Finite Group Databasebeta

← #10849288

Subgroups — #10849288

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×#5718C211yesmaximal, characteristic
2C2×C3×#490C211yesmaximal, characteristic, complemented
3#83041C211yesmaximal, characteristic, complemented
4C2×#5066C211yesmaximal, characteristic, complemented
5#71761C211yesmaximal, characteristic, complemented
6#70211C212yesmaximal, complemented
7#70211C212yesmaximal, complemented
8#70169C212yesmaximal, complemented
9#70167C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69812C211yesmaximal, characteristic, complemented
12#25655—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×#940C2212yes—
17C2×#939—22no—
18C2×#938—21no—
19C2×D4×Dic3C2212yes—
20#5718—24no—
21#5718—24no—
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#490—24no—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#97C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#821C2212yes—
31#5066—24no—
32#4981—21no—
33#4979—21no—
34#4976—22no—
35#4972C2212yes—
36#4972C2212yes—
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#550—31no—
47C2×#490D311yescharacteristic, complemented
48C2×#478—31no—
49#2584—31no—
50#2041—32no—
51#2041—32no—
52#1985—31no—
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1762—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×#67—22no—
61C22×#67—22no—
62C23×Dic3—22no—
63C2×#237—28no—
64C2×#237—48no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6D411yescharacteristic
68C22×C3×C8C2312yes—
69C3×#127—22no—
70C3×#127D414yes—
71C3×#125—22no—
72C3×#125—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79#940—24no—
80#940—24no—
81#939—216no—
82#938—28no—
83D4×Dic3—24no—
84D4×Dic3—24no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4D412yes—
92C2×C4×Dic3—21no—
93C2×C4×Dic3C2312yes—
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96C4×D12—22no—
97C4×D12D414yes—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C3×#102—24no—
101C3×#102—24no—
102C3×#97—22no—
103#832—24no—
104#832—24no—
105C3⋊#95—21no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108#821—24no—
109#809—22no—
110#809—22no—
111#808—22no—
112#807—22no—
113C4⋊(C3⋊C8)—22no—
114C22×#41—61no—
115#550—64no—
116#550—64no—
117C2×#127—32no—
118C2×#127C2×D311yescharacteristic
119C2×#125C2×D311yescharacteristic
120C2×#121—61no—
121C2×#121—62no—
122C2×C4×Q8—31no—
123C2×C4×D4—32no—
124C2×C4×D4—31no—
125#490—24no—
126#478—64no—
127C2×#107—31no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#102—31no—
135C2×#102C2×D312yes—
136C2×#102C2×D312yes—
137C2×#97C2×D311yescharacteristic
138#398—61no—
139#395—61no—
140#389—32no—
141#389—32no—
142#389—62no—
143C4×C4⋊C4—31no—
144#340—32no—
145#336—32no—
146#336—32no—
147C24×C3—22no—
148C23×D3—21no—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—44no—
157C2×C3×D4—81no—
158C2×C3×D4—81no—
159C2×C3×D4C2×D414yes—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—28no—
166C2×#67—28no—
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—82no—
170C2×#67—82no—
171C22×Dic3—28no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—44no—
175C22×Dic3—44no—
176C22×Dic3C2×D412yes—
177C22×Dic3C2×D412yes—
178#237—432no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12C2×D412yes—
183C2×C4×D3—24no—
184C2×C4×D3—44no—
185C2×Dic6—22no—
186C2×Dic6—24no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—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×C42C2×D412yes—
197C3×C42C2×D412yes—
198#217—28no—
199(C2×D3)⋊C4—24no—
200C4⋊Dic3—22no—
201C4⋊Dic3—21no—
202C4⋊Dic3—22no—
203C4⋊Dic3C2×D412yes—
204C4⋊Dic3#4114yescomplemented
205Dic3⋊C4—24no—
206Dic3⋊C4—44no—
207Dic3⋊C4—82no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C4×Dic3—81no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×#41—68no—
216C2×#41—128no—
217C22×Q8—31no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—31no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4C22×D312yes—
224C23×C4—62no—
225C23×C4—61no—
226C22×C8—31no—
227C22×C8—31no—
228C22×C8C22×D312yes—
229#127—22no—
230#127—64no—
231#127—64no—
232#127#6714yes—
233#125—22no—
234#125—22no—
235#121—68no—
236#121—616no—
237C4×Q8—62no—
238C4×Q8—62no—
239C4×D4—34no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—62no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C42—32no—
256C2×C42—61no—
257C2×C42C22×D311yescharacteristic
258#107—62no—
259#106—62no—
260C4⋊C8—62no—
261#102—24no—
262#102—24no—
263#102—62no—
264#102—62no—
265#102—64no—
266#102—64no—
267#102—64no—
268#97—22no—
269#95—62no—
270#95—62no—
271#95—61no—
272C23×C3—24no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—41no—
284C22×D3—41no—
285C3×D4—24no—
286C3×D4—24no—
287C3×D4—48no—
288C3×D4—42no—
289C3×D4—42no—
290C3×D4—84no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4#12012yes—
300C2×C3×C4#12012yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4112yes—
304C2×C3×C42−1+412yes—
305#67—416no—
306#67—88no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—24no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—416no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3AGL(1,ℤ/8)14yes—
319D12—22no—
320D12—22no—
321D12—42no—
322C4×D3—48no—
323Dic6—28no—
324C3×C8—42no—
325C3⋊C8—41no—
326C3⋊C8—41no—
327C24—22no—
328C24—61no—
329#41—1232no—
330C2×Q8—64no—
331C2×Q8—62no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—32no—
340C2×D4—44no—
341C2×D4—68no—
342C2×D4—68no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—61no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4—124no—
349C2×D4—124no—
350C2×D4—242no—
351C2×D4—242no—
352C2×D4C2×#6714yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—68no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4—64no—
361C22×C4—124no—
362C22×C4—124no—
363C22×C4—124no—
364C22×C4D3×D411yescharacteristic
365C22×C4D3×D411yescharacteristic
366C22×C4C23×D311yescharacteristic
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—61no—
371C2×C8—61no—
372C2×C8—62no—
373C2×C8—62no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—32no—
379C4⋊C4—34no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—64no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—61no—
386C4⋊C4—124no—
387C4⋊C4—242no—
388C22⋊C4—68no—
389C22⋊C4—64no—
390C42—62no—
391C42—62no—
392C42—241no—
393C42D3×D412yes—
394C42C2×#6712yes—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3C2×#12011yescharacteristic
403C22×C3#54812yes—
404C22×C3C2×AGL(1,ℤ/8)12yes—
405C22×C3#57611yescharacteristic
406C22×C3#57711yescharacteristic
407C2×D3—42no—
408C2×D3—42no—
409C2×D3—42no—
410C2×D3—82no—
411C2×D3—82no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4#53211yescharacteristic
417C3×C4#53611yescharacteristic
418C3×C4#54712yes—
419Dic3—24no—
420Dic3—44no—
421Dic3—42no—
422Dic3—42no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—62no—
431C23—122no—
432C23—122no—
433C23—121no—
434C23—121no—
435C23C2×D3×D411yescharacteristic
436Q8—68no—
437D4—24no—
438D4—24no—
439D4—48no—
440D4—42no—
441D4—42no—
442D4—62no—
443D4—62no—
444D4—84no—
445D4—1216no—
446D4—122no—
447D4—248no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—34no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—128no—
465C2×C4—1216no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4#93712yes—
470C2×C4#93712yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4D3×#4112yes—
473C2×C4#101012yes—
474C2×C4C22×#6711yescharacteristic
475C8—42no—
476C8—121no—
477C8—121no—
478C2×C3—84no—
479C2×C3#243211yescharacteristic
480C2×C3#300311yescharacteristic
481C2×C3#301211yescharacteristic
482C2×C3#336612yes—
483C2×C3#336812yes—
484D3—82no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—242no—
496C22—242no—
497C22#567712yes—
498C22D3×AGL(1,ℤ/8)12yes—
499C22#577411yescharacteristic
500C22#577511yescharacteristic
501C22C2×#93711yescharacteristic
502C4—22no—
503C4—22no—
504C4—64no—
505C4—81no—
506C4—81no—
507C4—124no—
508C4—122no—
509C4—122no—
510C4#566811yescharacteristic
511C4#567511yescharacteristic
512C4#582412yes—
513C3#2565511yescharacteristic, complemented, Sylow
514C2—84no—
515C2—242no—
516C2#8170812yes—
517C2#8170912yes—
518C2#8295911yescharacteristic
519C2#8394411yescharacteristic
520C2#8394911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#490C2
socleC23×C3C22×D4
prime radicalC2×#5718C2
hypercenterC2×#490D3
hypocenterC3#25655
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
2. C2×C3×#490C2semidirect
3. #83041C2semidirect
4. C2×#5066C2semidirect
5. #71761C2semidirect
6. #70211C2semidirect
7. #70211C2semidirect
8. #70169C2semidirect
9. #70167C2semidirect
10. #69838C2semidirect
11. #69812C2semidirect
47. C2×#490D3semidirect
204. C4⋊Dic3#41semidirect
513. C3#25655semidirect

Maximal subgroups

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

subgroupindex
C2×#57182
C2×C3×#4902
#830412
C2×#50662
#717612
#702112
#701692
#701672
#698382
#698122
#256553