Finite Group Databasebeta

← #10729049

Subgroups — #10729049

Order 768.

2150 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#924C211yesmaximal, characteristic
2C22×C3×#95C211yesmaximal, characteristic
3C2×#5210C212yesmaximal
4#72127C211yesmaximal, characteristic
5#72127C212yesmaximal
6#69424C212yesmaximal, complemented
7#69420C212yesmaximal, complemented
8#69409C214yesmaximal, complemented
9#14914—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×#217C2212yes—
15C22×#217C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C2×#924—24no—
18C2×#924—28no—
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2214yes—
21C2×C3×#95—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24#5210—28no—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C22×C22⋊Q8—31no—
31C2×#385—32no—
32C22×#95D311yescharacteristic
33#1446—32no—
34#1444—32no—
35#1441—34no—
36C22⋊#95—32no—
37C22⋊#95—31no—
38C25×C3C2311yescharacteristic
39C23×C3×C4—22no—
40C23×C3×C4—22no—
41C23×C3×C4—24no—
42C23×C3×C4—22no—
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—42no—
47C23×C3×C4—42no—
48C23×C3×C4—42no—
49C23×C3×C4—42no—
50C23×C3×C4D414yes—
51C23×C3×C4Q814yescomplemented
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2314yes—
59C22×Dic6—22no—
60C22×Dic6—24no—
61C22×Dic6—24no—
62C22×Dic6—42no—
63C2×#217—24no—
64C2×#217—24no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217D418yes—
68C2×#217Q818yescomplemented
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—24no—
71#924—232no—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—28no—
76C2×Dic3⋊C4—28no—
77C2×Dic3⋊C4—44no—
78C3×#95—24no—
79C3×#95—24no—
80C3⋊#95—24no—
81C3⋊#95—28no—
82C3⋊#95—42no—
83C3⋊#95—44no—
84C3⋊#95—44no—
85C23×Q8—31no—
86C24×C4C2×D311yescharacteristic
87C24×C4C2×D311yescharacteristic
88C24×C4C2×D311yescharacteristic
89C2×C22⋊Q8—68no—
90C2×C22⋊Q8—64no—
91C22×C4⋊C4—32no—
92C22×C4⋊C4—31no—
93C22×C4⋊C4—34no—
94C22×C22⋊C4—32no—
95C22×C22⋊C4—32no—
96#385—68no—
97C2×#95—24no—
98C2×#95—32no—
99C2×#95—34no—
100C2×#95—32no—
101C2×#95—34no—
102C2×#95—34no—
103C2×#95C2×D312yes—
104C2×#95C2×D312yes—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3C2×D414yes—
110C24×C3C2×D412yes—
111C24×C3C2×Q812yes—
112C24×C3C2×Q814yes—
113C24×C3#4112yes—
114C24×C3C2411yescharacteristic
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—28no—
118C22×C3×C4—28no—
119C22×C3×C4—28no—
120C22×C3×C4—24no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—28no—
132C22×C3×C4—44no—
133C22×C3×C4—48no—
134C22×C3×C4—416no—
135C22×C3×C4—416no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4—44no—
139C22×C3×C4—48no—
140C22×C3×C4—44no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×Q814yes—
144C22×C3×C4#4114yes—
145C22×C3×C4#4114yes—
146C22×Dic3—24no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C22×Dic3C2×D414yes—
161C22×Dic3C2×D414yes—
162C22×Dic3C2×Q818yes—
163C2×Dic6—28no—
164C2×Dic6—416no—
165C2×Dic6—44no—
166C2×Dic6—44no—
167#217—216no—
168#217—216no—
169C4⋊Dic3—28no—
170Dic3⋊C4—216no—
171Dic3⋊C4—416no—
172C25C22×D311yescharacteristic
173C22×Q8—64no—
174C22×Q8—64no—
175C22×Q8—62no—
176C22×Q8—122no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—32no—
185C23×C4—32no—
186C23×C4—34no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4Dic614yescomplemented
192C23×C4#6714yes—
193C23×C4C22×D312yes—
194C23×C4C22×D312yes—
195C23×C4C22×D311yescharacteristic
196C23×C4C22×D311yescharacteristic
197C22⋊Q8—632no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4—64no—
200C2×C4⋊C4—64no—
201C2×C4⋊C4—64no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—68no—
204C2×C4⋊C4—68no—
205C2×C4⋊C4—124no—
206C2×C22⋊C4—38no—
207C2×C22⋊C4—38no—
208C2×C22⋊C4—64no—
209C2×C22⋊C4—64no—
210C2×C22⋊C4—64no—
211C2×C22⋊C4—64no—
212#95—24no—
213#95—24no—
214#95—68no—
215#95—64no—
216#95—122no—
217#95—124no—
218#95—124no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—24no—
222C23×C3—24no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—24no—
226C23×C3—24no—
227C23×C3—28no—
228C23×C3—28no—
229C23×C3—28no—
230C23×C3—28no—
231C23×C3#12014yes—
232C23×C3C22⋊Q818yes—
233C23×C3C22⋊Q814yes—
234C23×C3C4⋊Q818yes—
235C23×C3C4⋊Q814yes—
236C23×C3C22×D412yes—
237C23×C3C22×D411yescharacteristic
238C23×C3C22×Q811yescharacteristic
239C23×C3C22×Q812yes—
240C23×C3C2×#4111yescharacteristic
241C23×C3C2×#4112yes—
242C23×C3C2×#4112yes—
243C23×C3H(5,𝔽2)12yes—
244C23×C3H(5,𝔽2)12yes—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—28no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—24no—
252C2×C3×C4—22no—
253C2×C3×C4—28no—
254C2×C3×C4—28no—
255C2×C3×C4—416no—
256C2×C3×C4—416no—
257C2×C3×C4—416no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—44no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4#12314yes—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—28no—
269C2×Dic3—28no—
270C2×Dic3—28no—
271C2×Dic3—28no—
272C2×Dic3—28no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—44no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—44no—
279C2×Dic3—44no—
280C2×Dic3—44no—
281C2×Dic3—44no—
282C2×Dic3—44no—
283Dic6—416no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C2×Dic612yes—
289C24#23512yes—
290C24D3×D414yes—
291C24D3×Q814yes—
292C24C2×#6712yes—
293C24C23×D311yescharacteristic
294C2×Q8—68no—
295C2×Q8—1216no—
296C2×Q8—124no—
297C2×Q8—124no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—28no—
315C22×C4—34no—
316C22×C4—34no—
317C22×C4—38no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—64no—
328C22×C4—64no—
329C22×C4—64no—
330C22×C4—64no—
331C22×C4—64no—
332C22×C4—64no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—64no—
336C22×C4—64no—
337C22×C4—64no—
338C22×C4—64no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4C2×Dic614yes—
342C22×C4#23514yes—
343C22×C4#23514yes—
344C22×C4C2×#6712yes—
345C22×C4C2×#6712yes—
346C4⋊C4—616no—
347C4⋊C4—68no—
348C4⋊C4—1216no—
349C22⋊C4—616no—
350C22⋊C4—616no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—216no—
357C22×C3—24no—
358C22×C3—216no—
359C22×C3#39518yes—
360C22×C3C2×#12011yescharacteristic
361C22×C3C2×C22⋊Q812yes—
362C22×C3C2×C22⋊Q811yescharacteristic
363C22×C3C2×#12311yescharacteristic
364C22×C3#52511yescharacteristic
365C22×C3#52912yes—
366C22×C3C2×C4⋊Q812yes—
367C22×C3C2×C4⋊Q811yescharacteristic
368C22×C3#53312yes—
369C22×C3#53411yescharacteristic
370C22×C3#53511yescharacteristic
371C22×C3#53812yes—
372C22×C3#54312yes—
373C22×C3#54614yes—
374C22×C3#54614yes—
375C22×C3#55414yes—
376C22×C3#55414yes—
377C3×C4—24no—
378C3×C4—44no—
379C3×C4—44no—
380C3×C4—42no—
381Dic3—44no—
382Dic3—44no—
383Dic3—48no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—28no—
388C23—28no—
389C23—24no—
390C23—28no—
391C23—28no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23#88818yes—
397C23#89618yes—
398C23#92414yes—
399C23#93714yes—
400C23#94414yes—
401C23C22×Dic611yescharacteristic
402C23C2×#23512yes—
403C23C2×#23512yes—
404C23C2×#23511yescharacteristic
405C23C2×D3×D412yes—
406C23#100412yes—
407C23#100412yes—
408C23C2×D3×Q812yes—
409C23C22×#6711yescharacteristic
410Q8—1216no—
411C2×C4—28no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—416no—
422C2×C4—416no—
423C2×C4—416no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—68no—
432C2×C4—68no—
433C2×C4—68no—
434C2×C4—68no—
435C2×C4—68no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4—124no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4#92914yes—
449C2×C3—28no—
450C2×C3C2×#39511yescharacteristic
451C2×C3#240411yescharacteristic
452C2×C3#240511yescharacteristic
453C2×C3#245014yes—
454C2×C3#250211yescharacteristic
455C2×C3#254914yes—
456C2×C3#258311yescharacteristic
457C2×C3#262611yescharacteristic
458C2×C3#285511yescharacteristic
459C22—24no—
460C22—24no—
461C22—216no—
462C22—216no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22#497918yes—
468C22#547012yes—
469C22#547912yes—
470C22#548312yes—
471C22#549014yes—
472C22#549014yes—
473C22#549412yes—
474C22C2×#88812yes—
475C22C2×#89612yes—
476C22#551114yes—
477C22#551114yes—
478C22C2×#92411yescharacteristic
479C22C2×#92911yescharacteristic
480C22#579111yescharacteristic
481C22C2×#93711yescharacteristic
482C22#580411yescharacteristic
483C22#580511yescharacteristic
484C22C2×#94411yescharacteristic
485C4—24no—
486C4—44no—
487C4—44no—
488C4—42no—
489C4—128no—
490C4—124no—
491C4—124no—
492C3#1491411yescharacteristic, complemented, Sylow
493C2—28no—
494C2#7335114yes—
495C2#7335714yes—
496C2#8207511yescharacteristic
497C2#8211711yescharacteristic
498C2#8212511yescharacteristic
499C2#8212611yescharacteristic
500C2C2×#497911yescharacteristic
501C2#8217811yescharacteristic
502C2#8218111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#95C2
prime radicalC25×C3C23
hypercenterC22×#95D3
hypocenterC3#14914
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
6. #69424C2semidirect
7. #69420C2semidirect
8. #69409C2semidirect
51. C23×C3×C4Q8semidirect
68. C2×#217Q8semidirect
191. C23×C4Dic6semidirect
492. C3#14914semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C22×C3×#952
C2×#52102
#721272
#694242
#694202
#694092
#149143