Finite Group Databasebeta

← #10720106

Subgroups — #10720106

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#926C211yesmaximal, characteristic
2C22×C3×#95C211yesmaximal, characteristic
3C2×#5210C212yesmaximal
4C2×#5211C211yesmaximal, characteristic
5#72127C212yesmaximal
6#69420C212yesmaximal, complemented
7#69415C212yesmaximal, complemented
8#69406C214yesmaximal, complemented
9#14128—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×#217C2212yes—
14C22×#217C2212yes—
15C2×#926—24no—
16C2×#926—28no—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2214yes—
20C22×C4×Dic3C2212yes—
21C2×C3×#95—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24#5211C418yescomplemented
25#5210—28no—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C22×#117—31no—
32C2×#385—32no—
33C2×#381—31no—
34C22×#95D311yescharacteristic
35#1444—32no—
36#1442—32no—
37#1436—34no—
38C22⋊#95—32no—
39C25×C3C2311yescharacteristic
40C23×C3×C4—24no—
41C23×C3×C4—24no—
42C23×C3×C4—22no—
43C23×C3×C4—22no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—42no—
49C23×C3×C4—42no—
50C23×C3×C4C2×C414yes—
51C23×C3×C4D414yes—
52C23×C3×C4Q814yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2314yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C2×#217—24no—
61C2×#217—24no—
62C2×#217—24no—
63C2×#217—24no—
64C2×#217C2×C418yes—
65C2×#217D418yes—
66#926—232no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—24no—
70C2×C4⋊Dic3—24no—
71C2×Dic3⋊C4—28no—
72C2×Dic3⋊C4—28no—
73C2×Dic3⋊C4—44no—
74C2×C4×Dic3—24no—
75C2×C4×Dic3—24no—
76C2×C4×Dic3—24no—
77C3×#95—24no—
78C3×#95—24no—
79C3⋊#95—28no—
80C3⋊#95—28no—
81C3⋊#95—44no—
82C3⋊#95C2×C418yescomplemented
83C3⋊#95C2×C418yescomplemented
84C24×C4C2×D311yescharacteristic
85C24×C4C2×D311yescharacteristic
86C24×C4C2×D311yescharacteristic
87C2×#117—68no—
88C2×#117—64no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4—31no—
91C22×C4⋊C4—34no—
92C22×C22⋊C4—32no—
93C22×C22⋊C4—32no—
94C22×C42—32no—
95#385—68no—
96#381—38no—
97C2×#95—24no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—34no—
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×C3C22×C412yes—
110C24×C3C2×D414yes—
111C24×C3C2×D412yes—
112C24×C3C2×Q812yes—
113C24×C3#4114yes—
114C24×C3C2411yescharacteristic
115C22×C3×C4—216no—
116C22×C3×C4—28no—
117C22×C3×C4—216no—
118C22×C3×C4—28no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—28no—
124C22×C3×C4—28no—
125C22×C3×C4—24no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—416no—
134C22×C3×C4—416no—
135C22×C3×C4—44no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4—44no—
139C22×C3×C4C4⋊C4116yescomplemented
140C22×C3×C4C22×C412yes—
141C22×C3×C4C22×C412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×Q812yes—
145C22×C3×C4C2×Q812yes—
146C22×C3×C4#4114yes—
147C22×C3×C4#4114yes—
148C22×Dic3—24no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3—24no—
162C22×Dic3C22×C418yes—
163C22×Dic3C2×D414yes—
164C22×Dic3C2×D414yes—
165#217—216no—
166#217—216no—
167C4⋊Dic3—28no—
168C4⋊Dic3—28no—
169Dic3⋊C4—416no—
170C4×Dic3—216no—
171C25C22×D311yescharacteristic
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—32no—
181C23×C4—34no—
182C23×C4—32no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4Dic614yes—
186C23×C4C4×D314yes—
187C23×C4#6714yes—
188C23×C4C22×D312yes—
189C23×C4C22×D312yes—
190C23×C4C22×D311yescharacteristic
191C23×C4C22×D311yescharacteristic
192#117—632no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—68no—
198C2×C4⋊C4—68no—
199C2×C4⋊C4—124no—
200C2×C22⋊C4—38no—
201C2×C22⋊C4—38no—
202C2×C22⋊C4—64no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—64no—
205C2×C22⋊C4—64no—
206C2×C42—64no—
207C2×C42—64no—
208C2×C42—64no—
209#95—24no—
210#95—24no—
211#95—38no—
212#95—38no—
213#95—68no—
214#95—68no—
215#95—124no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—24no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—24no—
224C23×C3—28no—
225C23×C3—28no—
226C23×C3—28no—
227C23×C3—28no—
228C23×C3C2×C4⋊C414yes—
229C23×C3C4×D418yes—
230C23×C3#12014yes—
231C23×C3C22⋊Q818yes—
232C23×C3#12314yes—
233C23×C3C23×C411yescharacteristic
234C23×C3C22×D412yes—
235C23×C3C22×D411yescharacteristic
236C23×C3C22×Q811yescharacteristic
237C23×C3C2×#4112yes—
238C23×C3C2×#4112yes—
239C23×C3C2×#4112yes—
240C23×C3H(5,𝔽2)12yes—
241C23×C32−1+412yes—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—232no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—28no—
254C2×C3×C4—28no—
255C2×C3×C4—416no—
256C2×C3×C4—416no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—44no—
263C2×C3×C4C2×C4⋊C414yes—
264C2×C3×C4C2×C4⋊C414yes—
265C2×Dic3—28no—
266C2×Dic3—28no—
267C2×Dic3—28no—
268C2×Dic3—24no—
269C2×Dic3—24no—
270C2×Dic3—28no—
271C2×Dic3—28no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—44no—
277C2×Dic3—44no—
278C2×Dic3—44no—
279C2×Dic3—44no—
280C2×Dic3—44no—
281C2×Dic3—44no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24C2×Dic612yes—
287C24C2×C4×D312yes—
288C24D3×D414yes—
289C24#23714yes—
290C24C2×#6712yes—
291C24C23×D311yescharacteristic
292C22×C4—216no—
293C22×C4—216no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—38no—
311C22×C4—34no—
312C22×C4—34no—
313C22×C4—416no—
314C22×C4—416no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—64no—
320C22×C4—64no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4—64no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—64no—
327C22×C4—64no—
328C22×C4—64no—
329C22×C4—64no—
330C22×C4—64no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4Dic3⋊C4116yescomplemented
334C22×C4C2×Dic612yes—
335C22×C4C2×Dic612yes—
336C22×C4C2×C4×D312yes—
337C22×C4C2×C4×D312yes—
338C22×C4#23514yes—
339C22×C4#23514yes—
340C22×C4C2×#6712yes—
341C22×C4C2×#6712yes—
342C4⋊C4—68no—
343C4⋊C4—68no—
344C4⋊C4—1216no—
345C22⋊C4—616no—
346C22⋊C4—616no—
347C42—616no—
348C22×C3—24no—
349C22×C3—216no—
350C22×C3—24no—
351C22×C3—24no—
352C22×C3—216no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3#38518yes—
357C22×C3C22×C4⋊C411yescharacteristic
358C22×C3C2×C4×D412yes—
359C22×C3C4×#4112yes—
360C22×C3#52012yes—
361C22×C3C2×#12011yescharacteristic
362C22×C3C2×C22⋊Q812yes—
363C22×C3C2×#12311yescharacteristic
364C22×C3#52511yescharacteristic
365C22×C3#52511yescharacteristic
366C22×C3#53312yes—
367C22×C3#53511yescharacteristic
368C22×C3#53611yescharacteristic
369C22×C3#54412yes—
370C22×C3#54614yes—
371C22×C3#54714yes—
372C22×C3#55114yes—
373C22×C3#55214yes—
374C3×C4—24no—
375C3×C4—24no—
376C3×C4—44no—
377C3×C4—44no—
378Dic3—48no—
379Dic3—44no—
380Dic3—44no—
381C23—24no—
382C23—24no—
383C23—28no—
384C23—28no—
385C23—24no—
386C23—24no—
387C23—28no—
388C23—28no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23#87818yes—
394C23#67⋊C418yes—
395C23C2×Dic3⋊C414yes—
396C23#93514yes—
397C23#93714yes—
398C23C22×Dic611yescharacteristic
399C23C22×C4×D311yescharacteristic
400C23C2×#23512yes—
401C23C2×#23512yes—
402C23C2×D3×D412yes—
403C23C2×#23712yes—
404C23#100412yes—
405C23#100712yes—
406C23C22×#6711yescharacteristic
407C2×C4—232no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—416no—
421C2×C4—416no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—68no—
429C2×C4—68no—
430C2×C4—68no—
431C2×C4—68no—
432C2×C4—68no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4C2×Dic3⋊C414yes—
446C2×C4C2×Dic3⋊C414yes—
447C2×C3—28no—
448C2×C3C2×#38511yescharacteristic
449C2×C3#229611yescharacteristic
450C2×C3#231711yescharacteristic
451C2×C3#235914yes—
452C2×C3#240411yescharacteristic
453C2×C3#240611yescharacteristic
454C2×C3#243814yes—
455C2×C3#248211yescharacteristic
456C2×C3#248411yescharacteristic
457C22—24no—
458C22—24no—
459C22—216no—
460C22—216no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22#494618yes—
466C22#547012yes—
467C22C4×#23512yes—
468C22C2×#87812yes—
469C22C2×#67⋊C412yes—
470C22#549014yes—
471C22#549514yes—
472C22#550012yes—
473C22#550412yes—
474C22#550714yes—
475C22#551314yes—
476C22C22×Dic3⋊C411yescharacteristic
477C22#579111yescharacteristic
478C22#579111yescharacteristic
479C22C2×#93511yescharacteristic
480C22C2×#93711yescharacteristic
481C22#580511yescharacteristic
482C22#581611yescharacteristic
483C4—24no—
484C4—24no—
485C4—44no—
486C4—44no—
487C4—124no—
488C4—128no—
489C4—124no—
490C3#1412811yescharacteristic, complemented, Sylow
491C2—28no—
492C2#7333714yes—
493C2#7335014yes—
494C2#8206911yescharacteristic
495C2C2×#494611yescharacteristic
496C2#8211311yescharacteristic
497C2#8212511yescharacteristic
498C2#8215711yescharacteristic
499C2#8217611yescharacteristic
500C2#8219211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×#95C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC22×#95D3
hypocenterC3#14128
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
6. #69420C2semidirect
7. #69415C2semidirect
8. #69406C2semidirect
24. #5211C4semidirect
82. C3⋊#95C2×C4semidirect
83. C3⋊#95C2×C4semidirect
139. C22×C3×C4C4⋊C4semidirect
333. C22×C4Dic3⋊C4semidirect
490. C3#14128semidirect

Maximal subgroups

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

subgroupindex
C22×#9262
C22×C3×#952
C2×#52102
C2×#52112
#721272
#694202
#694152
#694062
#141283