Finite Group Databasebeta

← #10848686

Subgroups — #10848686

Order 768.

1438 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic, complemented
3C2×#5085C211yesmaximal, characteristic, complemented
4C2×#5079C211yesmaximal, characteristic, complemented
5#71699C214yesmaximal, complemented
6C4×#830C211yesmaximal, characteristic, complemented
7#69687C214yesmaximal, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69580C211yesmaximal, characteristic, complemented
10#25597—31nomaximal, Sylow
11C22×#235—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C22⋊Q8C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×#215C2214yes—
16C2×#930—21no—
17C2×#930C2212yes—
18#5477—24no—
19#5477—24no—
20C2×#875—21no—
21C2×#874C2211yescharacteristic
22C2×C4×D12—21no—
23C2×C4×D12C2212yes—
24C2×#869C2211yescharacteristic
25C3×#391C2212yes—
26C3×#389—21no—
27C3×#389—21no—
28C3×#384C2211yescharacteristic
29C3×#380—21no—
30C3×#380—21no—
31C2×#835C2214yes—
32C2×#830C2212yes—
33#5085—24no—
34#5079—24no—
35C2×#809C2214yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4529C2214yes—
39#4524C2212yes—
40C2×#532—31no—
41C2×#492—31no—
42C2×#487—31no—
43#2857D311yescharacteristic, complemented
44#2000—34no—
45#1990—31no—
46#1957—31no—
47#1877—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8C2312yes—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×#67—22no—
54C2×#235—28no—
55C2×#235—44no—
56C2×#235—44no—
57C22×D12—21no—
58C22×D12—21no—
59C22×D12C2312yes—
60C22×C4×D3—22no—
61C22×Dic6—21no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C22⋊Q8—44no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—22no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42D411yescharacteristic
75C2×C3×C42D411yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#215—28no—
79C2×#215—28no—
80C2×#215—44no—
81#930—28no—
82#930—24no—
83#930—24no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88C22×C3⋊C8C2312yes—
89C22×C3⋊C8C2312yes—
90#875—28no—
91#874—22no—
92#874D414yes—
93C4×D12—28no—
94C4×D12—24no—
95C4×D12—24no—
96#869—22no—
97#869D414yes—
98C3×#95—22no—
99C3×#95—22no—
100#835—28no—
101#830—24no—
102#809—44no—
103C4×C3⋊C8—22no—
104C4×C3⋊C8—22no—
105C22×#41—61no—
106C22×SD16—34no—
107#532—64no—
108#532—64no—
109C2×C4⋊Q8—31no—
110C2×C4⋊Q8C2×D311yescharacteristic
111C2×#127—31no—
112C2×#124—61no—
113C2×C22⋊Q8C2×D312yes—
114C2×#121—32no—
115C2×#121—61no—
116C2×C4×D4—32no—
117C2×C4×D4—61no—
118C22×C42C2×D311yescharacteristic
119#492—64no—
120#487—64no—
121C2×#103—34no—
122C2×#102—34no—
123C2×C22⋊C8—32no—
124C2×C4×C8—31no—
125C2×C4×C8—31no—
126#391C2×D312yes—
127#389—21no—
128#389—21no—
129#384C2×D311yescharacteristic
130#380—21no—
131#380—21no—
132#340—34no—
133#336—32no—
134C24×C3C2×D411yescharacteristic
135C23×D3—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—24no—
139C2×C3×Q8—24no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4SD1614yescomplemented
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C2×#67—28no—
159C2×#67—44no—
160C2×#67—44no—
161C22×Dic3—22no—
162#235—432no—
163C2×D12—24no—
164C2×D12—24no—
165C2×D12—24no—
166C2×D12—24no—
167C2×D12—22no—
168C2×D12—42no—
169C2×D12—42no—
170C2×D12—42no—
171C2×D12—42no—
172C2×D12C2×D414yes—
173C2×C4×D3—28no—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C2×Dic6—24no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—44no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C22⋊C4—44no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—21no—
189C3×C42—21no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192#215—416no—
193(C2×D3)⋊C4—28no—
194(C2×D3)⋊C4—28no—
195C4⋊Dic3—24no—
196C4⋊Dic3—22no—
197C4⋊Dic3C2×D414yes—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8#4114yescomplemented
204C2×#41—68no—
205C2×#41—124no—
206C2×#41—124no—
207C22×Q8—61no—
208C22×Q8C22×D312yes—
209C22×D4—32no—
210C22×D4—62no—
211C22×D4—61no—
212C22×D4—61no—
213C23×C4—62no—
214C23×C4#6711yescharacteristic
215C23×C4#6711yescharacteristic
216C23×C4C22×D311yescharacteristic
217C2×SD16—68no—
218C2×SD16—68no—
219C2×SD16—124no—
220C22×C8—32no—
221C22×C8—32no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—34no—
225C4⋊Q8—62no—
226#127—34no—
227#127—62no—
228#124—68no—
229C22⋊Q8—44no—
230#121—64no—
231#121—64no—
232#121—68no—
233C4×D4—64no—
234C4×D4—64no—
235C4×D4—68no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4C22×D312yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42#6711yescharacteristic
250C2×C42#6711yescharacteristic
251C2×C42C22×D311yescharacteristic
252C2×C42C22×D311yescharacteristic
253#103—68no—
254#102—124no—
255C22⋊C8—64no—
256C4×C8—62no—
257C4×C8—62no—
258#95—22no—
259#95—22no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3C2×SD1612yes—
267C23×C3C22×D411yescharacteristic
268C22×D3—24no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C3×Q8—44no—
276C3×Q8—44no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—28no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—44no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4#12012yes—
296C2×C3×C4#12012yes—
297C2×C3×C4C2×SD1612yes—
298C2×C3×C4C2×SD1612yes—
299C2×C3×C4#13512yes—
300C2×C3×C4AGL(1,ℤ/8)12yes—
301C2×C3×C4#13712yes—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C2×#4112yes—
305C2×C3×C4H(5,𝔽2)12yes—
306#67—416no—
307C2×Dic3—24no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310D12—28no—
311D12—44no—
312D12—48no—
313D12—48no—
314C4×D3—416no—
315Dic6—48no—
316C3⋊C8—24no—
317C3⋊C8—42no—
318C24—62no—
319C24C2×#6711yescharacteristic
320#41—1232no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—64no—
326C2×Q8—122no—
327C2×Q8—122no—
328C2×D4—34no—
329C2×D4—68no—
330C2×D4—64no—
331C2×D4—64no—
332C2×D4—62no—
333C2×D4—64no—
334C2×D4—64no—
335C2×D4—124no—
336C2×D4—124no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—122no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—68no—
353C22×C4—62no—
354C22×C4—124no—
355C22×C4—124no—
356C22×C4#21514yescomplemented
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C23×D311yescharacteristic
363SD16—1216no—
364C2×C8—34no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—34no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—64no—
378C4⋊C4—62no—
379C22⋊C4—44no—
380C22⋊C4—68no—
381C22⋊C4—68no—
382C42—22no—
383C42—22no—
384C42—21no—
385C42—21no—
386C42D3×D412yes—
387C42D3×D412yes—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3C2×#12011yescharacteristic
396C22×C3#53211yescharacteristic
397C22×C3#53511yescharacteristic
398C22×C3C22×SD1611yescharacteristic
399C22×C3C2×#13511yescharacteristic
400C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
401C22×C3C2×#13711yescharacteristic
402C2×D3—44no—
403C2×D3—44no—
404C2×D3—44no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—82no—
408C2×D3—82no—
409C3×C4—22no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—82no—
413C3×C4—82no—
414C3×C4#45014yes—
415C3×C4#45214yes—
416C3×C4#54612yes—
417C3×C4#55012yes—
418Dic3—44no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—64no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23C2×#21512yes—
433C23C22×#6711yescharacteristic
434Q8—44no—
435Q8—44no—
436Q8—128no—
437D4—68no—
438D4—1216no—
439D4—128no—
440D4—128no—
441D4—124no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—64no—
461C2×C4—1216no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4#93712yes—
465C2×C4#93712yes—
466C2×C4C2×#21512yes—
467C2×C4C2×#21512yes—
468C2×C4#94212yes—
469C2×C4#94912yes—
470C2×C4#95012yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4#100412yes—
474C2×C4D3×#4112yes—
475C8—64no—
476C8—122no—
477C2×C3—42no—
478C2×C3#243111yescharacteristic
479C2×C3C2×#45011yescharacteristic
480C2×C3C2×#45211yescharacteristic
481C2×C3#316711yescharacteristic
482C2×C3#318211yescharacteristic
483C2×C3#320611yescharacteristic
484C2×C3#320711yescharacteristic
485D3—84no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—124no—
494C22—124no—
495C22—124no—
496C22—242no—
497C22—242no—
498C22—242no—
499C22—242no—
500C22C2×#93711yescharacteristic
501C22#580511yescharacteristic
502C22C22×#21511yescharacteristic
503C22C2×#94211yescharacteristic
504C22C2×#94911yescharacteristic
505C22C2×#95011yescharacteristic
506C22#583211yescharacteristic
507C4—22no—
508C4—41no—
509C4—41no—
510C4—82no—
511C4—82no—
512C4—124no—
513C4#504614yes—
514C4#504714yes—
515C4#559912yes—
516C4#561212yes—
517C3#2559711yescharacteristic, complemented, Sylow
518C2—42no—
519C2—244no—
520C2#8265211yescharacteristic
521C2C2×#504611yescharacteristic
522C2C2×#504711yescharacteristic
523C2#8321311yescharacteristic
524C2#8330311yescharacteristic
525C2#8334311yescharacteristic
526C2#8334711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenter#2857D3
hypocenterC3#25597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C3×#2857C2semidirect
3. C2×#5085C2semidirect
4. C2×#5079C2semidirect
5. #71699C2semidirect
6. C4×#830C2semidirect
7. #69687C2semidirect
8. #69582C2semidirect
9. #69580C2semidirect
43. #2857D3semidirect
151. C22×C3×C4SD16semidirect
203. C2×C3⋊C8#41semidirect
356. C22×C4#215semidirect
517. C3#25597semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C3×#28572
C2×#50852
C2×#50792
#716992
C4×#8302
#696872
#695822
#695802
#255973