Finite Group Databasebeta

← #10849281

Subgroups — #10849281

Order 768.

1324 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
1C2×#5577C211yesmaximal, characteristic
2C3×#2584C211yesmaximal, characteristic, complemented
3C2×#5066C211yesmaximal, characteristic, complemented
4C2×#5067C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69627C212yesmaximal, complemented
7#69627C212yesmaximal, complemented
8#69626C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69487C211yesmaximal, characteristic, complemented
11#69474C211yesmaximal, characteristic, complemented
12#25655—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×#125C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16#5577—24no—
17#5577—24no—
18C2×#895—22no—
19C2×#895—21no—
20C2×#874C2212yes—
21C2×#874C2211yescharacteristic
22C2×C4×D12C2212yes—
23C2×C4×D12C2211yescharacteristic
24C3×#398—21no—
25C3×#395—21no—
26C3×#389—22no—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30#5067—24no—
31#5066—24no—
32C2×#809C2212yes—
33C2×#809C2212yes—
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550—31no—
47C2×#490—31no—
48C2×#478—31no—
49#2584D311yescharacteristic, complemented
50#2041—32no—
51#2041—32no—
52#1985—31no—
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1762—31no—
57C22×C3×Q8D411yescharacteristic
58C2×#239—28no—
59C2×#239—48no—
60C22×D12—22no—
61C22×D12—21no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—22no—
67C22×C4×D3—21no—
68C3×#125—22no—
69C3×#125—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C42—21no—
82C2×C3×C42C2312yes—
83C2×C3×C42C2311yescharacteristic
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90#895—28no—
91#895—216no—
92#874—24no—
93#874—24no—
94#874—22no—
95#874D414yes—
96C4×D12—24no—
97C4×D12—22no—
98C4×D12—24no—
99C4×D12D414yes—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95—22no—
103#809—22no—
104#809—22no—
105#809—24no—
106#809—24no—
107#809—24no—
108#809—24no—
109#809—24no—
110#808—22no—
111#807—22no—
112C4⋊(C3⋊C8)—22no—
113#803—22no—
114C22×#41—61no—
115#550—64no—
116#550—64no—
117C2×#127—31no—
118C2×#127—32no—
119C2×#125C2×D311yescharacteristic
120C2×#121—62no—
121C2×#121—61no—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×D4—31no—
124C2×C4×D4—32no—
125#490—64no—
126#478—64no—
127C2×#107—31no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#102—31no—
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#97—31no—
138#398—21no—
139#395—21no—
140#389—22no—
141#389C2×D312yes—
142#389C2×D312yes—
143C4×C4⋊C4C2×D311yescharacteristic
144#340—32no—
145#336—32no—
146#336—32no—
147C23×D3—22no—
148C23×D3—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160#239—432no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—28no—
168C2×D12—24no—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12—28no—
172C2×D12—24no—
173C2×D12—44no—
174C2×D12—44no—
175C2×D12—44no—
176C2×D12—81no—
177C2×D12—81no—
178C2×D12—82no—
179C2×D12—82no—
180C2×D12C2×D414yes—
181C2×D12C2×D412yes—
182C2×C4×D3—28no—
183C2×C4×D3—24no—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3—44no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—44no—
197C3×C4⋊C4—82no—
198C3×C4⋊C4#4114yescomplemented
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—81no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—24no—
206C4⋊Dic3—21no—
207C4⋊Dic3C2×D412yes—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×#41—68no—
216C2×#41—128no—
217C22×Q8#6711yescharacteristic
218C22×D4—31no—
219C22×D4—32no—
220C22×D4—61no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4—61no—
224C23×C4—62no—
225C23×C4—61no—
226C22×C8—32no—
227C22×C8—31no—
228C22×C8—31no—
229#127—34no—
230#127—64no—
231#127—64no—
232#127—62no—
233#125—22no—
234#125—22no—
235#121—68no—
236#121—616no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—34no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—62no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4#6712yes—
249C2×C4⋊C4#6711yescharacteristic
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C4⋊C4C22×D312yes—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C42—21no—
256C2×C42C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#107—62no—
259#106—62no—
260C4⋊C8—62no—
261#102—62no—
262#102—62no—
263#102—64no—
264#102—64no—
265#102—64no—
266#102—64no—
267#102—64no—
268#97—62no—
269#95—21no—
270#95—22no—
271#95—22no—
272C23×C3C22×D411yescharacteristic
273C22×D3—24no—
274C22×D3—22no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—41no—
283C22×D3—42no—
284C22×D3—41no—
285C3×Q8—28no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12012yes—
304C2×C3×C4AGL(1,ℤ/8)14yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C42−1+412yes—
309C2×Dic3—22no—
310C2×Dic3—42no—
311D12—22no—
312D12—22no—
313D12—24no—
314D12—24no—
315D12—48no—
316D12—416no—
317D12—42no—
318D12—42no—
319D12—42no—
320D12—84no—
321D12—88no—
322C4×D3—48no—
323C4×D3—416no—
324C3⋊C8—41no—
325C3⋊C8—41no—
326C3⋊C8—42no—
327C24—62no—
328C24—61no—
329#41—1232no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×D4—32no—
333C2×D4—34no—
334C2×D4—68no—
335C2×D4—68no—
336C2×D4—64no—
337C2×D4—64no—
338C2×D4—62no—
339C2×D4—61no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4—124no—
349C2×D4—241no—
350C2×D4—241no—
351C2×D4—242no—
352C2×D4—242no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—68no—
357C22×C4—61no—
358C22×C4—64no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4—124no—
362C22×C4C2×#6712yes—
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6712yes—
366C22×C4C23×D311yescharacteristic
367C2×C8—62no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8—62no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—32no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—61no—
386C4⋊C4—82no—
387C4⋊C4#23914yescomplemented
388C22⋊C4—68no—
389C22⋊C4—64no—
390C42—22no—
391C42—22no—
392C42—81no—
393C42D3×D412yes—
394C42D3×D412yes—
395C22×C3C2×#12011yescharacteristic
396C22×C3#54812yes—
397C22×C3C2×AGL(1,ℤ/8)12yes—
398C22×C3#57611yescharacteristic
399C22×C3#57711yescharacteristic
400C2×D3—42no—
401C2×D3—44no—
402C2×D3—42no—
403C2×D3—44no—
404C2×D3—44no—
405C2×D3—42no—
406C2×D3—82no—
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—82no—
412C3×C4—22no—
413C3×C4—24no—
414C3×C4—22no—
415C3×C4—44no—
416C3×C4—42no—
417C3×C4—81no—
418C3×C4—81no—
419C3×C4#53211yescharacteristic
420C3×C4#53611yescharacteristic
421C3×C4#54712yes—
422Dic3—42no—
423C23—64no—
424C23—62no—
425C23—122no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23—121no—
433C23—122no—
434C23—121no—
435C23C22×#6711yescharacteristic
436Q8—28no—
437D4—62no—
438D4—62no—
439D4—64no—
440D4—64no—
441D4—128no—
442D4—1216no—
443D4—122no—
444D4—122no—
445D4—122no—
446D4—244no—
447D4—248no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—62no—
465C2×C4—128no—
466C2×C4—1216no—
467C2×C4—122no—
468C2×C4#93712yes—
469C2×C4#93712yes—
470C2×C4#94914yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×#23912yes—
474C2×C4#100712yes—
475C8—121no—
476C8—121no—
477C8—122no—
478C2×C3#243211yescharacteristic
479C2×C3#300311yescharacteristic
480C2×C3#301211yescharacteristic
481C2×C3#336612yes—
482C2×C3#336812yes—
483D3—82no—
484D3—84no—
485C22—122no—
486C22—124no—
487C22—124no—
488C22—122no—
489C22—122no—
490C22—124no—
491C22—242no—
492C22—242no—
493C22—242no—
494C22—242no—
495C22—242no—
496C22—242no—
497C22C2×#93711yescharacteristic
498C22#581712yes—
499C22C2×#94912yes—
500C22#583511yescharacteristic
501C22#583711yescharacteristic
502C4—22no—
503C4—22no—
504C4—24no—
505C4—44no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—122no—
510C4#572411yescharacteristic
511C4#572511yescharacteristic
512C4#572612yes—
513C3#2565511yescharacteristic, complemented, Sylow
514C2—242no—
515C2—244no—
516C2#8292711yescharacteristic
517C2#8325511yescharacteristic
518C2#8342411yescharacteristic
519C2#8363812yes—
520C2#8369012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2584C2
socleC23×C3C22×D4
prime radicalC2×#5577C2
hypercenter#2584D3
hypocenterC3#25655
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C3×#2584C2semidirect
3. C2×#5066C2semidirect
4. C2×#5067C2semidirect
5. #71635C2semidirect
6. #69627C2semidirect
7. #69627C2semidirect
8. #69626C2semidirect
9. #69625C2semidirect
10. #69487C2semidirect
11. #69474C2semidirect
49. #2584D3semidirect
198. C3×C4⋊C4#41semidirect
387. C4⋊C4#239semidirect
513. C3#25655semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
C3×#25842
C2×#50662
C2×#50672
#716352
#696272
#696262
#696252
#694872
#694742
#256553