Finite Group Databasebeta

← #10944171

Subgroups — #10944171

Order 768.

1402 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic, complemented
2C2×#5245C212yesmaximal, complemented
3#82526C212yesmaximal, complemented
4#82476C211yesmaximal, characteristic, complemented
5#71667C214yesmaximal, complemented
6#71661C212yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71637C211yesmaximal, characteristic, complemented
9#32921—31nomaximal, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C3×#520—22no—
12C3×#520—22no—
13C3×#520C2212yes—
14C3×#519—22no—
15C3×#519C2211yescharacteristic
16C3×C4×#41—22no—
17C3×C4×#41—22no—
18C3×C4×#41C2212yes—
19C2×C3×C4×Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#949C2214yes—
22C2×#947C2212yes—
23C2×#874C2211yescharacteristic
24#5245—28no—
25C2×#837C2214yes—
26C2×#836C2212yes—
27#5027—22no—
28#5027—22no—
29#5015—22no—
30#5015—22no—
31#5001C2214yes—
32#4991—22no—
33#4988C2212yes—
34#4939—22no—
35C4×C4⋊Dic3C2211yescharacteristic
36#4924C2212yes—
37#4532C2214yes—
38#4530C2212yes—
39C4×2−1+4D311yescharacteristic, complemented
40#3038—32no—
41C2×#454—32no—
42#2548—31no—
43#1888—32no—
44#1883—34no—
45#1806—32no—
46#1801—31no—
47C3×2−1+4—24no—
48C3×2−1+4—22no—
49C3×2−1+4D414yescomplemented
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41C2312yes—
53C22×C3×Q8—22no—
54C22×C3×Q8C2311yescharacteristic
55C22×D12—21no—
56C22×D12—21no—
57C22×D12C2312yes—
58C3×C4×Q8—22no—
59C3×C4×Q8—24no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8D412yes—
67C3×C4×Q8D412yes—
68C3×C4×D4—22no—
69C3×C4×D4—24no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4—24no—
74C3×C4×D4—22no—
75C3×C4×D4D412yes—
76C3×C4×D4D412yes—
77C3×#117—24no—
78C3×#117—24no—
79C3×#117—22no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117C2312yes—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4D412yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—24no—
88C2×C3×C42—22no—
89C2×C3×C42D412yes—
90C2×C3×C42C2311yescharacteristic
91#949—28no—
92#949—28no—
93#947—24no—
94#947—24no—
95C2×#215—24no—
96C2×#213—24no—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—24no—
99C2×C4⋊Dic3—21no—
100C2×C4×Dic3C2312yes—
101C2×#208C2314yes—
102#874—22no—
103#874—22no—
104#837—28no—
105#836—24no—
106C3⋊#95—21no—
107#809—24no—
108#809—24no—
109C4⋊(C3⋊C8)—22no—
110C4⋊(C3⋊C8)—22no—
111C2×2−1+4C2×D311yescharacteristic
112C2×AGL(1,ℤ/8)—34no—
113C2×#127—31no—
114C2×#124—32no—
115#520—22no—
116#520—22no—
117#520C2×D312yes—
118#519—22no—
119#519C2×D311yescharacteristic
120C4×#41—22no—
121C4×#41—22no—
122C4×#41C2×D312yes—
123C2×C4×Q8—22no—
124C2×C4×Q8C2×D311yescharacteristic
125#460—62no—
126#460—62no—
127#459—62no—
128#459—62no—
129#454—68no—
130#423—32no—
131C2×#104—34no—
132#418—34no—
133C2×#101—32no—
134#394—62no—
135#390—32no—
136#390—62no—
137C4×C4⋊C4—31no—
138#343—32no—
139#339—34no—
140C23×D3—22no—
141C3×#41—28no—
142C3×#41—28no—
143C3×#41—28no—
144C3×#41—28no—
145C3×#41—22no—
146C3×#41—22no—
147C3×#41C2×D414yes—
148C3×#41#4114yes—
149C2×C3×Q8—22no—
150C2×C3×Q8—28no—
151C2×C3×Q8—28no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8C2×D412yes—
155C2×C3×Q8C2×D412yes—
156C2×C3×Q8#4112yes—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4C2×D412yes—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—22no—
171C2×D12—24no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—24no—
175C2×D12—44no—
176C2×D12—41no—
177C2×D12—42no—
178C2×D12—41no—
179C2×D12—82no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4C2×D412yes—
188C3×C4⋊C4C2×D412yes—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C42—24no—
194C3×C42—24no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202#215—48no—
203#213—48no—
204(C2×D3)⋊C4—48no—
205(C2×D3)⋊C4—84no—
206C4⋊Dic3—42no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209#208—24no—
210#208—24no—
211C2×C3⋊C8—24no—
2122−1+4—24no—
2132−1+4—22no—
2142−1+4D1214yescomplemented
215C2×#41—22no—
216C2×#41—22no—
217C2×#41C22×D312yes—
218C22×Q8—22no—
219C22×Q8C22×D311yescharacteristic
220C22×D4—32no—
221C22×D4—61no—
222C22×D4—61no—
223AGL(1,ℤ/8)—68no—
224AGL(1,ℤ/8)—68no—
225C2×SD16—64no—
226C2×D8—64no—
227C2×M16—34no—
228#127—62no—
229#127—62no—
230#124—64no—
231#124—64no—
232C4×Q8—22no—
233C4×Q8—24no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8#6712yes—
241C4×Q8#6712yes—
242C4×D4—22no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4#6712yes—
250C4×D4#6712yes—
251#117—24no—
252#117—24no—
253#117—22no—
254#117—22no—
255#117—22no—
256#117C22×D312yes—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4#6712yes—
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—32no—
267C2×C42#6712yes—
268C2×C42C22×D311yescharacteristic
269C4⋊C8—62no—
270C4⋊C8—62no—
271#104—68no—
272#102—64no—
273#102—64no—
274#101—64no—
275#95—61no—
276C23×C3—22no—
277C23×C3C22×D411yescharacteristic
278C22×D3—44no—
279C22×D3—44no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C3×Q8—28no—
284C3×Q8—24no—
285C3×Q8—22no—
286C3×Q8—24no—
287C3×Q8#12114yes—
288C3×D4—28no—
289C3×D4—24no—
290C3×D4—24no—
291C3×D4—22no—
292C3×D4#12114yes—
293C2×C3×C4—216no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—44no—
305C2×C3×C4—42no—
306C2×C3×C4—82no—
307C2×C3×C4—84no—
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12012yes—
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12114yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D412yes—
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4111yescharacteristic
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319D12—44no—
320D12—44no—
321D12—82no—
322D12—84no—
323D12—82no—
324C3⋊C8—44no—
325C24—62no—
326#41—28no—
327#41—28no—
328#41—28no—
329#41—28no—
330#41—22no—
331#41—22no—
332#41C2×D1214yes—
333#41#23514yes—
334C2×Q8—22no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8C2×D1212yes—
340C2×Q8#23512yes—
341C2×Q8C2×#6712yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—64no—
347C2×D4—64no—
348C2×D4—124no—
349C2×D4—121no—
350C2×D4—122no—
351C2×D4—121no—
352C2×D4—242no—
353C2×D4C2×#6712yes—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—62no—
360C22×C4C2×#6712yes—
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6712yes—
363C22×C4C2×#6711yescharacteristic
364C22×C4C23×D311yescharacteristic
365SD16—128no—
366D8—128no—
367M16—64no—
368M16—64no—
369C2×C8—64no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—122no—
378C4⋊C4C2×#6712yes—
379C4⋊C4C2×#6712yes—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—128no—
385C22⋊C4—244no—
386C42—24no—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—62no—
394C42—62no—
395C42C2×#6712yes—
396C42C2×#6712yes—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—24no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3C2×#12011yescharacteristic
403C22×C3C2×#12111yescharacteristic
404C22×C3#52511yescharacteristic
405C2×D3—84no—
406C2×D3—84no—
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—82no—
410C3×C4—24no—
411C3×C4—24no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4#45412yescomplemented
418C3×C4C2×#12112yes—
419C3×C4#52512yes—
420Dic3—82no—
421C23—22no—
422C23—124no—
423C23—124no—
424C23—122no—
425C23—122no—
426C23—122no—
427C23C22×#6711yescharacteristic
428Q8—28no—
429Q8—24no—
430Q8—24no—
431Q8—22no—
432Q8#93014yes—
433D4—28no—
434D4—24no—
435D4—24no—
436D4—22no—
437D4—124no—
438D4—124no—
439D4—242no—
440D4—244no—
441D4—242no—
442D4#93014yes—
443C2×C4—216no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—84no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4#93014yes—
462C2×C4#95311yescharacteristic
463C2×C4#95312yes—
464C2×C4#95311yescharacteristic
465C2×C4C22×D1211yescharacteristic
466C2×C4C2×#23512yes—
467C2×C4C2×#23511yescharacteristic
468C2×C4C22×#6712yes—
469C8—124no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—44no—
473C2×C3#284711yescharacteristic
474C2×C3C2×#45411yescharacteristic
475C2×C3#302111yescharacteristic
476D3—162no—
477C22—21no—
478C22—21no—
479C22—24no—
480C22—42no—
481C22—42no—
482C22—244no—
483C22—244no—
484C22—242no—
485C22—242no—
486C22—242no—
487C22C2×#93011yescharacteristic
488C22#579111yescharacteristic
489C22C2×#95311yescharacteristic
490C4—24no—
491C4—24no—
492C4—21no—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—242no—
498C4#524512yescomplemented
499C4C2×#93012yes—
500C4#579112yes—
501C3#3292111yescharacteristic, complemented, Sylow
502C2—21no—
503C2—21no—
504C2—44no—
505C2—482no—
506C2#8403211yescharacteristic
507C2C2×#524511yescharacteristic
508C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×2−1+4C2
socleC22×C3C2×#120
hypercenterC4×2−1+4D3
hypocenterC3#32921
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×C4×2−1+4C2semidirect
2. C2×#5245C2semidirect
3. #82526C2semidirect
4. #82476C2semidirect
5. #71667C2semidirect
6. #71661C2semidirect
7. #71646C2semidirect
8. #71637C2semidirect
39. C4×2−1+4D3semidirect
49. C3×2−1+4D4semidirect
214. 2−1+4D12semidirect
417. C3×C4#454semidirect
498. C4#5245semidirect
501. C3#32921semidirect

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C2×#52452
#825262
#824762
#716672
#716612
#716462
#716372
#329213