Finite Group Databasebeta

← #10944159

Subgroups — #10944159

Order 768.

1590 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1004C211yesmaximal, characteristic, complemented
2C3×#2541C211yesmaximal, characteristic, complemented
3C2×#5077C212yesmaximal, complemented
4#83165C212yesmaximal, complemented
5#71763C212yesmaximal, complemented
6#71617C211yesmaximal, characteristic, complemented
7#69663C214yesmaximal, complemented
8#69494C212yesmaximal, complemented
9#32920—31nomaximal, Sylow
10C2×#1004C2211yescharacteristic
11C2×C3×#127C2212yes—
12C2×C3×#127C2211yescharacteristic, complemented
13C2×C3×#121—22no—
14C2×C3×C4×D4C2211yescharacteristic
15#5798C2211yescharacteristic
16C2×#932C2214yes—
17C2×C4×#67—22no—
18#5546—22no—
19#5545—22no—
20C4×D3×D4—22no—
21C4×#237—22no—
22#5504C2212yes—
23#5486—22no—
24C4×#235C2212yes—
25C3×#392—22no—
26C3×#390C2212yescomplemented
27C3×C4×C4⋊C4C2211yescharacteristic, complemented
28C2×#833C2212yes—
29#5215C2214yescomplemented
30#5077—28no—
31#5075—22no—
32#5075—22no—
33#5072—22no—
34#5072—22no—
35C2×#805C2214yes—
36#4924C2212yescomplemented
37#4530C2212yescomplemented
38#4523C2214yescomplemented
39C4×H(5,𝔽2)—31no—
40#3038—32no—
41C2×#453—32no—
42#2541D311yescharacteristic, complemented
43#1887—32no—
44#1883—34no—
45#1806—32no—
46#1800—31no—
47C22×C3×D4—24no—
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4C2312yes—
51C22×C3×D4C2311yescharacteristic
52C23×C3×C4D412yes—
53C22×#67—22no—
54#1004—22no—
55#1004—24no—
56#1004D414yescomplemented
57C2×#237—22no—
58C2×D3×D4—22no—
59C2×#235C2312yes—
60C22×C4×D3—22no—
61C3×#127—24no—
62C3×#127—24no—
63C3×#127—22no—
64C3×#127—22no—
65C3×#121—48no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4D412yes—
72C2×C3×C42C2312yes—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75D4×Dic3—22no—
76D4×Dic3—22no—
77C2×#214—24no—
78#932—28no—
79#932—28no—
80C2×#213—24no—
81C4×#67—24no—
82C4×#67—24no—
83C4×#67—22no—
84C4×#67—22no—
85C4×#67—24no—
86C2×(C2×D3)⋊C4—22no—
87#926C2312yes—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—22no—
90C2×#208C2314yes—
91#893—22no—
92#892—22no—
93D3×C4⋊C4—22no—
94#887—22no—
95#67⋊C4—24no—
96#67⋊C4—24no—
97D3×C22⋊C4—24no—
98#877—24no—
99C4×D12D412yescomplemented
100C4×D12D412yescomplemented
101#872—22no—
102C42×D3—22no—
103C4×Dic6D412yescomplemented
104C4×Dic6D412yescomplemented
105C3×#95—21no—
106#833—24no—
107#832—24no—
108#832—24no—
109#805—28no—
110C4⋊(C3⋊C8)—22no—
111C4⋊(C3⋊C8)—22no—
112C2×H(5,𝔽2)—31no—
113C2×AGL(1,ℤ/8)—34no—
114C2×#127C2×D312yes—
115C2×#127C2×D311yescharacteristic, complemented
116C2×#121—22no—
117#520—32no—
118#520—62no—
119#518—31no—
120#518—62no—
121#518—62no—
122C4×#41—32no—
123C4×#41—62no—
124C2×C4×D4—62no—
125C2×C4×D4—62no—
126C2×C4×D4C2×D311yescharacteristic
127#460—62no—
128#460—62no—
129#459—62no—
130#459—62no—
131#453—68no—
132#423—32no—
133C2×#104—34no—
134#418—34no—
135C2×#100—32no—
136#392—22no—
137#390C2×D312yescomplemented
138C4×C4⋊C4C2×D311yescharacteristic, complemented
139#343—32no—
140#339—34no—
141C24×C3—22no—
142C24×C3C2×D412yes—
143C23×D3—22no—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—44no—
151C2×C3×D4—48no—
152C2×C3×D4—48no—
153C2×C3×D4—42no—
154C2×C3×D4—41no—
155C2×C3×D4—41no—
156C2×C3×D4—82no—
157C2×C3×D4—84no—
158C2×C3×D4C2×D412yes—
159C2×C3×D4#4112yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—44no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×#67—24no—
169C2×#67—28no—
170C2×#67—28no—
171C2×#67—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175#237—28no—
176#237—28no—
177D3×D4—28no—
178D3×D4—28no—
179#235—22no—
180#235—22no—
181#235C2×D414yes—
182#235#4114yes—
183C2×D12C2×D412yes—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—24no—
187C2×C4×D3—22no—
188C2×C4×D3—84no—
189C2×Dic6C2×D412yes—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—22no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—48no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198#217—24no—
199#217—22no—
200#214—48no—
201#213—48no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—22no—
204(C2×D3)⋊C4—22no—
205C4⋊Dic3C2×D412yes—
206C4⋊Dic3C2×D412yes—
207Dic3⋊C4—24no—
208Dic3⋊C4—22no—
209Dic3⋊C4—22no—
210C4×Dic3—24no—
211C4×Dic3—22no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214#208—24no—
215#208—24no—
216C2×C3⋊C8—24no—
217H(5,𝔽2)—34no—
218H(5,𝔽2)—64no—
219H(5,𝔽2)—62no—
220C2×#41—32no—
221C2×#41—62no—
222C22×D4—24no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—62no—
226C22×D4—62no—
227C22×D4C22×D312yes—
228C22×D4C22×D311yescharacteristic
229C23×C4—62no—
230C23×C4#6712yes—
231AGL(1,ℤ/8)—68no—
232AGL(1,ℤ/8)—68no—
233C2×SD16—64no—
234C2×D8—64no—
235C2×M16—34no—
236#127—24no—
237#127—24no—
238#127—22no—
239#127—22no—
240#121—48no—
241C4×Q8—32no—
242C4×Q8—32no—
243C4×Q8—62no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—32no—
247C4×D4—32no—
248C4×D4—64no—
249C4×D4—64no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4—64no—
254C4×D4—64no—
255C4×D4—62no—
256C4×D4—64no—
257C4×D4—62no—
258#117—32no—
259#117—64no—
260#117—62no—
261#117—62no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C22⋊C4—24no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4#6712yes—
271C2×C42—62no—
272C2×C42—62no—
273C2×C42C22×D312yes—
274C2×C42C22×D311yescharacteristic
275C4⋊C8—62no—
276C4⋊C8—62no—
277#104—68no—
278#102—64no—
279#102—64no—
280#100—64no—
281#95—21no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—44no—
287C23×C3—44no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3#12114yes—
292C23×C3C22×D411yescharacteristic
293C22×D3—22no—
294C22×D3—28no—
295C22×D3—82no—
296C22×D3—82no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—44no—
300C3×D4—44no—
301C3×D4—82no—
302C3×D4—84no—
303C3×D4—816no—
304C3×D4—82no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—42no—
313C2×C3×C4—82no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D412yes—
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4112yes—
321#67—216no—
322#67—24no—
323#67—24no—
324C2×Dic3—28no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—82no—
329C2×Dic3—82no—
330D12—22no—
331D12#12114yescomplemented
332C4×D3—24no—
333C4×D3—24no—
334C4×D3—44no—
335C4×D3—82no—
336C4×D3—84no—
337Dic6—22no—
338Dic6#12114yescomplemented
339C3⋊C8—44no—
340C24—22no—
341C24—62no—
342C24C2×#6712yes—
343#41—34no—
344#41—34no—
345#41—68no—
346#41—68no—
347#41—62no—
348#41—62no—
349C2×Q8—32no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—32no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—68no—
364C2×D4—68no—
365C2×D4—64no—
366C2×D4—68no—
367C2×D4—68no—
368C2×D4—62no—
369C2×D4—82no—
370C2×D4—84no—
371C2×D4D3×D412yes—
372C2×D4#23712yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—64no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—244no—
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6712yes—
387C22×C4C2×#6711yescharacteristic
388C22×C4C23×D311yescharacteristic
389SD16—128no—
390D8—128no—
391M16—64no—
392M16—64no—
393C2×C8—64no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—32no—
397C4⋊C4—32no—
398C4⋊C4—64no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C22⋊C4—24no—
402C22⋊C4—48no—
403C22⋊C4—64no—
404C22⋊C4—64no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—62no—
408C42—22no—
409C42—22no—
410C42—64no—
411C42—62no—
412C42—62no—
413C42—62no—
414C42D3×D412yes—
415C42D3×D412yes—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—24no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—82no—
423C22×C3—84no—
424C22×C3—84no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3C2×#12011yescharacteristic
429C22×C3C2×#12111yescharacteristic
430C22×C3#52511yescharacteristic
431C2×D3—24no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—88no—
435C3×C4—21no—
436C3×C4—22no—
437C3×C4—44no—
438C3×C4—42no—
439C3×C4#45312yescomplemented
440C3×C4C2×#12112yes—
441C3×C4#52512yes—
442Dic3—24no—
443Dic3—42no—
444Dic3—42no—
445C23—22no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—68no—
455C23—62no—
456C23—242no—
457C23—242no—
458C23#93814yes—
459C23C22×#6711yescharacteristic
460Q8—34no—
461Q8—62no—
462D4—24no—
463D4—24no—
464D4—34no—
465D4—44no—
466D4—44no—
467D4—616no—
468D4—64no—
469D4—64no—
470D4—62no—
471D4—82no—
472D4—84no—
473D4—816no—
474D4—82no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—68no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—82no—
490C2×C4—124no—
491C2×C4—242no—
492C2×C4—244no—
493C2×C4—242no—
494C2×C4—242no—
495C2×C4#93712yes—
496C2×C4#93711yescharacteristic
497C2×C4#93711yescharacteristic
498C2×C4C2×D3×D411yescharacteristic
499C2×C4C2×D3×D412yes—
500C2×C4C2×#23711yescharacteristic
501C2×C4D3×#4112yes—
502C8—124no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—44no—
506C2×C3—162no—
507C2×C3#284711yescharacteristic
508C2×C3C2×#45311yescharacteristic
509C2×C3#302111yescharacteristic
510D3—44no—
511C22—21no—
512C22—21no—
513C22—24no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—64no—
518C22—82no—
519C22—84no—
520C22—84no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—122no—
525C22—122no—
526C22—248no—
527C22C2×#93711yescharacteristic
528C22C2×#93811yescharacteristic
529C22#582811yescharacteristic
530C4—22no—
531C4—21no—
532C4—44no—
533C4—42no—
534C4—64no—
535C4—122no—
536C4—122no—
537C4#507712yescomplemented
538C4D3×#12112yes—
539C4#560612yes—
540C3#3292011yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—44no—
544C2—124no—
545C2—162no—
546C2#8272311yescharacteristic
547C2C2×#507711yescharacteristic
548C2#8339511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2541C2
socleC22×C3C2×#120
hypercenter#2541D3
hypocenterC3#32920
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C4×#1004C2semidirect
2. C3×#2541C2semidirect
3. C2×#5077C2semidirect
4. #83165C2semidirect
5. #71763C2semidirect
6. #71617C2semidirect
7. #69663C2semidirect
8. #69494C2semidirect
12. C2×C3×#127C22semidirect
26. C3×#390C22semidirect
27. C3×C4×C4⋊C4C22semidirect
29. #5215C22semidirect
36. #4924C22semidirect
37. #4530C22semidirect
38. #4523C22semidirect
42. #2541D3semidirect
56. #1004D4semidirect
99. C4×D12D4semidirect
100. C4×D12D4semidirect
103. C4×Dic6D4semidirect
104. C4×Dic6D4semidirect
115. C2×#127C2×D3semidirect
137. #390C2×D3semidirect
138. C4×C4⋊C4C2×D3semidirect
331. D12#121semidirect
338. Dic6#121semidirect
439. C3×C4#453semidirect
537. C4#5077semidirect
540. C3#32920semidirect

Maximal subgroups

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

subgroupindex
C4×#10042
C3×#25412
C2×#50772
#831652
#717632
#716172
#696632
#694942
#329203