Finite Group Databasebeta

← #10944167

Subgroups — #10944167

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
1C4×#1007C211yesmaximal, characteristic, complemented
2C3×#2548C211yesmaximal, characteristic, complemented
3#83165C212yesmaximal, complemented
4C2×#5062C212yesmaximal, complemented
5#71724C212yesmaximal, complemented
6#71637C211yesmaximal, characteristic, complemented
7#69663C214yesmaximal, complemented
8#69494C212yesmaximal, complemented
9#32921—31nomaximal, Sylow
10C2×#1007C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C2×C3×#124C2212yes—
13C2×C3×C4×Q8C2211yescharacteristic
14#5812C2211yescharacteristic
15C2×#932C2214yes—
16#5574—22no—
17C4×#239—22no—
18#5572—22no—
19C4×D3×Q8—22no—
20#5504—22no—
21#5504C2212yes—
22C4×#235—22no—
23C4×#235C2212yes—
24C3×#394—22no—
25C3×#390—22no—
26C3×#390C2212yes—
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#836C2212yes—
29#5215C2214yes—
30#5075—22no—
31#5075—22no—
32#5072—22no—
33#5072—22no—
34#5062—28no—
35C2×#805C2214yes—
36#4924C2212yes—
37#4530C2212yes—
38#4523C2214yes—
39C4×2−1+4—31no—
40#3038—32no—
41C2×#454—32no—
42#2548D311yescharacteristic, complemented
43#1888—32no—
44#1883—34no—
45#1806—32no—
46#1801—31no—
47C22×C3×Q8C2311yescharacteristic
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4C2312yes—
51#1007—22no—
52#1007—24no—
53#1007D414yescomplemented
54C2×#239—22no—
55C2×D3×Q8—22no—
56C2×#235—22no—
57C2×#235C2312yes—
58C3×#127—22no—
59C3×#127—22no—
60C3×#124—24no—
61C3×#124—24no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D412yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C42D412yes—
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72Q8×Dic3—22no—
73Q8×Dic3—22no—
74C2×#214—24no—
75#932—28no—
76#932—28no—
77C2×#213—24no—
78C4×#67—22no—
79C4×#67—24no—
80C4×#67—22no—
81#926—22no—
82#926C2312yes—
83C2×#208C2314yes—
84#893—24no—
85#893—22no—
86#892—24no—
87#892—22no—
88D3×C4⋊C4—24no—
89D3×C4⋊C4—22no—
90#887—24no—
91#887—22no—
92C4×D12—22no—
93C4×D12—22no—
94C4×D12D412yes—
95C4×D12D412yes—
96#872—24no—
97#872—22no—
98C42×D3—24no—
99C42×D3—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6D412yes—
103C4×Dic6D412yes—
104C3×#95—21no—
105#836—24no—
106#832—24no—
107#832—24no—
108#805—28no—
109C4⋊(C3⋊C8)—22no—
110C4⋊(C3⋊C8)—22no—
111C2×2−1+4—31no—
112C2×AGL(1,ℤ/8)—34no—
113C2×#127C2×D311yescharacteristic
114C2×#124C2×D312yes—
115#520—32no—
116#520—62no—
117#520—62no—
118#519—31no—
119#519—62no—
120C4×#41—32no—
121C4×#41—62no—
122C4×#41—62no—
123C2×C4×Q8—62no—
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—22no—
135#390—22no—
136#390C2×D312yes—
137C4×C4⋊C4C2×D311yescharacteristic
138#343—32no—
139#339—34no—
140C24×C3—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8C2×D412yes—
144C2×C3×Q8#4112yes—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—44no—
150C2×C3×D4—42no—
151C2×C3×D4—41no—
152C2×C3×D4—41no—
153C2×C3×D4—82no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C2×#67—22no—
161#239—28no—
162#239—28no—
163D3×Q8—28no—
164D3×Q8—28no—
165#235—28no—
166#235—28no—
167#235—22no—
168#235—22no—
169#235C2×D414yes—
170#235#4114yes—
171C2×D12—22no—
172C2×D12C2×D412yes—
173C2×C4×D3—24no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—24no—
177C2×C4×D3—22no—
178C2×Dic6—22no—
179C2×Dic6C2×D412yes—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C22⋊C4—48no—
184C3×C22⋊C4—84no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42C2×D412yes—
190C3×C42C2×D412yes—
191#217—22no—
192#214—48no—
193#213—48no—
194(C2×D3)⋊C4—24no—
195(C2×D3)⋊C4—22no—
196(C2×D3)⋊C4—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3C2×D412yes—
200C4⋊Dic3C2×D412yes—
201Dic3⋊C4—24no—
202Dic3⋊C4—22no—
203Dic3⋊C4—22no—
204C4×Dic3—24no—
205C4×Dic3—24no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209#208—24no—
210#208—24no—
211C2×C3⋊C8—24no—
2122−1+4—34no—
2132−1+4—64no—
2142−1+4—62no—
215C2×#41—32no—
216C2×#41—62no—
217C2×#41—62no—
218C22×Q8—62no—
219C22×Q8C22×D311yescharacteristic
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C22×D312yes—
223AGL(1,ℤ/8)—68no—
224AGL(1,ℤ/8)—68no—
225C2×SD16—64no—
226C2×D8—64no—
227C2×M16—34no—
228#127—22no—
229#127—22no—
230#124—24no—
231#124—24no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—32no—
235C4×Q8—32no—
236C4×Q8—62no—
237C4×Q8—62no—
238C4×Q8—62no—
239C4×Q8—64no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×D4—32no—
243C4×D4—32no—
244C4×D4—62no—
245C4×D4—62no—
246C4×D4—62no—
247C4×D4—64no—
248C4×D4—62no—
249C4×D4—62no—
250C4×D4—64no—
251#117—32no—
252#117—64no—
253#117—62no—
254#117—64no—
255#117—62no—
256#117—62no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4#6712yes—
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C42—64no—
265C2×C42—62no—
266C2×C42#6712yes—
267C2×C42C22×D312yes—
268C2×C42C22×D311yescharacteristic
269C4⋊C8—62no—
270C4⋊C8—62no—
271#104—68no—
272#102—64no—
273#102—64no—
274#101—64no—
275#95—21no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3C22×D411yescharacteristic
282C22×D3—22no—
283C3×Q8—24no—
284C3×Q8—24no—
285C3×D4—44no—
286C3×D4—44no—
287C3×D4—82no—
288C3×D4—84no—
289C3×D4—82no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4#12012yes—
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4#12114yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D412yes—
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4112yes—
307#67—24no—
308#67—24no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312D12—28no—
313D12—22no—
314D12#12114yes—
315C4×D3—216no—
316C4×D3—24no—
317C4×D3—24no—
318C4×D3—44no—
319C4×D3—82no—
320C4×D3—84no—
321Dic6—28no—
322Dic6—22no—
323Dic6#12114yes—
324C3⋊C8—44no—
325C24—22no—
326#41—34no—
327#41—34no—
328#41—68no—
329#41—68no—
330#41—68no—
331#41—68no—
332#41—62no—
333#41—62no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—32no—
337C2×Q8—62no—
338C2×Q8—68no—
339C2×Q8—68no—
340C2×Q8D3×D412yes—
341C2×Q8#23712yes—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—32no—
347C2×D4—44no—
348C2×D4—41no—
349C2×D4—42no—
350C2×D4—41no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—82no—
354C22×C4—22no—
355C22×C4—64no—
356C22×C4—64no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6712yes—
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—32no—
373C4⋊C4—32no—
374C4⋊C4—42no—
375C4⋊C4—64no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C22⋊C4—48no—
381C22⋊C4—64no—
382C22⋊C4—62no—
383C22⋊C4—62no—
384C22⋊C4—62no—
385C22⋊C4—84no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—64no—
391C42—64no—
392C42—62no—
393C42—62no—
394C42—62no—
395C42D3×D412yes—
396C42D3×D412yes—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—84no—
400C22×C3—84no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3C2×#12011yescharacteristic
405C22×C3C2×#12111yescharacteristic
406C22×C3#52511yescharacteristic
407C2×D3—24no—
408C2×D3—42no—
409C2×D3—42no—
410C3×C4—24no—
411C3×C4—21no—
412C3×C4—22no—
413C3×C4—42no—
414C3×C4—82no—
415C3×C4#45412yescomplemented
416C3×C4C2×#12112yes—
417C3×C4#52512yes—
418Dic3—24no—
419Dic3—42no—
420Dic3—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—62no—
427C23C22×#6711yescharacteristic
428Q8—24no—
429Q8—24no—
430Q8—34no—
431Q8—68no—
432Q8—62no—
433D4—34no—
434D4—44no—
435D4—44no—
436D4—68no—
437D4—64no—
438D4—64no—
439D4—62no—
440D4—82no—
441D4—84no—
442D4—82no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—616no—
453C2×C4—62no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—124no—
459C2×C4—242no—
460C2×C4—244no—
461C2×C4#93711yescharacteristic
462C2×C4#93712yes—
463C2×C4#93711yescharacteristic
464C2×C4#93814yes—
465C2×C4C2×D3×D411yescharacteristic
466C2×C4C2×D3×D412yes—
467C2×C4C2×#23711yescharacteristic
468C2×C4D3×#4112yes—
469C8—124no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—162no—
473C2×C3#284711yescharacteristic
474C2×C3C2×#45411yescharacteristic
475C2×C3#302111yescharacteristic
476D3—44no—
477C22—21no—
478C22—21no—
479C22—64no—
480C22—84no—
481C22—84no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—122no—
486C22—122no—
487C22C2×#93711yescharacteristic
488C22C2×#93811yescharacteristic
489C22#582811yescharacteristic
490C4—21no—
491C4—24no—
492C4—22no—
493C4—42no—
494C4—64no—
495C4—82no—
496C4—122no—
497C4—122no—
498C4#506212yescomplemented
499C4D3×#12112yes—
500C4#560612yes—
501C3#3292111yescharacteristic, complemented, Sylow
502C2—21no—
503C2—21no—
504C2—124no—
505C2—162no—
506C2#8272311yescharacteristic
507C2C2×#506211yescharacteristic
508C2#8339511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2548C2
socleC22×C3C2×#120
hypercenter#2548D3
hypocenterC3#32921
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C4×#1007C2semidirect
2. C3×#2548C2semidirect
3. #83165C2semidirect
4. C2×#5062C2semidirect
5. #71724C2semidirect
6. #71637C2semidirect
7. #69663C2semidirect
8. #69494C2semidirect
42. #2548D3semidirect
53. #1007D4semidirect
415. C3×C4#454semidirect
498. C4#5062semidirect
501. C3#32921semidirect

Maximal subgroups

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

subgroupindex
C4×#10072
C3×#25482
#831652
C2×#50622
#717242
#716372
#696632
#694942
#329213