Finite Group Databasebeta

← #10845416

Subgroups — #10845416

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2547C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic, complemented
3#82417C211yesmaximal, characteristic
4C2×#5012C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6#69524C212yesmaximal, complemented
7#69519C212yesmaximal, complemented
8#69517C212yesmaximal, complemented
9#69513C212yesmaximal, complemented
10C4×#809C211yesmaximal, characteristic, complemented
11#69467C211yesmaximal, characteristic, complemented
12#25347—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×#938—21no—
17C2×#930C2212yes—
18C22×#208C2212yes—
19C2×C4×D12C2211yescharacteristic
20C3×#397—22no—
21C3×#391—21no—
22C3×#380C2212yes—
23C3×#380C2212yes—
24C3×C4×C4⋊C4C2211yescharacteristic
25#5212C2212yes—
26C2×#830C2212yes—
27#5087—24no—
28#5012—24no—
29#4987—21no—
30C2×#809C2212yes—
31C2×#809C2211yescharacteristic
32C2×#809C2211yescharacteristic
33C2×#808C2212yes—
34C2×#807C2212yes—
35#4961—22no—
36#4935C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4529C2212yes—
40#4529C2211yescharacteristic
41#4529C2211yescharacteristic
42#4529C2212yes—
43#4529C2212yes—
44C2×#474—31no—
45C2×#436—31no—
46#2547D311yescharacteristic, complemented
47#2537—31no—
48#2076—32no—
49#2034—32no—
50#1990—31no—
51#1948—31no—
52#1895—32no—
53#1809—32no—
54C4×#102—31no—
55C22×C3×Q8—21no—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67—22no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×#124—44no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77#938—44no—
78#930—24no—
79#930—42no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×#208—24no—
87C2×#208—24no—
88C2×#208—42no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92C4×D12—22no—
93C4×D12—22no—
94C3×#95—21no—
95C3×#95—22no—
96C3×#95—22no—
97C3⋊#95—22no—
98#830—24no—
99#809—22no—
100#809—24no—
101#809—22no—
102#808—42no—
103#807—24no—
104C4⋊(C3⋊C8)D414yescomplemented
105C4×C3⋊C8—22no—
106C22×M16—32no—
107C2×C4⋊Q8C2×D311yescharacteristic
108C2×#124—21no—
109C2×#121—32no—
110C2×#121—61no—
111C2×C4×D4—31no—
112C2×C4×D4C2×D311yescharacteristic
113#474—64no—
114#436—64no—
115C2×#107—32no—
116C2×#106—32no—
117C2×C4⋊C8—31no—
118C2×#102—31no—
119C2×#102—32no—
120C2×#102—31no—
121C2×C22⋊C8—32no—
122C2×C4×C8—31no—
123#397—22no—
124#394—62no—
125#391—21no—
126#384—31no—
127#380—32no—
128#380—61no—
129#380C2×D312yes—
130#380C2×D312yes—
131C4×C4⋊C4C2×D311yescharacteristic
132#340—31no—
133#340—32no—
134#340—31no—
135#340—32no—
136#340—32no—
137C24×C3C2×D412yes—
138C23×D3—21no—
139C2×C3×Q8—42no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C2×#67—44no—
159C2×#67—44no—
160C2×#67—82no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12#4112yes—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C2×C4×D3—42no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—42no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—44no—
183C3×C42—42no—
184C3×C42C2×D412yes—
185C3×C42#4112yes—
186#217—44no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—42no—
194C4⋊Dic3—81no—
195#208—48no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8C2×D414yes—
204C2×C3⋊C8#4112yes—
205C22×Q8—21no—
206C22×D4—31no—
207C22×D4—62no—
208C22×D4#6711yescharacteristic
209C23×C4—61no—
210C23×C4C22×D312yes—
211C2×M16—64no—
212C2×M16—64no—
213C2×M16—122no—
214C22×C8—31no—
215C22×C8—32no—
216C22×C8—31no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219#124—44no—
220#121—64no—
221#121—122no—
222#121—124no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—62no—
226C4×D4—62no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4#6711yescharacteristic
233C2×C4⋊C4C22×D312yes—
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4#6712yes—
242C2×C42—21no—
243C2×C42C22×D311yescharacteristic
244#107—64no—
245#106—122no—
246C4⋊C8—34no—
247#102—62no—
248#102—64no—
249#102—62no—
250C22⋊C8—64no—
251C4×C8—62no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—62no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3C22×D411yescharacteristic
264C22×D3—22no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—41no—
271C3×Q8—84no—
272C3×D4—44no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4#12112yes—
292C2×C3×C4#12112yes—
293C2×C3×C4AGL(1,ℤ/8)12yes—
294C2×C3×C4AGL(1,ℤ/8)12yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4H(5,𝔽2)11yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301#67—88no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308D12—22no—
309D12—22no—
310D12—41no—
311D12—41no—
312C4×D3—84no—
313C3⋊C8—24no—
314C3⋊C8—22no—
315C3⋊C8—41no—
316C24—61no—
317C24C2×#6712yes—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—32no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—124no—
331C2×D4—124no—
332C2×D4—242no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—61no—
343C22×C4—62no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4—122no—
347C22×C4C2×#6712yes—
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C23×D311yescharacteristic
351M16—128no—
352C2×C8—34no—
353C2×C8—32no—
354C2×C8—61no—
355C2×C8—62no—
356C2×C8—62no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—61no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—42no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C4⋊C4—122no—
374C4⋊C4—241no—
375C22⋊C4—24no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—124no—
379C22⋊C4—124no—
380C22⋊C4—122no—
381C42—42no—
382C42D3×D412yes—
383C42#23712yes—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3C2×#12111yescharacteristic
392C22×C3#54611yescharacteristic
393C22×C3#54711yescharacteristic
394C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3#57711yescharacteristic
397C22×C3#57811yescharacteristic
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C2×D3—81no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4—82no—
410C3×C4#46914yes—
411C3×C4#52911yescharacteristic
412C3×C4#54211yescharacteristic
413C3×C4#54611yescharacteristic
414C3×C4#54811yescharacteristic
415Dic3—81no—
416Dic3—82no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—62no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23C22×#6711yescharacteristic
432Q8—84no—
433D4—44no—
434D4—62no—
435D4—62no—
436D4—121no—
437D4—121no—
438D4—248no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—244no—
464C2×C4#93212yes—
465C2×C4#93912yes—
466C2×C4#93912yes—
467C2×C4#94912yes—
468C2×C4C2×D3×D411yescharacteristic
469C2×C4C2×#23711yescharacteristic
470C2×C4D3×#4111yescharacteristic
471C2×C4D3×#4111yescharacteristic
472C2×C4#100911yescharacteristic
473C2×C4#101011yescharacteristic
474C8—62no—
475C8—64no—
476C8—121no—
477C2×C3—82no—
478C2×C3#242711yescharacteristic
479C2×C3C2×#46911yescharacteristic
480C2×C3#306311yescharacteristic
481C2×C3#332411yescharacteristic
482C2×C3#333011yescharacteristic
483C2×C3#335311yescharacteristic
484C2×C3#335411yescharacteristic
485D3—82no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—241no—
497C22—241no—
498C22—241no—
499C22—241no—
500C22C2×#93211yescharacteristic
501C22C2×#93911yescharacteristic
502C22C2×#94911yescharacteristic
503C22#582411yescharacteristic
504C22#582911yescharacteristic
505C22#583711yescharacteristic
506C22#583811yescharacteristic
507C4—22no—
508C4—42no—
509C4—81no—
510C4—81no—
511C4—82no—
512C4—241no—
513C4—242no—
514C4#503914yes—
515C4#567611yescharacteristic
516C4#567711yescharacteristic
517C4#568011yescharacteristic
518C4#568211yescharacteristic
519C3#2534711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—242no—
522C2C2×#503911yescharacteristic
523C2#8299611yescharacteristic
524C2#8340011yescharacteristic
525C2#8352111yescharacteristic
526C2#8352711yescharacteristic
527C2#8366211yescharacteristic
528C2#8366611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2547C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenter#2547D3
hypocenterC3#25347
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C3×#2547C2semidirect
2. C2×#5087C2semidirect
4. C2×#5012C2semidirect
5. #71787C2semidirect
6. #69524C2semidirect
7. #69519C2semidirect
8. #69517C2semidirect
9. #69513C2semidirect
10. C4×#809C2semidirect
11. #69467C2semidirect
46. #2547D3semidirect
104. C4⋊(C3⋊C8)D4semidirect
519. C3#25347semidirect

Maximal subgroups

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

subgroupindex
C3×#25472
C2×#50872
#824172
C2×#50122
#717872
#695242
#695192
#695172
#695132
C4×#8092
#694672
#253473