Finite Group Databasebeta

← #10949318

Subgroups — #10949318

Order 768.

1248 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1248 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86252C211yesmaximal, characteristic, complemented
2C3×#3018C211yesmaximal, characteristic, complemented
3#84126C212yesmaximal, complemented
4#71819C211yesmaximal, characteristic, complemented
5#70991C214yesmaximal, complemented
6#70984C214yesmaximal, complemented
7#70146C212yesmaximal, complemented
8#33559—31nomaximal, Sylow
9C3×#585C2211yescharacteristic, complemented
10C3×#575—21no—
11C3×#532C2211yescharacteristic
12#5830—21no—
13#5830—22no—
14#5830C2212yes—
15#5734—21no—
16#5717C2212yescomplemented
17#5683—21no—
18#5683C2212yescomplemented
19#5607—24no—
20#5563—22no—
21D4×D12—22no—
22#5477—21no—
23C3×#456—21no—
24C3×#455C2212yescomplemented
25C3×#454—21no—
26C3×#453C2212yes—
27C2×C3×#104—21no—
28C3×#413C2211yescharacteristic, complemented
29#5244—24no—
30#5244C2218yescomplemented
31#5240—22no—
32#5240C2212yescomplemented
33#4545C2212yescomplemented
34#4541C2212yescomplemented
35#4475C2218yescomplemented
36#3549—31no—
37#3027—32no—
38#3018D311yescharacteristic, complemented
39#2199—34no—
40#2196—34no—
41#2126—31no—
42#2122—32no—
43C3×2−1+4—21no—
44C3×2−1+4—41no—
45C3×H(5,𝔽2)—22no—
46C3×H(5,𝔽2)—41no—
47C3×H(5,𝔽2)C2312yes—
48C2×C3×#41—22no—
49C2×C3×#41—21no—
50C2×C3×#41—42no—
51C2×C3×#41D411yescharacteristic, complemented
52C2×C3×#41C2311yescharacteristic
53C2×D3×D4—24no—
54C2×#235—22no—
55C22×D12—21no—
56C22×D12D412yescomplemented
57C3×#137—22no—
58C3×#137—41no—
59C3×AGL(1,ℤ/8)—22no—
60C3×AGL(1,ℤ/8)—41no—
61C3×#135—42no—
62C2×C3×M16—21no—
63C3×C4⋊Q8D411yescharacteristic, complemented
64C3×#127C2311yescharacteristic
65C3×#124D411yescharacteristic, complemented
66C3×#124C2311yescharacteristic
67C3×#121—22no—
68C3×C4×D4—22no—
69C2×C3×C42D411yescharacteristic, complemented
70#947—22no—
71#947—22no—
72#947—42no—
73#940—22no—
74#940—22no—
75#940—42no—
76#938—24no—
77#937—22no—
78#937—22no—
79#937—28no—
80#937D414yescomplemented
81#937C2314yes—
82#930—22no—
83#930—22no—
84#930—22no—
85#930—22no—
86#930—44no—
87#926—21no—
88#926—22no—
89#926D412yescomplemented
90#895—24no—
91#892—24no—
92#885—24no—
93#885—24no—
94#882—24no—
95#882—24no—
96#875—22no—
97#874—21no—
98C4×D12—22no—
99C4×D12—22no—
100#869—21no—
101C3×#104—22no—
102C3×#104—22no—
103C3×#101—21no—
104C3×#100C2312yes—
105#834—22no—
106#834—28no—
107#834C2314yes—
108#585C2×D311yescharacteristic, complemented
109#575—21no—
110H(3,𝔽4)—32no—
111H(3,𝔽4)—61no—
112#555—62no—
113#555—61no—
114D42—32no—
115D42—62no—
116#535—32no—
117#535—64no—
118#535—61no—
119#535—62no—
120#532—61no—
121#532C2×D311yescharacteristic
122B(4,𝔽2)—38no—
123B(4,𝔽2)—64no—
124#456—21no—
125#455C2×D312yescomplemented
126#454—21no—
127#453C2×D312yes—
128C2×#104—21no—
129#413C2×D311yescharacteristic, complemented
130#410—32no—
131#410—62no—
132#354—32no—
133#353—32no—
134AΣL(1,𝔽16)—38no—
135C23×D3—22no—
136C23×D3C2×D414yes—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—41no—
140C3×#41—41no—
141C3×#41—42no—
142C3×#41—42no—
143C3×#41—42no—
144C3×#41—41no—
145C3×#41—44no—
146C3×#41—41no—
147C3×#41—41no—
148C3×#41—42no—
149C3×#41—41no—
150C3×#41—42no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—42no—
154C2×C3×Q8C2×D411yescharacteristic
155C2×C3×Q8C2×D411yescharacteristic
156C2×C3×D4—21no—
157C2×C3×D4—22no—
158C2×C3×D4—24no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—42no—
162C2×C3×D4—44no—
163C2×C3×D4—41no—
164C2×C3×D4C2×D412yes—
165C2×C3×D4C2×D411yescharacteristic
166C2×C3×D4C2411yescharacteristic
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—41no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C2×#67—24no—
174C2×#67—24no—
175C2×#67—24no—
176C2×#67—48no—
177D3×D4—88no—
178#235—84no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12—81no—
187C2×D12—81no—
188C2×C4×D3—24no—
189C2×Dic6—22no—
190C3×Q16—42no—
191C3×SD16—42no—
192C3×SD16—42no—
193C3×D8—42no—
194C3×M16—22no—
195C3×M16—41no—
196C2×C3×C8—41no—
197C3×C4⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—21no—
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203#217—22no—
204#217—24no—
205#217C2×D414yes—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—48no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—42no—
214C4×Dic3—22no—
215C4×Dic3—22no—
216C4×Dic3—42no—
2172−1+4—21no—
2182−1+4—41no—
219H(5,𝔽2)—22no—
220H(5,𝔽2)—41no—
221H(5,𝔽2)C22×D312yes—
222C2×#41—22no—
223C2×#41—21no—
224C2×#41—42no—
225C2×#41—62no—
226C2×#41#6711yescharacteristic, complemented
227C2×#41C22×D311yescharacteristic
228C22×D4—32no—
229C22×D4—61no—
230C22×D4—64no—
231#137—22no—
232#137—41no—
233AGL(1,ℤ/8)—22no—
234AGL(1,ℤ/8)—41no—
235#135—42no—
236C2×M16—21no—
237C4⋊Q8—61no—
238C4⋊Q8#6711yescharacteristic, complemented
239#127—61no—
240#127—62no—
241#127—62no—
242#127—122no—
243#127C22×D311yescharacteristic
244#124—64no—
245#124—64no—
246#124—62no—
247#124—62no—
248#124—62no—
249#124—122no—
250#124#6711yescharacteristic, complemented
251#124C22×D311yescharacteristic
252#121—22no—
253#121—64no—
254#121—64no—
255#121—62no—
256#121—62no—
257#121—62no—
258#121—62no—
259#121—124no—
260#120—34no—
261#120—34no—
262#120—64no—
263#120—64no—
264#120—62no—
265#120—62no—
266#120—68no—
267C4×D4—22no—
268C4×D4—62no—
269C4×D4—62no—
270#117—32no—
271#117—64no—
272#117—61no—
273#117—62no—
274C2×C42#6711yescharacteristic, complemented
275#104—22no—
276#104—22no—
277#101—21no—
278#100C22×D312yes—
279#99—34no—
280#99—68no—
281#99—62no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3#12011yescharacteristic
286C23×C3C22×D412yes—
287C22×D3—24no—
288C22×D3—44no—
289C22×D3—84no—
290C22×D3—82no—
291C22×D3—84no—
292C3×Q8—22no—
293C3×Q8—41no—
294C3×Q8—41no—
295C3×Q8—42no—
296C3×Q8—41no—
297C3×D4—22no—
298C3×D4—42no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—44no—
302C3×D4—42no—
303C3×D4—44no—
304C3×D4—42no—
305C3×D4—42no—
306C3×D4—41no—
307C3×D4—81no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—42no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326#67—84no—
327#67—84no—
328#67—84no—
329#67—168no—
330C2×Dic3—24no—
331C2×Dic3—44no—
332D12—44no—
333D12—42no—
334D12—42no—
335D12—82no—
336D12—82no—
337D12—82no—
338C4×D3—48no—
339Dic6—82no—
340C3×C8—42no—
341C24—34no—
342C24—62no—
343#41—22no—
344#41—22no—
345#41—41no—
346#41—41no—
347#41—42no—
348#41—42no—
349#41—42no—
350#41—41no—
351#41—44no—
352#41—41no—
353#41—41no—
354#41—42no—
355#41—42no—
356#41—41no—
357#41—244no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—62no—
362C2×Q8C2×#6711yescharacteristic
363C2×Q8C2×#6711yescharacteristic
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—41no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—64no—
376C2×D4—64no—
377C2×D4—64no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—122no—
381C2×D4—122no—
382C2×D4—128no—
383C2×D4—241no—
384C2×D4—241no—
385C2×D4—248no—
386C2×D4D3×D412yes—
387C2×D4C2×#6711yescharacteristic
388C2×D4C23×D311yescharacteristic
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—64no—
394C22×C4D3×D412yes—
395C22×C4C2×#6711yescharacteristic
396Q16—42no—
397SD16—42no—
398SD16—42no—
399D8—42no—
400M16—22no—
401M16—41no—
402C2×C8—41no—
403C4⋊C4—22no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—122no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—34no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—62no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—128no—
417C42—21no—
418C42—62no—
419C42—62no—
420C42—122no—
421C42C2×#6711yescharacteristic
422C42C2×#6711yescharacteristic
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—41no—
430C22×C3—84no—
431C22×C3—161no—
432C22×C3C2×#12011yescharacteristic
433C2×D3—44no—
434C2×D3—44no—
435C2×D3—44no—
436C2×D3—88no—
437C2×D3—164no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—42no—
444C3×C4B(4,𝔽2)11yescharacteristic
445C3×C4B(4,𝔽2)11yescharacteristic
446Dic3—44no—
447Dic3—84no—
448C23—24no—
449C23—21no—
450C23—41no—
451C23—64no—
452C23—124no—
453C23—244no—
454C23—244no—
455C23—242no—
456C23#93711yescharacteristic
457C23C2×D3×D412yes—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461Q8—42no—
462Q8—41no—
463Q8—242no—
464D4—22no—
465D4—42no—
466D4—41no—
467D4—41no—
468D4—42no—
469D4—44no—
470D4—44no—
471D4—42no—
472D4—42no—
473D4—41no—
474D4—81no—
475D4—124no—
476D4—122no—
477D4—122no—
478D4—242no—
479D4—242no—
480D4—242no—
481D4—244no—
482D4—244no—
483D4—244no—
484D4—488no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—64no—
501C2×C4—128no—
502C2×C4—124no—
503C2×C4#93712yes—
504C2×C4#93711yescharacteristic
505C2×C4C22×#6711yescharacteristic
506C8—42no—
507C2×C3—21no—
508C2×C3—42no—
509C2×C3—41no—
510C2×C3—81no—
511C2×C3—82no—
512C2×C3C2×B(4,𝔽2)11yescharacteristic
513D3—84no—
514C22—22no—
515C22—21no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—41no—
521C22—84no—
522C22—124no—
523C22—124no—
524C22—124no—
525C22—161no—
526C22—248no—
527C22—484no—
528C22C2×#93711yescharacteristic
529C4—22no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—42no—
534C4—41no—
535C4—124no—
536C4—244no—
537C4#503211yescharacteristic
538C4#503211yescharacteristic
539C3#3355911yescharacteristic, complemented, Sylow
540C2—21no—
541C2—42no—
542C2—41no—
543C2—81no—
544C2—82no—
545C2—244no—
546C2C2×#503211yescharacteristic
subgroupG/H
centerC2C2×#5032
commutator, cosocleC2×C3×D4C24
FrattiniC2×D4C23×D3
FittingC3×#3018C2
socleC2×C3C2×B(4,𝔽2)
hypercenter#3018D3
hypocenterC3#33559
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3018

Complements

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

NH ≅ G/Ntype
1. #86252C2semidirect
2. C3×#3018C2semidirect
3. #84126C2semidirect
4. #71819C2semidirect
5. #70991C2semidirect
6. #70984C2semidirect
7. #70146C2semidirect
9. C3×#585C22semidirect
16. #5717C22semidirect
18. #5683C22semidirect
24. C3×#455C22semidirect
28. C3×#413C22semidirect
30. #5244C22semidirect
32. #5240C22semidirect
33. #4545C22semidirect
34. #4541C22semidirect
35. #4475C22semidirect
38. #3018D3semidirect
51. C2×C3×#41D4semidirect
56. C22×D12D4semidirect
63. C3×C4⋊Q8D4semidirect
65. C3×#124D4semidirect
69. C2×C3×C42D4semidirect
80. #937D4semidirect
89. #926D4semidirect
108. #585C2×D3semidirect
125. #455C2×D3semidirect
129. #413C2×D3semidirect
226. C2×#41#67semidirect
238. C4⋊Q8#67semidirect
250. #124#67semidirect
274. C2×C42#67semidirect
539. C3#33559semidirect

Maximal subgroups

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

subgroupindex
#862522
C3×#30182
#841262
#718192
#709912
#709842
#701462
#335593