Finite Group Databasebeta

← #10848637

Subgroups — #10848637

Order 768.

1234 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3508C211yesmaximal, characteristic, complemented
2#83338C211yesmaximal, characteristic, complemented
3#83181C211yesmaximal, characteristic, complemented
4#82527C211yesmaximal, characteristic, complemented
5#71857C212yesmaximal, complemented
6#71856C212yesmaximal, complemented
7#71840C211yesmaximal, characteristic, complemented
8#71766C211yesmaximal, characteristic, complemented
9#71650C212yesmaximal, complemented
10#71649C212yesmaximal, complemented
11#71643C211yesmaximal, characteristic, complemented
12#25593—31nomaximal, Sylow
13C3×#559—22no—
14C3×#550—22no—
15C3×#546—24no—
16C3×#540—22no—
17C3×#535C2212yes—
18C3×#533C2211yescharacteristic
19C3×#532—22no—
20C2×C3×#127C2211yescharacteristic
21C3×#518C2212yes—
22C3×C4×#41C2211yescharacteristic
23C2×#951C2211yescharacteristic
24C2×#949C2211yescharacteristic
25C2×#940C2212yes—
26C2×#936C2212yes—
27C2×#875C2211yescharacteristic
28C2×#837C2212yes—
29#5223—24no—
30C2×#833C2212yes—
31C2×#833C2212yes—
32#5089—22no—
33#5068—22no—
34#5028—22no—
35#5016—22no—
36#5004C2211yescharacteristic
37#5004C2212yes—
38#5001C2211yescharacteristic
39#5001C2212yes—
40#4932C2211yescharacteristic
41#4924C2211yescharacteristic
42C4×#208C2211yescharacteristic
43#4532C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#3508D311yescharacteristic, complemented
47#3140—31no—
48#3040—31no—
49#2657—31no—
50#2004—32no—
51#2002—32no—
52#1995—31no—
53#1965—31no—
54#1887—32no—
55#1885—32no—
56#1759—31no—
57C2×C3×#41—24no—
58C2×C3×#41C2311yescharacteristic
59C22×C3×D4D414yes—
60C22×C3×D4C2312yes—
61C22×#67—24no—
62C22×D12C2311yescharacteristic
63C22×Dic6C2311yescharacteristic
64C3×C4⋊Q8D412yescomplemented
65C3×#127—22no—
66C3×#127—22no—
67C3×#127—21no—
68C3×#127—21no—
69C3×#125D412yescomplemented
70C3×#124—24no—
71C3×#124—22no—
72C3×#123—28no—
73C3×C22⋊Q8—22no—
74C3×#121—24no—
75C3×#121—28no—
76C3×#121—24no—
77C3×#120—24no—
78C3×C4×Q8—22no—
79C3×C4×Q8D412yescomplemented
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4—24no—
83C3×C4×D4—24no—
84C3×C4×D4D412yescomplemented
85C3×#117—21no—
86C3×#117C2311yescharacteristic
87C3×#117C2312yes—
88C2×C3×C22⋊C4—24no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—24no—
92#951—42no—
93#949—42no—
94C2×#216—21no—
95C2×#215—21no—
96#940—24no—
97#940—42no—
98#936—24no—
99#936—42no—
100C2×#214—21no—
101C2×#213—21no—
102C2×(C2×D3)⋊C4—22no—
103C2×C4×Dic3C2312yes—
104C2×#208C2312yes—
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107#875—22no—
108#875—21no—
109#875—21no—
110#837—24no—
111#833—42no—
112#833—42no—
113C3⋊#95—22no—
114#810—22no—
115#810—24no—
116#809—22no—
117#809—24no—
118C4⋊(C3⋊C8)—22no—
119#803—21no—
120C4×C3⋊C8—21no—
121C2×#137—31no—
122C2×AGL(1,ℤ/8)—31no—
123#559—22no—
124#550—22no—
125#546—24no—
126#540—22no—
127#535C2×D312yes—
128#533C2×D311yescharacteristic
129#532—22no—
130C2×#127—32no—
131C2×#127C2×D311yescharacteristic
132C2×#124—32no—
133C2×#124—31no—
134#518C2×D312yes—
135C4×#41C2×D311yescharacteristic
136#489—62no—
137#488—62no—
138#464—62no—
139#463—62no—
140#423—31no—
141C2×#104—32no—
142#419—32no—
143#419—31no—
144#418—32no—
145#418—31no—
146C2×#100—32no—
147C2×#100—32no—
148C4×M16—31no—
149#392—64no—
150#376—31no—
151#339—32no—
152#339—32no—
153#339—32no—
154C24×C3C2×D414yes—
155C23×D3—21no—
156C3×#41—22no—
157C3×#41—22no—
158C3×#41—88no—
159C2×C3×Q8—22no—
160C2×C3×Q8C2×D411yescharacteristic
161C2×C3×D4—21no—
162C2×C3×D4—28no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—84no—
172C2×C3×D4C2×D411yescharacteristic
173C22×C3×C4—28no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—48no—
179C2×#67—88no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—22no—
185C2×D12—22no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—21no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—22no—
201C3×C42—21no—
202C3×C42—22no—
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D412yes—
206#217—48no—
207#216—42no—
208#215—42no—
209#214—42no—
210#213—42no—
211(C2×D3)⋊C4—42no—
212(C2×D3)⋊C4—42no—
213C4×Dic3—22no—
214C4×Dic3#4114yescomplemented
215#208—22no—
216#208—22no—
217#208—22no—
218#208—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×#41—24no—
223C2×#41C22×D311yescharacteristic
224C22×Q8—31no—
225C22×D4—31no—
226C22×D4—64no—
227C22×D4#6714yes—
228C22×D4C22×D312yes—
229#137—122no—
230AGL(1,ℤ/8)—122no—
231C2×Q16—61no—
232C2×SD16—61no—
233C2×SD16—61no—
234C2×D8—61no—
235C2×M16—31no—
236C2×M16—31no—
237C2×M16—32no—
238C4⋊Q8#6712yescomplemented
239#127—22no—
240#127—21no—
241#127—21no—
242#127—22no—
243#127—64no—
244#127—122no—
245#125#6712yescomplemented
246#124—24no—
247#124—22no—
248#124—64no—
249#124—62no—
250#124—61no—
251#124—61no—
252#124—122no—
253#123—28no—
254C22⋊Q8—22no—
255#121—24no—
256#121—28no—
257#121—24no—
258#120—24no—
259C4×Q8—22no—
260C4×Q8#6712yescomplemented
261C4×D4—24no—
262C4×D4—22no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4#6712yescomplemented
266#117—21no—
267#117C22×D311yescharacteristic
268#117C22×D312yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—62no—
272C2×C42—21no—
273C2×C42—32no—
274C2×C42C22×D311yescharacteristic
275C4⋊C8—62no—
276#104—64no—
277#103—62no—
278#103—64no—
279#102—62no—
280#102—64no—
281#100—122no—
282#100—122no—
283#97—61no—
284C4×C8—61no—
285#95—62no—
286C23×C3—24no—
287C23×C3—21no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—48no—
291C23×C3C22×D411yescharacteristic
292C22×D3—44no—
293C22×D3—41no—
294C22×D3—42no—
295C3×Q8—21no—
296C3×Q8—21no—
297C3×Q8—44no—
298C3×D4—21no—
299C3×D4—21no—
300C3×D4—48no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—42no—
305C3×D4—42no—
306C3×D4—84no—
307C3×D4—88no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—42no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—88no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4112yes—
320C2×C3×C4H(5,𝔽2)12yes—
321#67—816no—
322C2×Dic3—24no—
323C2×Dic3—44no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326D12—41no—
327D12—41no—
328D12—42no—
329Dic6—41no—
330Dic6—41no—
331Dic6—42no—
332C3⋊C8—41no—
333C3⋊C8—41no—
334C3⋊C8—42no—
335C24—61no—
336C24C2×#6714yes—
337#41—22no—
338#41—22no—
339#41—88no—
340C2×Q8—22no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8C2×#6711yescharacteristic
346C2×D4—21no—
347C2×D4—28no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—84no—
361C2×D4—128no—
362C2×D4—248no—
363C2×D4C2×#6711yescharacteristic
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4C2×#6712yes—
370C22×C4C23×D311yescharacteristic
371Q16—122no—
372SD16—122no—
373SD16—122no—
374D8—122no—
375M16—62no—
376M16—62no—
377M16—62no—
378M16—62no—
379C2×C8—62no—
380C2×C8—61no—
381C2×C8—61no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4C2×#6712yes—
386C4⋊C4C2×#6714yes—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—21no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—128no—
393C22⋊C4—122no—
394C22⋊C4—122no—
395C42—22no—
396C42—21no—
397C42—22no—
398C42—34no—
399C42—62no—
400C42C2×#6711yescharacteristic
401C42C2×#6711yescharacteristic
402C42C2×#6712yes—
403C22×C3—22no—
404C22×C3—41no—
405C22×C3—44no—
406C22×C3—42no—
407C22×C3—48no—
408C22×C3—84no—
409C22×C3—84no—
410C22×C3C2×#12011yescharacteristic
411C22×C3#54612yes—
412C2×D3—84no—
413C2×D3—81no—
414C2×D3—82no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—41no—
418C3×C4—42no—
419C3×C4—44no—
420C3×C4#53211yescharacteristic
421C3×C4#53511yescharacteristic
422C3×C4#55012yes—
423Dic3—44no—
424Dic3—81no—
425C23—24no—
426C23—21no—
427C23—24no—
428C23—24no—
429C23—48no—
430C23—124no—
431C23—122no—
432C23—121no—
433C23C22×#6711yescharacteristic
434Q8—21no—
435Q8—21no—
436Q8—44no—
437Q8—121no—
438Q8—121no—
439Q8—122no—
440D4—21no—
441D4—21no—
442D4—48no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—84no—
449D4—88no—
450D4—121no—
451D4—121no—
452D4—122no—
453D4—2416no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—64no—
461C2×C4—88no—
462C2×C4—124no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4#95312yes—
466C2×C4#95312yes—
467C2×C4D3×#4112yes—
468C2×C4#100912yes—
469C2×C4C22×#6711yescharacteristic
470C2×C4C22×#6711yescharacteristic
471C8—121no—
472C8—121no—
473C8—122no—
474C2×C3—22no—
475C2×C3—81no—
476C2×C3—84no—
477C2×C3#243111yescharacteristic
478C2×C3#301912yes—
479D3—161no—
480C22—22no—
481C22—41no—
482C22—44no—
483C22—42no—
484C22—48no—
485C22—84no—
486C22—84no—
487C22—244no—
488C22—241no—
489C22—242no—
490C22#582912yes—
491C22C2×#95311yescharacteristic
492C4—22no—
493C4—22no—
494C4—41no—
495C4—42no—
496C4—44no—
497C4—124no—
498C4—241no—
499C4#583011yescharacteristic
500C4#583211yescharacteristic
501C4#583412yes—
502C3#2559311yescharacteristic, complemented, Sylow
503C2—22no—
504C2—84no—
505C2—81no—
506C2—481no—
507C2#8409211yescharacteristic
508C2#8410412yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3508C2
socleC22×C3C2×#120
hypercenter#3508D3
hypocenterC3#25593
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. C3×#3508C2semidirect
2. #83338C2semidirect
3. #83181C2semidirect
4. #82527C2semidirect
5. #71857C2semidirect
6. #71856C2semidirect
7. #71840C2semidirect
8. #71766C2semidirect
9. #71650C2semidirect
10. #71649C2semidirect
11. #71643C2semidirect
46. #3508D3semidirect
64. C3×C4⋊Q8D4semidirect
69. C3×#125D4semidirect
79. C3×C4×Q8D4semidirect
84. C3×C4×D4D4semidirect
214. C4×Dic3#41semidirect
238. C4⋊Q8#67semidirect
245. #125#67semidirect
260. C4×Q8#67semidirect
265. C4×D4#67semidirect
502. C3#25593semidirect

Maximal subgroups

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

subgroupindex
C3×#35082
#833382
#831812
#825272
#718572
#718562
#718402
#717662
#716502
#716492
#716432
#255933