Finite Group Databasebeta

← #10830418

Subgroups — #10830418

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
1#85786C211yesmaximal, characteristic, complemented
2C3×#2328C211yesmaximal, characteristic, complemented
3C2×#5077C212yesmaximal, complemented
4#82307C212yesmaximal, complemented
5#71617C211yesmaximal, characteristic, complemented
6#69664C212yesmaximal, complemented
7#69662C212yesmaximal, complemented
8#69563C212yesmaximal, complemented
9#69440C212yesmaximal, complemented
10#24226—31nomaximal, Sylow
11C2×#1004C2212yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×C4×D4C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic, complemented
15C2×#932C2214yescomplemented
16C2×#930—22no—
17#5555—22no—
18#5554—24no—
19D4×D12—22no—
20#5510C2212yes—
21#5493—22no—
22#5477C2212yes—
23C3×#390C2211yescharacteristic
24C3×#390C2211yescharacteristic, complemented
25C3×#386—22no—
26C3×#386—22no—
27C3×#382C2211yescharacteristic, complemented
28C3×#376C2211yescharacteristic, complemented
29C2×#833C2212yes—
30#5077—28no—
31#5019—22no—
32#5016—24no—
33#5015—22no—
34#4966C2214yescomplemented
35C2×#805C2214yescomplemented
36C2×#805C2214yescomplemented
37#4924C2212yescomplemented
38#4530C2212yescomplemented
39#3496—31no—
40#3036—32no—
41C2×#453—32no—
42#2328D311yescharacteristic, complemented
43#2040—32no—
44#2006—32no—
45#1963—32no—
46#1800—31no—
47#1792—32no—
48C22×C3×D4—22no—
49C22×C3×D4C2311yescharacteristic
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4D412yes—
53C22×#67—24no—
54#1004—24no—
55#1004—24no—
56#1004—44no—
57C2×#237—24no—
58C2×D3×D4—24no—
59C2×#235C2314yes—
60C22×D12—22no—
61C3×#127—22no—
62C3×#127D414yescomplemented
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4D412yes—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#214—24no—
77#932—28no—
78#932—44no—
79C2×#213—24no—
80#930—24no—
81#930—24no—
82#930—22no—
83#930—22no—
84#930—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×C4⋊Dic3—22no—
87C2×#208C2314yes—
88#897—24no—
89#895—24no—
90#886—28no—
91#882—28no—
92#875D414yescomplemented
93#874D412yescomplemented
94C4×D12—22no—
95C4×D12—22no—
96#869D412yescomplemented
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—22no—
100#833—24no—
101#810—24no—
102#809—24no—
103#805—28no—
104#805—44no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×H(5,𝔽2)—32no—
108C2×AGL(1,ℤ/8)—34no—
109#547—62no—
110#546—64no—
111D42—62no—
112#537—32no—
113#534—62no—
114#532—32no—
115C2×#127C2×D311yescharacteristic
116C2×#121—62no—
117C2×C4×D4C2×D311yescharacteristic
118C2×C4×D4C2×D311yescharacteristic, complemented
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#390C2×D311yescharacteristic
129#390C2×D311yescharacteristic, complemented
130#386—22no—
131#386—22no—
132#382C2×D311yescharacteristic, complemented
133#376C2×D311yescharacteristic, complemented
134#343—32no—
135C24×C3—21no—
136C24×C3C2×D412yes—
137C23×D3—24no—
138C2×C3×D4—28no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—82no—
145C2×C3×D4—82no—
146C2×C3×D4C2×D412yes—
147C2×C3×D4C2×D412yescomplemented
148C2×C3×D4#4112yes—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—42no—
155C22×C3×C4—82no—
156C22×C3×C4—82no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C2×#67—28no—
162C2×#67—24no—
163C2×#67—48no—
164C2×#67—48no—
165C22×Dic3—24no—
166#237—48no—
167#237—48no—
168D3×D4—48no—
169D3×D4—48no—
170#235—24no—
171#235—24no—
172#235—24no—
173#235—24no—
174C2×D12—22no—
175C2×D12—24no—
176C2×D12—22no—
177C2×D12—84no—
178C2×D12C2×D414yes—
179C2×C4×D3—24no—
180C2×Dic6C2×D414yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—82no—
187C3×C22⋊C4—82no—
188C3×C42—21no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192C3×C42C2×D412yescomplemented
193#214—48no—
194#213—48no—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—24no—
198C4⋊Dic3—24no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201#208—24no—
202#208—24no—
203C2×C3⋊C8—24no—
204H(5,𝔽2)—64no—
205H(5,𝔽2)—64no—
206H(5,𝔽2)—124no—
207C2×#41—34no—
208C2×#41—64no—
209C22×D4—22no—
210C22×D4—62no—
211C22×D4—64no—
212C22×D4—64no—
213C22×D4C22×D311yescharacteristic
214C22×D4C22×D311yescharacteristic
215C23×C4—21no—
216C23×C4#6712yes—
217AGL(1,ℤ/8)—68no—
218AGL(1,ℤ/8)—124no—
219C2×SD16—64no—
220C2×D8—64no—
221C2×M16—34no—
222C4⋊Q8—32no—
223#127—22no—
224#127—32no—
225#127D1214yescomplemented
226#124—34no—
227#123—68no—
228C22⋊Q8—64no—
229#121—64no—
230#121—64no—
231#121—64no—
232#121—62no—
233#121—64no—
234#121—62no—
235#120—68no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—62no—
241C4×D4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4C22×D311yescharacteristic
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4#6712yes—
250C2×C42C22×D311yescharacteristic
251C2×C42C22×D311yescharacteristic
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—21no—
261#95—21no—
262#95—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3#12114yes—
273C23×C3C22×D411yescharacteristic
274C22×D3—24no—
275C22×D3—48no—
276C22×D3—84no—
277C22×D3—84no—
278C3×D4—24no—
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—42no—
282C3×D4—88no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—44no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—82no—
297C2×C3×C4—84no—
298C2×C3×C4#12011yescharacteristic
299C2×C3×C4#12012yes—
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D412yes—
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4H(5,𝔽2)12yes—
305#67—416no—
306#67—44no—
307#67—44no—
308C2×Dic3—24no—
309C2×Dic3—48no—
310D12—24no—
311D12—24no—
312D12—44no—
313D12—82no—
314D12—84no—
315C4×D3—44no—
316C4×D3—44no—
317Dic6—24no—
318Dic6—24no—
319C3⋊C8—44no—
320C24—21no—
321C24—64no—
322C24C2×#6712yes—
323#41—64no—
324#41—64no—
325#41—64no—
326#41—64no—
327#41—128no—
328#41—128no—
329C2×Q8—34no—
330C2×D4—28no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—34no—
337C2×D4—64no—
338C2×D4—68no—
339C2×D4—64no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—128no—
345C2×D4—128no—
346C2×D4—128no—
347C2×D4—128no—
348C2×D4—244no—
349C2×D4C2×D1212yes—
350C2×D4C2×D1212yescomplemented
351C2×D4#23512yes—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—42no—
358C22×C4—64no—
359C22×C4—64no—
360C22×C4—82no—
361C22×C4—82no—
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—22no—
372C4⋊C4—22no—
373C4⋊C4—64no—
374C4⋊C4—62no—
375C4⋊C4—62no—
376C22⋊C4—24no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—68no—
380C22⋊C4—64no—
381C22⋊C4—64no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C42—21no—
385C42—22no—
386C42C2×D1212yescomplemented
387C42D3×D412yes—
388C42D3×D412yes—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—24no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3C2×#12011yescharacteristic
402C22×C3C2×#12111yescharacteristic
403C22×C3#53511yescharacteristic
404C2×D3—44no—
405C2×D3—44no—
406C2×D3—44no—
407C2×D3—168no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—81no—
414C3×C4—81no—
415C3×C4D4212yes—
416C3×C4#54612yes—
417Dic3—44no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—64no—
428C23—128no—
429C23—244no—
430C23—244no—
431C23#93014yes—
432C23C22×#6711yescharacteristic
433Q8—64no—
434Q8—64no—
435D4—24no—
436D4—24no—
437D4—24no—
438D4—42no—
439D4—64no—
440D4—64no—
441D4—88no—
442D4—124no—
443D4—1216no—
444D4—124no—
445D4—124no—
446D4—242no—
447D4—244no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—64no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—128no—
465C2×C4—124no—
466C2×C4—124no—
467C2×C4#93712yes—
468C2×C4#93711yescharacteristic
469C2×C4#93711yescharacteristic
470C2×C4C22×D1211yescharacteristic
471C2×C4C2×#23511yescharacteristic
472C2×C4C2×D3×D412yes—
473C2×C4#100912yes—
474C8—124no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—44no—
478C2×C3—82no—
479C2×C3#240811yescharacteristic
480C2×C3#301911yescharacteristic
481C2×C3#302111yescharacteristic
482D3—84no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—82no—
493C22—81no—
494C22—81no—
495C22—124no—
496C22—124no—
497C22—124no—
498C22—488no—
499C22C2×#93011yescharacteristic
500C22C2×#93711yescharacteristic
501C22#583011yescharacteristic
502C4—21no—
503C4—21no—
504C4—22no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—124no—
510C4D4×D1212yes—
511C4#555012yes—
512C3#2422611yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—44no—
516C2—82no—
517C2—244no—
518C2#8240311yescharacteristic
519C2#8339411yescharacteristic
520C2#8339511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2328C2
socleC22×C3C2×#120
hypercenter#2328D3
hypocenterC3#24226
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. #85786C2semidirect
2. C3×#2328C2semidirect
3. C2×#5077C2semidirect
4. #82307C2semidirect
5. #71617C2semidirect
6. #69664C2semidirect
7. #69662C2semidirect
8. #69563C2semidirect
9. #69440C2semidirect
14. C2×C3×C4×D4C22semidirect
15. C2×#932C22semidirect
24. C3×#390C22semidirect
27. C3×#382C22semidirect
28. C3×#376C22semidirect
34. #4966C22semidirect
35. C2×#805C22semidirect
36. C2×#805C22semidirect
37. #4924C22semidirect
38. #4530C22semidirect
42. #2328D3semidirect
62. C3×#127D4semidirect
92. #875D4semidirect
93. #874D4semidirect
96. #869D4semidirect
118. C2×C4×D4C2×D3semidirect
129. #390C2×D3semidirect
132. #382C2×D3semidirect
133. #376C2×D3semidirect
147. C2×C3×D4C2×D4semidirect
192. C3×C42C2×D4semidirect
225. #127D12semidirect
350. C2×D4C2×D12semidirect
386. C42C2×D12semidirect
512. C3#24226semidirect

Maximal subgroups

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

subgroupindex
#857862
C3×#23282
C2×#50772
#823072
#716172
#696642
#696622
#695632
#694402
#242263