Finite Group Databasebeta

← #10830412

Subgroups — #10830412

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3496C211yesmaximal, characteristic, complemented
2C2×#5241C212yesmaximal, complemented
3#83297C212yesmaximal, complemented
4#82329C211yesmaximal, characteristic, complemented
5#72140C212yesmaximal, complemented
6#71879C212yesmaximal, complemented
7#71877C212yesmaximal, complemented
8#71816C212yesmaximal, complemented
9#71617C211yesmaximal, characteristic, complemented
10#24226—31nomaximal, Sylow
11C2×C3×H(5,𝔽2)C2212yes—
12C3×#547—22no—
13C3×#546—24no—
14C3×D42—22no—
15C3×#537C2212yes—
16C3×#534—22no—
17C3×#532C2212yes—
18C2×C3×#121—22no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×#949C2214yescomplemented
21C2×#940C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic, complemented
23#5241—28no—
24C2×#837C2214yescomplemented
25C2×#837C2214yescomplemented
26#5232C2214yescomplemented
27#5221C2211yescharacteristic
28C2×#833C2212yes—
29#5213—22no—
30#5083—22no—
31#5072—22no—
32#5057—24no—
33#4988C2211yescharacteristic, complemented
34#4977C2211yescharacteristic, complemented
35#4956—22no—
36#4932C2211yescharacteristic, complemented
37#4924C2212yescomplemented
38#4530C2212yescomplemented
39#3496D311yescharacteristic, complemented
40#3036—32no—
41C2×#453—32no—
42#2328—31no—
43#2040—32no—
44#2006—32no—
45#1963—32no—
46#1800—31no—
47#1792—32no—
48C3×H(5,𝔽2)—24no—
49C3×H(5,𝔽2)—24no—
50C3×H(5,𝔽2)—44no—
51C2×C3×#41—24no—
52C2×C3×#41C2314yes—
53C22×C3×D4—24no—
54C22×C3×D4—22no—
55C22×C3×D4—24no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C22×#67—22no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C3×C4⋊Q8D412yescomplemented
62C3×#127D412yescomplemented
63C3×#124D414yescomplemented
64C3×#123—28no—
65C3×C22⋊Q8—24no—
66C3×#121—24no—
67C3×#121—24no—
68C3×#121—24no—
69C3×#121—22no—
70C3×#121—24no—
71C3×#121—22no—
72C3×#120—28no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4D412yes—
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83#949—28no—
84#949—44no—
85C2×#215—24no—
86#940—22no—
87#940D414yescomplemented
88C2×#213—24no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C2×#208C2314yes—
95C4×D12—22no—
96C4×D12—22no—
97#837—28no—
98#837—44no—
99#835—24no—
100#833—24no—
101#832—24no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—22no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×H(5,𝔽2)C2×D312yes—
108C2×AGL(1,ℤ/8)—34no—
109#547—22no—
110#546—24no—
111D42—22no—
112#537C2×D312yes—
113#534—22no—
114#532C2×D312yes—
115C2×#127—31no—
116C2×#121—22no—
117C2×C4×D4—31no—
118C2×C4×D4C2×D311yescharacteristic
119#463—64no—
120#461—62no—
121#459—62no—
122#453—68no—
123#423—32no—
124C2×#104—34no—
125C2×#104—34no—
126#417—34no—
127C2×#100—32no—
128#390—31no—
129#390—31no—
130#386—62no—
131#386—62no—
132#382—31no—
133#376—31no—
134#343—32no—
135C24×C3—24no—
136C24×C3C2×D412yes—
137C23×D3—21no—
138C3×#41—24no—
139C3×#41—24no—
140C3×#41—24no—
141C3×#41—24no—
142C3×#41—48no—
143C3×#41—48no—
144C2×C3×Q8C2×D414yes—
145C2×C3×D4—28no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—48no—
153C2×C3×D4—48no—
154C2×C3×D4—48no—
155C2×C3×D4—48no—
156C2×C3×D4—84no—
157C2×C3×D4C2×D412yes—
158C2×C3×D4C2×D414yes—
159C2×C3×D4#4112yes—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—82no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—28no—
170C2×#67—82no—
171C2×#67—82no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12C2×D412yescomplemented
179C2×C4×D3—42no—
180C2×C4×D3—82no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C22⋊C4—28no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C42C2×D412yes—
190C3×C42C2×D412yes—
191#217—82no—
192#215—48no—
193#213—48no—
194(C2×D3)⋊C4—41no—
195(C2×D3)⋊C4—41no—
196(C2×D3)⋊C4—82no—
197C4⋊Dic3—22no—
198C4×Dic3—21no—
199C4×Dic3—22no—
200C4×Dic3C2×D412yescomplemented
201#208—24no—
202#208—24no—
203C2×C3⋊C8—24no—
204H(5,𝔽2)—24no—
205H(5,𝔽2)—24no—
206H(5,𝔽2)—44no—
207C2×#41—24no—
208C2×#41C22×D314yes—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—31no—
213C22×D4—62no—
214C22×D4C22×D311yescharacteristic
215C23×C4—61no—
216C23×C4#6712yes—
217AGL(1,ℤ/8)—68no—
218AGL(1,ℤ/8)—124no—
219C2×SD16—64no—
220C2×D8—64no—
221C2×M16—34no—
222C4⋊Q8#6712yescomplemented
223#127—34no—
224#127—62no—
225#127#6712yescomplemented
226#124#6714yescomplemented
227#123—28no—
228C22⋊Q8—24no—
229#121—24no—
230#121—24no—
231#121—24no—
232#121—22no—
233#121—24no—
234#121—22no—
235#120—28no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—62no—
241C4×D4—62no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C22⋊C4—24no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4#6712yes—
250C2×C42—31no—
251C2×C42—31no—
252C2×C42C22×D311yescharacteristic
253C4⋊C8—62no—
254C4⋊C8—62no—
255#104—68no—
256#104—124no—
257#103—64no—
258#102—64no—
259#100—64no—
260#95—61no—
261#95—61no—
262#95—62no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—48no—
269C23×C3—84no—
270C23×C3—84no—
271C23×C3#12114yes—
272C23×C3C22×D411yescharacteristic
273C22×D3—22no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—41no—
277C22×D3—41no—
278C3×Q8—24no—
279C3×Q8—24no—
280C3×D4—24no—
281C3×D4—24no—
282C3×D4—24no—
283C3×D4—24no—
284C3×D4—44no—
285C3×D4—416no—
286C3×D4—44no—
287C3×D4—44no—
288C3×D4—82no—
289C3×D4—84no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—44no—
295C2×C3×C4—48no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—42no—
299C2×C3×C4—82no—
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D412yes—
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4H(5,𝔽2)12yes—
307#67—88no—
308C2×Dic3—22no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316D12—24no—
317D12—42no—
318C4×D3—84no—
319C3⋊C8—44no—
320C24—24no—
321C24—61no—
322C24C2×#6712yes—
323#41—24no—
324#41—24no—
325#41—24no—
326#41—24no—
327#41—48no—
328#41—48no—
329C2×Q8C2×#6714yes—
330C2×D4—28no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—32no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—68no—
343C2×D4—61no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—84no—
347C2×D4—242no—
348C2×D4—242no—
349C2×D4D3×D412yes—
350C2×D4#23712yes—
351C2×D4C2×#6714yes—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—82no—
360C22×C4—122no—
361C22×C4—242no—
362C22×C4C2×#6712yes—
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C23×D311yescharacteristic
366SD16—128no—
367D8—128no—
368M16—64no—
369M16—64no—
370C2×C8—64no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—62no—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—242no—
383C22⋊C4—242no—
384C42—32no—
385C42—61no—
386C42—62no—
387C42C2×#6712yes—
388C42C2×#6712yes—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—24no—
392C22×C3—44no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—82no—
399C22×C3—168no—
400C22×C3C2×#12011yescharacteristic
401C22×C3C2×#12111yescharacteristic
402C22×C3#53511yescharacteristic
403C2×D3—44no—
404C2×D3—42no—
405C2×D3—82no—
406C2×D3—81no—
407C2×D3—81no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4D4212yes—
414C3×C4#54612yes—
415Dic3—42no—
416Dic3—81no—
417Dic3—81no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—48no—
424C23—62no—
425C23—84no—
426C23—84no—
427C23—122no—
428C23—122no—
429C23—121no—
430C23—121no—
431C23#93814yes—
432C23C22×#6711yescharacteristic
433Q8—24no—
434Q8—24no—
435D4—24no—
436D4—24no—
437D4—24no—
438D4—24no—
439D4—44no—
440D4—416no—
441D4—44no—
442D4—44no—
443D4—64no—
444D4—82no—
445D4—84no—
446D4—122no—
447D4—248no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—62no—
458C2×C4—82no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—244no—
467C2×C4#95311yescharacteristic
468C2×C4#95312yes—
469C2×C4#95311yescharacteristic
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C2×#23711yescharacteristic
472C2×C4#100412yes—
473C2×C4C22×#6712yes—
474C8—124no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—44no—
478C2×C3—84no—
479C2×C3#240811yescharacteristic
480C2×C3#301911yescharacteristic
481C2×C3#302111yescharacteristic
482D3—82no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—44no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—82no—
493C22—124no—
494C22—122no—
495C22—168no—
496C22—242no—
497C22—241no—
498C22—241no—
499C22C2×#93811yescharacteristic
500C22#580511yescharacteristic
501C22C2×#95311yescharacteristic
502C4—21no—
503C4—21no—
504C4—22no—
505C4—44no—
506C4—42no—
507C4—122no—
508C4—241no—
509C4—241no—
510C4D4×#6712yes—
511C4#580612yes—
512C3#2422611yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—44no—
516C2—84no—
517C2—242no—
518C2#8405311yescharacteristic
519C2#8410411yescharacteristic
520C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3496C2
socleC22×C3C2×#120
hypercenter#3496D3
hypocenterC3#24226
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C3×#3496C2semidirect
2. C2×#5241C2semidirect
3. #83297C2semidirect
4. #82329C2semidirect
5. #72140C2semidirect
6. #71879C2semidirect
7. #71877C2semidirect
8. #71816C2semidirect
9. #71617C2semidirect
20. C2×#949C22semidirect
22. C2×C4×D12C22semidirect
24. C2×#837C22semidirect
25. C2×#837C22semidirect
26. #5232C22semidirect
33. #4988C22semidirect
34. #4977C22semidirect
36. #4932C22semidirect
37. #4924C22semidirect
38. #4530C22semidirect
39. #3496D3semidirect
61. C3×C4⋊Q8D4semidirect
62. C3×#127D4semidirect
63. C3×#124D4semidirect
87. #940D4semidirect
178. C2×D12C2×D4semidirect
200. C4×Dic3C2×D4semidirect
222. C4⋊Q8#67semidirect
225. #127#67semidirect
226. #124#67semidirect
512. C3#24226semidirect

Maximal subgroups

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

subgroupindex
C3×#34962
C2×#52412
#832972
#823292
#721402
#718792
#718772
#718162
#716172
#242263