Finite Group Databasebeta

← #10943892

Subgroups — #10943892

Order 768.

3606 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×2−1+4C211yesmaximal, characteristic, complemented
2C2×#5245C214yesmaximal, complemented
3#84082C212yesmaximal, complemented
4#84066C211yesmaximal, characteristic, complemented
5#72143C214yesmaximal, complemented
6#72138C212yesmaximal, complemented
7#72133C211yesmaximal, characteristic, complemented
8#32889—31nomaximal, Sylow
9C2×C3×2−1+4—216no—
10C2×C3×2−1+4—24no—
11C2×C3×2−1+4—22no—
12C2×C3×2−1+4C2214yes—
13C22×C3×#41—22no—
14C22×C3×#41—22no—
15C22×C3×#41C2212yes—
16C23×C3×Q8—22no—
17C23×C3×Q8C2211yescharacteristic
18C2×#949C2214yescomplemented
19C2×#947C2212yescomplemented
20C2×#930C2211yescharacteristic
21#5245—216no—
22#5238—24no—
23C2×#837C2218yescomplemented
24#5232C2214yescomplemented
25#5231—22no—
26#5229C2212yescomplemented
27#5227—22no—
28C2×#836C2214yescomplemented
29#5217—22no—
30#5213—22no—
31C4×#217C2211yescharacteristic
32#5207C2212yes—
33C22×2−1+4D311yescharacteristic, complemented
34C2×#454—34no—
35#3006—32no—
36#2436—31no—
37#1793—34no—
38#1789—32no—
39#1784—31no—
40C3×2−1+4—232no—
41C3×2−1+4—232no—
42C3×2−1+4—232no—
43C3×2−1+4—216no—
44C3×2−1+4—28no—
45C3×2−1+4D4116yescomplemented
46C2×C3×#41—232no—
47C2×C3×#41—28no—
48C2×C3×#41—28no—
49C2×C3×#41—232no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—28no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41D414yes—
56C2×C3×#41C2314yes—
57C22×C3×Q8—22no—
58C22×C3×Q8—232no—
59C22×C3×Q8—28no—
60C22×C3×Q8—28no—
61C22×C3×Q8—28no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D412yes—
65C22×C3×Q8D412yes—
66C22×C3×Q8C2312yes—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4D412yes—
70C23×C3×C4—24no—
71C23×C3×C4—22no—
72C23×C3×C4D412yes—
73C23×C3×C4C2311yescharacteristic
74C22×#67—21no—
75C22×D12C2311yescharacteristic
76C2×#217—21no—
77#949—216no—
78#947—28no—
79C2×#215—24no—
80C2×#213—24no—
81#930—22no—
82#930—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4×Dic3C2312yes—
87C2×#208C2314yes—
88#837—216no—
89#836—28no—
90#835—24no—
91#832—24no—
92C3⋊#95—21no—
93C3⋊#95—24no—
94#830—22no—
95#830—22no—
96C2×2−1+4—216no—
97C2×2−1+4—24no—
98C2×2−1+4—22no—
99C2×2−1+4C2×D314yes—
100C22×#41—22no—
101C22×#41—22no—
102C22×#41C2×D312yes—
103C23×Q8—22no—
104C23×Q8C2×D311yescharacteristic
105C2×AGL(1,ℤ/8)—34no—
106C2×#124—32no—
107C2×#121—31no—
108#454—616no—
109#449—64no—
110#448—62no—
111#447—62no—
112C2×#104—38no—
113#417—34no—
114C2×#101—34no—
115#407—32no—
116#393—62no—
117#391—32no—
118#386—62no—
119C4×C22⋊C4—31no—
120C24×C3—22no—
121C24×C3C2×D411yescharacteristic
122C23×D3—21no—
123C3×#41—264no—
124C3×#41—264no—
125C3×#41—232no—
126C3×#41—216no—
127C3×#41—232no—
128C3×#41—216no—
129C3×#41—28no—
130C3×#41—232no—
131C3×#41—28no—
132C3×#41—232no—
133C3×#41—28no—
134C3×#41C2×D4116yes—
135C2×C3×Q8—216no—
136C2×C3×Q8—216no—
137C2×C3×Q8—28no—
138C2×C3×Q8—264no—
139C2×C3×Q8—232no—
140C2×C3×Q8—216no—
141C2×C3×Q8—28no—
142C2×C3×Q8—232no—
143C2×C3×Q8—22no—
144C2×C3×Q8—24no—
145C2×C3×Q8—28no—
146C2×C3×Q8—24no—
147C2×C3×Q8—24no—
148C2×C3×Q8C2×D414yes—
149C2×C3×Q8C2×D418yes—
150C2×C3×D4—216no—
151C2×C3×D4—216no—
152C2×C3×D4—28no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—28no—
156C2×C3×D4—22no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—232no—
159C22×C3×C4—216no—
160C22×C3×C4—216no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—28no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—44no—
174C2×#67—41no—
175C2×#67—41no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C2×D12—22no—
179C2×D12—24no—
180#217—42no—
181#215—48no—
182#213—48no—
183(C2×D3)⋊C4—42no—
184(C2×D3)⋊C4—48no—
185C4⋊Dic3—22no—
186C4×Dic3—24no—
187#208—28no—
188C2×C3⋊C8—24no—
1892−1+4—232no—
1902−1+4—232no—
1912−1+4—232no—
1922−1+4—216no—
1932−1+4—28no—
1942−1+4#67116yescomplemented
195C2×#41—232no—
196C2×#41—28no—
197C2×#41—28no—
198C2×#41—232no—
199C2×#41—216no—
200C2×#41—28no—
201C2×#41—28no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41#6714yes—
205C2×#41C22×D314yes—
206C22×Q8—22no—
207C22×Q8—232no—
208C22×Q8—28no—
209C22×Q8—28no—
210C22×Q8—28no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8#6712yes—
214C22×Q8#6712yes—
215C22×Q8C22×D312yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—31no—
219C22×D4—61no—
220C22×D4#6712yes—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4#6712yes—
224C23×C4C22×D311yescharacteristic
225AGL(1,ℤ/8)—616no—
226C2×SD16—64no—
227C2×D8—64no—
228C2×M16—34no—
229#124—68no—
230#121—62no—
231#121—62no—
232C2×C4⋊C4—31no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—64no—
236C2×C42—32no—
237#104—616no—
238#103—64no—
239#102—64no—
240#101—68no—
241C22⋊C8—62no—
242C22⋊C8—62no—
243#95—61no—
244#95—64no—
245C23×C3—28no—
246C23×C3—22no—
247C23×C3—24no—
248C23×C3—22no—
249C23×C3—41no—
250C23×C3—44no—
251C23×C3—44no—
252C23×C3#12011yescharacteristic
253C23×C3#12011yescharacteristic
254C23×C3C22×D411yescharacteristic
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—42no—
258C22×D3—41no—
259C3×Q8—232no—
260C3×Q8—28no—
261C3×Q8—216no—
262C3×Q8—24no—
263C3×Q8—216no—
264C3×Q8#12018yes—
265C3×D4—216no—
266C3×D4—232no—
267C3×D4—28no—
268C3×D4—216no—
269C3×D4—24no—
270C3×D4#12018yes—
271C2×C3×C4—28no—
272C2×C3×C4—28no—
273C2×C3×C4—264no—
274C2×C3×C4—216no—
275C2×C3×C4—28no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—216no—
282C2×C3×C4—24no—
283C2×C3×C4#12012yes—
284C2×C3×C4#12018yes—
285C2×C3×C4C22×D412yes—
286C2×C3×C4C22×D414yes—
287#67—82no—
288#67—82no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294D12—44no—
295C3⋊C8—44no—
296C24—22no—
297C24—61no—
298C24C2×#6711yescharacteristic
299#41—264no—
300#41—264no—
301#41—232no—
302#41—216no—
303#41—232no—
304#41—216no—
305#41—28no—
306#41—232no—
307#41—28no—
308#41—232no—
309#41—28no—
310#41C2×#67116yes—
311C2×Q8—216no—
312C2×Q8—216no—
313C2×Q8—28no—
314C2×Q8—264no—
315C2×Q8—232no—
316C2×Q8—216no—
317C2×Q8—28no—
318C2×Q8—232no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—28no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8C2×#6714yes—
325C2×Q8C2×#6718yes—
326C2×D4—216no—
327C2×D4—216no—
328C2×D4—28no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—22no—
333C2×D4—62no—
334C2×D4—64no—
335C2×D4—124no—
336C2×D4—121no—
337C2×D4—121no—
338C2×D4C2×#6714yes—
339C22×C4—232no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4C2×#6714yes—
352C22×C4C2×#6711yescharacteristic
353C22×C4C2×#6712yes—
354C22×C4C2×#6712yes—
355C22×C4C23×D311yescharacteristic
356SD16—128no—
357D8—128no—
358M16—68no—
359C2×C8—64no—
360C4⋊C4—62no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C22⋊C4—128no—
364C42—64no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—28no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—44no—
374C22×C3—416no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3#45412yescomplemented
378C22×C3C2×#12012yes—
379C22×C3C2×#12011yescharacteristic
380C2×D3—84no—
381C2×D3—81no—
382C2×D3—81no—
383C2×D3—81no—
384C3×C4—28no—
385C3×C4—28no—
386C3×C4—22no—
387C3×C4C2×#12014yes—
388Dic3—82no—
389Dic3—81no—
390C23—28no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—41no—
395C23—44no—
396C23—44no—
397C23—122no—
398C23—122no—
399C23—122no—
400C23—121no—
401C23#95311yescharacteristic
402C23#95311yescharacteristic
403C23C22×#6711yescharacteristic
404Q8—232no—
405Q8—28no—
406Q8—216no—
407Q8—24no—
408Q8—216no—
409Q8#95318yes—
410D4—216no—
411D4—232no—
412D4—28no—
413D4—216no—
414D4—24no—
415D4—124no—
416D4—242no—
417D4—242no—
418D4#95318yes—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—264no—
422C2×C4—216no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—216no—
430C2×C4—24no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4#95318yes—
437C2×C4#95312yes—
438C2×C4C22×#6714yes—
439C2×C4C22×#6712yes—
440C8—124no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—41no—
444C2×C3—48no—
445C2×C3#284611yescharacteristic
446C2×C3C2×#45411yescharacteristic
447C2×C3#302011yescharacteristic
448D3—161no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—28no—
457C22—44no—
458C22—416no—
459C22—44no—
460C22—44no—
461C22—244no—
462C22—241no—
463C22—241no—
464C22—241no—
465C22#524512yescomplemented
466C22C2×#95312yes—
467C22C2×#95311yescharacteristic
468C4—28no—
469C4—28no—
470C4—22no—
471C4—242no—
472C4—241no—
473C4C2×#95314yes—
474C3#3288911yescharacteristic, complemented, Sylow
475C2—22no—
476C2—22no—
477C2—41no—
478C2—48no—
479C2—481no—
480C2C2×#524511yescharacteristic
481C2#8411611yescharacteristic
482C2#8412811yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×2−1+4C2
socleC22×C3C2×#120
hypercenterC22×2−1+4D3
hypocenterC3#32889
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×2−1+4

Complements

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

NH ≅ G/Ntype
1. C22×C3×2−1+4C2semidirect
2. C2×#5245C2semidirect
3. #84082C2semidirect
4. #84066C2semidirect
5. #72143C2semidirect
6. #72138C2semidirect
7. #72133C2semidirect
18. C2×#949C22semidirect
19. C2×#947C22semidirect
23. C2×#837C22semidirect
24. #5232C22semidirect
26. #5229C22semidirect
28. C2×#836C22semidirect
33. C22×2−1+4D3semidirect
45. C3×2−1+4D4semidirect
194. 2−1+4#67semidirect
377. C22×C3#454semidirect
465. C22#5245semidirect
474. C3#32889semidirect

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C2×#52452
#840822
#840662
#721432
#721382
#721332
#328893