristic
109#835—24no—
110#835—22no—
111#835—42no—
112#835—42no—
113#835—42no—
114#832—22no—
115#832—42no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119C3⋊#95—21no—
120#830—22no—
121#830—22no—
122C22×#41C2×D311yescharacteristic
123C23×Q8C2×D311yescharacteristic
124C22×Q16—32no—
125C22×Q16—31no—
126C22×SD16—31no—
127#536—24no—
128#536—24no—
129C2×C4⋊Q8C2×D312yes—
130C2×#124C2×D312yes—
131C2×#123—22no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8—31no—
135C2×#117C2×D311yescharacteristic
136#452—64no—
137#451—64no—
138C2×#103—32no—
139C2×#103—31no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#102—31no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—31no—
147#400—61no—
148#398—61no—
149#397—62no—
150#391—32no—
151#389—32no—
152#380—31no—
153#336—32no—
154#336—32no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C3×#41—44no—
158C3×#41—44no—
159C3×#41—48no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—48no—
170C2×C3×Q8—44no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8C2×D414yes—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—24no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—44no—
180C2×C3×D4C2×D412yes—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—24no—
187C22×C3×C4—44no—
188C22×C3×C4—44no—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C2×Dic6—21no—
199C2×Dic6—21no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—24no—
209C3×C42—22no—
210C3×C42—22no—
211#217—42no—
212#217—81no—
213#216—48no—
214#216—44no—
215#214—44no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220Dic3⋊C4—42no—
221Dic3⋊C4—82no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×#41—22no—
230C2×#41—42no—
231C2×#41—42no—
232C2×#41#6714yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—31no—
236C22×Q8—42no—
237C22×Q8—42no—
238C22×Q8#6714yes—
239C22×Q8C22×D312yes—
240C22×Q8C22×D311yescharacteristic
241C22×D4—21no—
242C22×D4C22×D311yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C22×D311yescharacteristic
246C2×Q16—64no—
247C2×Q16—62no—
248C2×Q16—122no—
249C2×Q16—122no—
250C2×Q16—122no—
251C2×SD16—62no—
252C2×SD16—122no—
253C22×C8—31no—
254C22×C8—32no—
255C22×C8—31no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258#124—24no—
259#124—24no—
260#123—216no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—62no—
264C22⋊Q8—62no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4#6711yescharacteristic
274C2×C4⋊C4#6711yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4#6712yes—
280C2×C42C22×D312yes—
281#103—64no—
282#103—62no—
283#103—122no—
284#103—122no—
285#103—122no—
286#102—62no—
287#102—122no—
288C22⋊C8—62no—
289C22⋊C8—62no—
290#95—61no—
291#95—62no—
292#95—62no—
293#95—62no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3#12012yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—24no—
306C3×Q8—22no—
307C3×Q8—24no—
308C3×Q8—22no—
309C3×Q8—48no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—48no—
316C3×D4—42no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—48no—
329C2×C3×C4—48no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4#12012yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4H(5,𝔽2)12yes—
337C2×C3×C4H(5,𝔽2)12yes—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—41no—
346Dic6—41no—
347Dic6—41no—
348Dic6—42no—
349C3⋊C8—41no—
350C3⋊C8—41no—
351C3⋊C8—42no—
352C24—21no—
353C24C2×#6711yescharacteristic
354#41—44no—
355#41—44no—
356#41—48no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—61no—
373C2×Q8—61no—
374C2×Q8C2×#6714yes—
375C2×Q8C2×#6712yes—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—44no—
381C2×D4C2×#6712yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4C2×#6712yes—
394C22×C4C2×#6712yes—
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6712yes—
397C22×C4C2×#6712yes—
398C22×C4C23×D311yescharacteristic
399Q16—128no—
400Q16—124no—
401SD16—124no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—62no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—122no—
418C4⋊C4—242no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—122no—
423C22⋊C4—241no—
424C42—22no—
425C42—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#53512yes—
438C22×C3#57711yescharacteristic
439C22×C3#57811yescharacteristic
440C22×C3#57812yes—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—44no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4#53411yescharacteristic
447C3×C4#53411yescharacteristic
448C3×C4#53412yes—
449Dic3—81no—
450Dic3—82no—
451Dic3—81no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23#95312yes—
462C23C22×#6711yescharacteristic
463Q8—24no—
464Q8—22no—
465Q8—24no—
466Q8—22no—
467Q8—48no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—121no—
472Q8—121no—
473Q8—122no—
474D4—22no—
475D4—22no—
476D4—48no—
477D4—42no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4#95312yes—
503C2×C4#100412yes—
504C2×C4#100412yes—
505C2×C4C22×#6711yescharacteristic
506C2×C4C22×#6711yescharacteristic
507C8—121no—
508C8—121no—
509C8—122no—
510C2×C3—42no—
511C2×C3—82no—
512C2×C3#261311yescharacteristic
513C2×C3#301111yescharacteristic
514C2×C3#301311yescharacteristic
515C2×C3#322312yes—
516C2×C3#322512yes—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22#580512yes—
528C22#583711yescharacteristic
529C22#583811yescharacteristic
530C22#583812yes—
531C22C2×#95311yescharacteristic
532C4—22no—
533C4—42no—
534C4—44no—
535C4—42no—
536C4—42no—
537C4—241no—
538C4—241no—
539C4—242no—
540C4#580411yescharacteristic
541C4#580411yescharacteristic
542C4#580412yes—
543C3#2963311yescharacteristic, complemented, Sylow
544C2—42no—
545C2—82no—
546C2#8341012yes—
547C2#8341412yes—
548C2#8405411yescharacteristic
549C2#8411411yescharacteristic
550C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#536D3
hypocenterC3#29633
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. C2×#5228C2semidirect
4. #82263C2semidirect
6. #72132C2semidirect
7. #71793C2semidirect
8. #71781C2semidirect
9. #71729C2semidirect
10. #71722C2semidirect
11. #71584C2semidirect
543. C3#29633semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×#52392
C2×#52282
#822632
#721352
#721322
#717932
#717812
#717292
#717222
#715842
#296333