Finite Group Databasebeta

← #10728219

Subgroups — #10728219

Order 768.

2046 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×#5210C211yesmaximal, characteristic
3C2×#4985C212yesmaximal, complemented
4#72127C211yesmaximal, characteristic
5#72127C212yesmaximal
6#72127C212yesmaximal
7#71569C212yesmaximal, complemented
8#71567C212yesmaximal, complemented
9#71566C212yesmaximal, complemented
10#14843—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2212yes—
13C22×C3×C4⋊C4—24no—
14C22×C3×C4⋊C4—28no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22#5210—24no—
23C2×C3⋊#95C2211yescharacteristic
24C2×C3⋊#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31#4985—28no—
32C23×C4⋊C4D311yescharacteristic
33C2×#400—32no—
34C2×#385—31no—
35#1445—32no—
36#1442—32no—
37#1441—32no—
38C22⋊#95—32no—
39C22⋊#95—31no—
40C22⋊#95—32no—
41C25×C3C2311yescharacteristic
42C23×C3×C4—24no—
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—28no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—44no—
50C23×C3×C4D414yes—
51C23×C3×C4C2314yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2314yes—
55C23×Dic3C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C2×C3×C4⋊C4—232no—
60C2×C3×C4⋊C4—28no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4D418yes—
65C2×#217—22no—
66C2×#217—22no—
67C2×#217—22no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217Q814yes—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—28no—
80C3⋊#95—24no—
81C3⋊#95—28no—
82C3⋊#95—24no—
83C3⋊#95—44no—
84C3⋊#95—44no—
85C24×C4C2×D311yescharacteristic
86C24×C4C2×D312yes—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4—31no—
91C22×C4⋊C4C2×D312yes—
92C22×C4⋊C4C2×D312yes—
93C22×C22⋊C4—31no—
94C22×C22⋊C4—32no—
95C22×C22⋊C4—31no—
96#400—68no—
97#385—64no—
98C2×#95—34no—
99C2×#95—34no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—31no—
103C2×#95—31no—
104C2×#95—34no—
105C2×#95—34no—
106C24×C3—22no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3C2×D412yes—
111C24×C3C2×Q812yes—
112C24×C3#4114yes—
113C24×C3#4114yes—
114C24×C3#4112yes—
115C24×C3C2411yescharacteristic
116C22×C3×C4—232no—
117C22×C3×C4—28no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—28no—
123C22×C3×C4—28no—
124C22×C3×C4—216no—
125C22×C3×C4—24no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—416no—
132C22×C3×C4—416no—
133C22×C3×C4—48no—
134C22×C3×C4—48no—
135C22×C3×C4C2×D418yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D414yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4#4114yes—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—22no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—24no—
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—24no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—22no—
160C22×Dic3—24no—
161C22×Dic3—22no—
162C22×Dic3—24no—
163C22×Dic3C2×Q812yes—
164C22×Dic3C2×Q812yes—
165C3×C4⋊C4—216no—
166C3×C4⋊C4—216no—
167#217—28no—
168#217—44no—
169#217—48no—
170Dic3⋊C4—44no—
171Dic3⋊C4—44no—
172C25C22×D311yescharacteristic
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—31no—
181C23×C4—31no—
182C23×C4—31no—
183C23×C4—34no—
184C23×C4—31no—
185C23×C4—44no—
186C23×C4#6714yes—
187C23×C4C22×D314yes—
188C23×C4C22×D311yescharacteristic
189C23×C4C22×D311yescharacteristic
190C2×C4⋊C4—232no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4#6718yes—
200C2×C22⋊C4—34no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—64no—
205C2×C22⋊C4—64no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—62no—
208#95—64no—
209#95—62no—
210#95—62no—
211#95—68no—
212#95—64no—
213#95—68no—
214#95—124no—
215#95—124no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—28no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—28no—
222C23×C3—28no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—28no—
226C23×C3—24no—
227C23×C3—24no—
228C23×C3C22⋊Q814yes—
229C23×C3#12314yes—
230C23×C3#12414yes—
231C23×C3#12518yes—
232C23×C3#12618yes—
233C23×C3C22×D411yescharacteristic
234C23×C3C22×D412yes—
235C23×C3C22×Q811yescharacteristic
236C23×C3C2×#4112yes—
237C23×C3C2×#4112yes—
238C23×C3C2×#4112yes—
239C23×C3C2×#4111yescharacteristic
240C23×C3H(5,𝔽2)12yes—
241C23×C32−1+412yes—
242C2×C3×C4—232no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—28no—
248C2×C3×C4—28no—
249C2×C3×C4—28no—
250C2×C3×C4—28no—
251C2×C3×C4—28no—
252C2×C3×C4—432no—
253C2×C3×C4—44no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4#12014yes—
258C2×C3×C4#12314yes—
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—44no—
274C2×Dic3—44no—
275C2×Dic3—44no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—44no—
279C2×Dic3—44no—
280C2×Dic3—44no—
281C2×Dic3—44no—
282C2×Dic3—44no—
283C2×Dic3—44no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24#23514yes—
289C24#23714yes—
290C24D3×Q812yes—
291C24#23912yes—
292C24C2×#6712yes—
293C24C23×D311yescharacteristic
294C22×C4—232no—
295C22×C4—28no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—216no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—32no—
310C22×C4—32no—
311C22×C4—416no—
312C22×C4—416no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—64no—
324C22×C4—64no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—64no—
329C22×C4—64no—
330C22×C4—62no—
331C22×C4—64no—
332C22×C4—64no—
333C22×C4—62no—
334C22×C4—64no—
335C22×C4—61no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4#23514yes—
339C22×C4C2×#6718yes—
340C22×C4C2×#6712yes—
341C22×C4C2×#6712yes—
342C22×C4C2×#6714yes—
343C4⋊C4—216no—
344C4⋊C4—216no—
345C4⋊C4—124no—
346C4⋊C4—124no—
347C22⋊C4—68no—
348C22⋊C4—128no—
349C22⋊C4—124no—
350C22×C3—24no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—216no—
355C22×C3—216no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3#40018yescomplemented
359C22×C3C2×#12011yescharacteristic
360C22×C3C2×C22⋊Q811yescharacteristic
361C22×C3C2×#12311yescharacteristic
362C22×C3C2×#12311yescharacteristic
363C22×C3#52514yes—
364C22×C3C2×#12411yescharacteristic
365C22×C3C2×#12512yes—
366C22×C3C2×#12612yes—
367C22×C3#53511yescharacteristic
368C22×C3#53611yescharacteristic
369C22×C3#53914yes—
370C22×C3#54614yes—
371C22×C3#54812yes—
372C22×C3D4×Q812yes—
373C22×C3#55214yes—
374C22×C3#55312yes—
375C22×C3#55412yes—
376C3×C4—24no—
377C3×C4—24no—
378C3×C4—48no—
379Dic3—42no—
380Dic3—42no—
381Dic3—81no—
382Dic3—81no—
383Dic3—84no—
384C23—24no—
385C23—28no—
386C23—24no—
387C23—28no—
388C23—22no—
389C23—28no—
390C23—22no—
391C23—28no—
392C23—24no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23#88918yes—
397C23#89818yes—
398C23#92914yes—
399C23#93614yes—
400C23#94614yes—
401C23C2×#23512yes—
402C23C2×#23512yes—
403C23C2×#23712yes—
404C23#100412yes—
405C23C2×D3×Q811yescharacteristic
406C23C2×#23911yescharacteristic
407C23#100712yes—
408C23C22×#6711yescharacteristic
409C23C22×#6712yes—
410C2×C4—232no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—432no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—64no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—64no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—121no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—124no—
444C2×C4—124no—
445C2×C4—121no—
446C2×C4—124no—
447C2×C4—121no—
448C2×C4—124no—
449C2×C4—121no—
450C2×C4#92914yes—
451C2×C4#95314yes—
452C2×C3—28no—
453C2×C3C2×#40011yescharacteristic
454C2×C3#240911yescharacteristic
455C2×C3#243511yescharacteristic
456C2×C3#247314yes—
457C2×C3#248114yes—
458C2×C3#248711yescharacteristic
459C2×C3#250311yescharacteristic
460C2×C3#262311yescharacteristic
461C2×C3#284911yescharacteristic
462C22—24no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—216no—
469C22—216no—
470C22#498518yescomplemented
471C22C2×#88912yes—
472C22#551112yes—
473C22C2×#89812yes—
474C22#551314yes—
475C22#551412yes—
476C22#551514yes—
477C22C2×#92911yescharacteristic
478C22C2×#92911yescharacteristic
479C22#579114yes—
480C22C2×#93611yescharacteristic
481C22#580511yescharacteristic
482C22#580614yes—
483C22C2×#94611yescharacteristic
484C22Q8×#6712yes—
485C22#581611yescharacteristic
486C22#581712yes—
487C22C2×#95311yescharacteristic
488C4—24no—
489C4—24no—
490C4—48no—
491C4—122no—
492C4—122no—
493C4—241no—
494C4—241no—
495C4—244no—
496C3#1484311yescharacteristic, complemented, Sylow
497C2—28no—
498C2C2×#498511yescharacteristic
499C2#8217414yes—
500C2#8217514yes—
501C2#8217911yescharacteristic
502C2#8219711yescharacteristic
503C2#8220011yescharacteristic
504C2#8403311yescharacteristic
505C2#8405011yescharacteristic
506C2#8406811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC23×C3×C4⋊C4C2
prime radicalC25×C3C23
hypercenterC23×C4⋊C4D3
hypocenterC3#14843
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#4985C2semidirect
7. #71569C2semidirect
8. #71567C2semidirect
9. #71566C2semidirect
358. C22×C3#400semidirect
470. C22#4985semidirect
496. C3#14843semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×#52102
C2×#49852
#721272
#715692
#715672
#715662
#148433