Finite Group Databasebeta

← #10726609

Subgroups — #10726609

Order 768.

2494 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#924C211yesmaximal, characteristic
2C23×Dic3⋊C4C211yesmaximal, characteristic
3C2×#4950C212yesmaximal
4C2×#4656C212yesmaximal, complemented
5C3×C22⋊#95C211yesmaximal, characteristic
6#71544C212yesmaximal
7#69420C214yesmaximal, complemented
8#69410C212yesmaximal, complemented
9#14720—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×Dic3C2211yescharacteristic
12C24×Dic3C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2214yes—
18C2×#924—24no—
19C2×#924—28no—
20C22×Dic3⋊C4—24no—
21C22×Dic3⋊C4—28no—
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30#4950—28no—
31#4656—28no—
32C22×C22⋊Q8—31no—
33C23×C4⋊C4—31no—
34C2×#395—32no—
35C2×#385—32no—
36#1444—34no—
37#1441—32no—
38C22⋊#95—32no—
39C22⋊#95D311yescharacteristic
40C25×C3C2311yescharacteristic
41C23×C3×C4—24no—
42C23×C3×C4—24no—
43C23×C3×C4—42no—
44C23×C3×C4D414yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2314yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—22no—
49C23×Dic3—22no—
50C23×Dic3—24no—
51C23×Dic3—24no—
52C23×Dic3—22no—
53C23×Dic3—24no—
54C23×Dic3—24no—
55C23×Dic3—42no—
56C23×Dic3Q814yescomplemented
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C22×Dic6—22no—
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—42no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4Q818yescomplemented
68C2×#217—24no—
69C2×#217—24no—
70C2×#217D418yes—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—28no—
74C2×C4⋊Dic3—28no—
75C2×C4⋊Dic3—44no—
76#924—232no—
77C2×Dic3⋊C4—232no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4D418yes—
87C3×#95—24no—
88C3×#95—42no—
89C3⋊#95—28no—
90C3⋊#95—28no—
91C3⋊#95—28no—
92C23×Q8—31no—
93C24×C4—31no—
94C24×C4—31no—
95C24×C4C2×D311yescharacteristic
96C2×C22⋊Q8—68no—
97C2×C22⋊Q8—64no—
98C22×C4⋊C4—32no—
99C22×C4⋊C4—32no—
100C22×C4⋊C4—32no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—34no—
103C22×C4⋊C4—68no—
104C22×C4⋊C4—64no—
105C22×C22⋊C4—32no—
106C22×C22⋊C4C2×D312yes—
107#395—68no—
108#385—68no—
109C2×#95—34no—
110C2×#95—34no—
111C2×#95—34no—
112C2×#95C2×D312yes—
113C2×#95C2×D312yes—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3C2×D412yes—
119C24×C3C2×D414yes—
120C24×C3C2×Q814yes—
121C24×C3C2×Q812yes—
122C24×C3#4112yes—
123C24×C3C2411yescharacteristic
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—28no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—28no—
137C22×C3×C4—44no—
138C22×C3×C4—416no—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D414yes—
142C22×C3×C4C2×Q818yes—
143C22×Dic3—28no—
144C22×Dic3—216no—
145C22×Dic3—28no—
146C22×Dic3—28no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—216no—
152C22×Dic3—28no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—24no—
162C22×Dic3—24no—
163C22×Dic3—24no—
164C22×Dic3—24no—
165C22×Dic3—416no—
166C22×Dic3—44no—
167C22×Dic3—44no—
168C22×Dic3C2×D414yes—
169C22×Dic3C2×D414yes—
170C22×Dic3C2×D414yes—
171C22×Dic3C2×D414yes—
172C22×Dic3C2×Q814yes—
173C22×Dic3#4114yes—
174C2×Dic6—28no—
175C2×Dic6—416no—
176C2×Dic6—44no—
177C2×Dic6—44no—
178C3×C22⋊C4—216no—
179#217—216no—
180C4⋊Dic3—28no—
181C4⋊Dic3—416no—
182Dic3⋊C4—216no—
183Dic3⋊C4—216no—
184Dic3⋊C4—216no—
185C25C22×D311yescharacteristic
186C22×Q8—64no—
187C22×Q8—62no—
188C22×Q8—64no—
189C22×Q8—122no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—32no—
193C23×C4—34no—
194C23×C4—32no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—42no—
198C23×C4—64no—
199C23×C4—64no—
200C23×C4—64no—
201C23×C4—62no—
202C23×C4—62no—
203C23×C4—62no—
204C23×C4—64no—
205C23×C4—122no—
206C23×C4D1214yes—
207C23×C4C22×D314yes—
208C23×C4C22×D311yescharacteristic
209C23×C4C22×D311yescharacteristic
210C22⋊Q8—632no—
211C2×C4⋊C4—38no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—64no—
214C2×C4⋊C4—64no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—632no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4—64no—
223C2×C4⋊C4—68no—
224C2×C4⋊C4—68no—
225C2×C4⋊C4—124no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—38no—
229C2×C22⋊C4—64no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4Dic618yescomplemented
232#95—24no—
233#95—42no—
234#95—68no—
235#95—68no—
236#95—68no—
237C23×C3—22no—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—24no—
244C23×C3—24no—
245C23×C3—28no—
246C23×C3—28no—
247C23×C3—28no—
248C23×C3—28no—
249C23×C3#12014yes—
250C23×C3C22⋊Q818yes—
251C23×C3C22⋊Q814yes—
252C23×C3C4⋊Q814yes—
253C23×C3C4⋊Q818yes—
254C23×C3C22×D411yescharacteristic
255C23×C3C22×D412yes—
256C23×C3C22×D412yes—
257C23×C3C22×Q812yes—
258C23×C3C22×Q811yescharacteristic
259C23×C3C2×#4111yescharacteristic
260C23×C3C2×#4112yes—
261C23×C3H(5,𝔽2)12yes—
262C23×C32−1+412yes—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—28no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—28no—
269C2×C3×C4—22no—
270C2×C3×C4—28no—
271C2×C3×C4—28no—
272C2×C3×C4—416no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4#12014yes—
279C2×Dic3—232no—
280C2×Dic3—24no—
281C2×Dic3—24no—
282C2×Dic3—24no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—24no—
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—24no—
289C2×Dic3—28no—
290C2×Dic3—28no—
291C2×Dic3—28no—
292C2×Dic3—416no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—44no—
300C2×Dic3—44no—
301C2×Dic3—44no—
302C2×Dic3C22⋊Q818yescomplemented
303Dic6—416no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24C2×Dic614yes—
309C24C2×D1212yes—
310C24D3×D414yes—
311C24#23712yes—
312C24D3×Q812yes—
313C24C23×D311yescharacteristic
314C2×Q8—68no—
315C2×Q8—1216no—
316C2×Q8—124no—
317C2×Q8—124no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—28no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—28no—
331C22×C4—34no—
332C22×C4—34no—
333C22×C4—34no—
334C22×C4—34no—
335C22×C4—34no—
336C22×C4—34no—
337C22×C4—44no—
338C22×C4—416no—
339C22×C4—616no—
340C22×C4—68no—
341C22×C4—68no—
342C22×C4—616no—
343C22×C4—68no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—68no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—64no—
353C22×C4—64no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—64no—
358C22×C4—64no—
359C22×C4—64no—
360C22×C4—64no—
361C22×C4—1216no—
362C22×C4—124no—
363C22×C4—124no—
364C22×C4C2×Dic618yes—
365C22×C4C2×D1212yes—
366C22×C4C2×D1212yes—
367C22×C4D3×D414yes—
368C4⋊C4—616no—
369C4⋊C4—616no—
370C4⋊C4—68no—
371C4⋊C4—616no—
372C4⋊C4—1216no—
373C22⋊C4—216no—
374C22⋊C4—616no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—216no—
382C22×C3—216no—
383C22×C3#39518yescomplemented
384C22×C3C2×#12011yescharacteristic
385C22×C3C2×#12011yescharacteristic
386C22×C3C2×C22⋊Q812yes—
387C22×C3C2×C22⋊Q811yescharacteristic
388C22×C3C2×C22⋊Q812yes—
389C22×C3#52511yescharacteristic
390C22×C3#52512yes—
391C22×C3C2×C4⋊Q811yescharacteristic
392C22×C3C2×C4⋊Q812yes—
393C22×C3#53411yescharacteristic
394C22×C3#53611yescharacteristic
395C22×C3#53912yes—
396C22×C3#54412yes—
397C22×C3D4214yes—
398C22×C3#54714yes—
399C22×C3D4×Q814yes—
400C22×C3#55414yes—
401C3×C4—24no—
402C3×C4—42no—
403C3×C4—48no—
404Dic3—28no—
405Dic3—44no—
406Dic3—44no—
407Dic3—44no—
408C23—28no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—28no—
414C23—28no—
415C23—28no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23#86914yes—
421C23#87818yes—
422C23#88214yes—
423C23#88818yes—
424C23#89714yes—
425C23C22×Dic612yes—
426C23C22×D1211yescharacteristic
427C23C2×D3×D412yes—
428C23C2×D3×D412yes—
429C23C2×#23711yescharacteristic
430C23#100412yes—
431C23C2×D3×Q811yescharacteristic
432C23D3×#4112yes—
433C23#101012yes—
434Q8—1216no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—38no—
445C2×C4—416no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—632no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—64no—
457C2×C4—64no—
458C2×C4—68no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—68no—
463C2×C4—68no—
464C2×C4—1216no—
465C2×C4—124no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—124no—
473C2×C4—124no—
474C2×C4#88214yes—
475C2×C3—28no—
476C2×C3C2×#39511yescharacteristic
477C2×C3#240311yescharacteristic
478C2×C3#240611yescharacteristic
479C2×C3#243411yescharacteristic
480C2×C3#244014yes—
481C2×C3#244914yes—
482C2×C3#258311yescharacteristic
483C2×C3#262711yescharacteristic
484C2×C3#284811yescharacteristic
485C22—24no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—216no—
491C22—216no—
492C22—24no—
493C22#465618yescomplemented
494C22C2×#86911yescharacteristic
495C22C2×#87812yes—
496C22C2×#88211yescharacteristic
497C22C2×#88211yescharacteristic
498C22#549311yescharacteristic
499C22C2×#88812yes—
500C22C2×#89711yescharacteristic
501C22#552411yescharacteristic
502C22#552611yescharacteristic
503C22D4×Dic614yes—
504C22#553914yes—
505C22#559014yes—
506C22#560212yes—
507C22#560612yes—
508C22#562312yes—
509C22D3×C22⋊Q812yes—
510C22#565514yes—
511C4—24no—
512C4—42no—
513C4—48no—
514C4—68no—
515C4—124no—
516C4—124no—
517C4—124no—
518C3#1472011yescharacteristic, complemented, Sylow
519C2—28no—
520C2C2×#465611yescharacteristic
521C2#7349011yescharacteristic
522C2#7354711yescharacteristic
523C2#7385211yescharacteristic
524C2#7395311yescharacteristic
525C2#7404911yescharacteristic
526C2#7406311yescharacteristic
527C2#7441614yes—
528C2#7470914yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C22⋊#95C2
prime radicalC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#14720
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. C2×#4656C2semidirect
7. #69420C2semidirect
8. #69410C2semidirect
56. C23×Dic3Q8semidirect
67. C2×C3×C22⋊C4Q8semidirect
231. C2×C22⋊C4Dic6semidirect
302. C2×Dic3C22⋊Q8semidirect
383. C22×C3#395semidirect
493. C22#4656semidirect
518. C3#14720semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C23×Dic3⋊C42
C2×#49502
C2×#46562
C3×C22⋊#952
#715442
#694202
#694102
#147203