Finite Group Databasebeta

← #10849351

Subgroups — #10849351

Order 768.

1048 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5568C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3#83239C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic
5#71631C211yesmaximal, characteristic, complemented
6#70642C212yesmaximal, complemented
7#70638C212yesmaximal
8#70473C211yesmaximal, characteristic, complemented
9#70458C211yesmaximal, characteristic
10#70284C212yesmaximal, complemented
11#70279C212yesmaximal
12#25661—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947C2211yescharacteristic
16C2×#944—21no—
17C2×#944C2211yescharacteristic
18#5568—24no—
19#5568—24no—
20C2×#898—21no—
21C2×#896—21no—
22C2×#893C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×#876—21no—
25C2×#875C2211yescharacteristic
26C2×#872—21no—
27C2×C4×Dic6C2211yescharacteristic
28C3×#477—24no—
29C2×C3×#107C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32#5229—21no—
33C2×#835C2212yes—
34C2×#821C2212yes—
35C2×#816C2212yes—
36#5089—24no—
37#4983—21no—
38#4983—21no—
39#4972C2212yes—
40#4972C2212yes—
41C2×#810C2212yes—
42C2×#809C2212yes—
43C4×C4⋊Dic3C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#556—31no—
49C2×#489—31no—
50C2×#477D311yescharacteristic, complemented
51#2546—31no—
52#2043—32no—
53#2042—32no—
54#1985—31no—
55#1983—31no—
56#1863—32no—
57#1862—32no—
58#1763—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6D411yescharacteristic
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76#947—22no—
77#947—22no—
78#944—22no—
79#944—22no—
80#944—44no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4D411yescharacteristic
94C2×Dic3⋊C4D411yescharacteristic
95C2×C4×Dic3—21no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3C2311yescharacteristic
98C2×C4×Dic3C2311yescharacteristic
99C22×C3⋊C8C2312yes—
100#898—28no—
101#896—28no—
102#893—22no—
103#893—22no—
104#887—22no—
105#887—22no—
106#876—28no—
107#875—22no—
108#875D414yes—
109#872—28no—
110C4×Dic6—22no—
111C4×Dic6—22no—
112C3×#107—24no—
113C3×C4⋊C8—22no—
114C3×#103—24no—
115#835—24no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#821—24no—
120#816—24no—
121#810—24no—
122#809—24no—
123#803—22no—
124#556—64no—
125#556—64no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—61no—
128C2×C4⋊Q8C2×D311yescharacteristic
129C2×#126—61no—
130C2×#126—61no—
131C2×#124—31no—
132C2×#124—31no—
133C2×C22⋊Q8—61no—
134C2×C4×Q8—31no—
135C2×C4×Q8—31no—
136C2×C4×Q8C2×D311yescharacteristic
137C2×C4×D4—31no—
138C2×#117—61no—
139#489—64no—
140#477—24no—
141C2×#107C2×D312yes—
142C2×C4⋊C8C2×D311yescharacteristic
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#103C2×D312yes—
147C2×#102—32no—
148C2×#102—32no—
149C2×#97—31no—
150#398—61no—
151#398—61no—
152#391—61no—
153#389—32no—
154#389—32no—
155C4×C4⋊C4—31no—
156#340—32no—
157#336—32no—
158#336—32no—
159C23×D3—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—42no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8C2×D412yes—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12C2×D412yes—
183C2×C4×D3—24no—
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6—22no—
191C2×Dic6—41no—
192C2×Dic6C2×D412yes—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4C2×D412yes—
202C3×C42—24no—
203C3×C42C2×D412yes—
204C3×C42C2×D412yes—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3#4114yes—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215Dic3⋊C4—24no—
216Dic3⋊C4—22no—
217Dic3⋊C4—22no—
218Dic3⋊C4—44no—
219Dic3⋊C4—44no—
220C4×Dic3—24no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C4×Dic3—42no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C22×Q8—21no—
230C22×Q8—31no—
231C22×Q8—31no—
232C22×Q8C22×D311yescharacteristic
233C22×D4—31no—
234C23×C4—61no—
235C22×C8—32no—
236C22×C8C22×D312yes—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—62no—
240C4⋊Q8—62no—
241C4⋊Q8—124no—
242#126—68no—
243#126—68no—
244#124—34no—
245#124—62no—
246#124—62no—
247#124—62no—
248C22⋊Q8—68no—
249C4×Q8—22no—
250C4×Q8—62no—
251C4×Q8—62no—
252C4×Q8—62no—
253C4×Q8—62no—
254C4×Q8#6714yes—
255C4×D4—62no—
256C4×D4—62no—
257#117—68no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C42—21no—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42—61no—
280C2×C42—61no—
281C2×C42C22×D311yescharacteristic
282#107—24no—
283C4⋊C8—22no—
284#103—24no—
285#103—64no—
286#103—64no—
287#103—64no—
288#102—64no—
289#102—64no—
290#97—62no—
291#95—62no—
292#95—62no—
293#95—61no—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—42no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—42no—
300C3×Q8—22no—
301C3×Q8—22no—
302C3×Q8—42no—
303C3×Q8—84no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4112yes—
320C2×C3×C42−1+412yes—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—22no—
327C2×Dic3—24no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3AGL(1,ℤ/8)12yes—
336D12—24no—
337D12—41no—
338D12—41no—
339C4×D3—48no—
340Dic6—24no—
341Dic6—44no—
342Dic6—41no—
343Dic6—41no—
344C3×C8—42no—
345C3⋊C8—42no—
346C24—61no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—32no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—61no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—121no—
360C2×Q8C2×#6712yes—
361C2×D4—32no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—61no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—64no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—122no—
376C22×C4—122no—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D411yescharacteristic
379C22×C4C23×D311yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—24no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—34no—
391C4⋊C4—42no—
392C4⋊C4—64no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—124no—
400C4⋊C4—124no—
401C4⋊C4C2×#6712yes—
402C22⋊C4—64no—
403C22⋊C4—64no—
404C22⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C42—24no—
408C42—64no—
409C42—61no—
410C42—61no—
411C42—61no—
412C42—61no—
413C42—122no—
414C42D3×D412yes—
415C42C2×#6712yes—
416C22×C3C2×#12011yescharacteristic
417C22×C3#54812yes—
418C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
419C22×C3#57511yescharacteristic
420C22×C3#57812yes—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—82no—
425C2×D3—82no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—82no—
431C3×C4#53211yescharacteristic
432C3×C4#53611yescharacteristic
433C3×C4#54712yes—
434Dic3—22no—
435Dic3—44no—
436Dic3—42no—
437Dic3—41no—
438Dic3—41no—
439Dic3—41no—
440C23—62no—
441C23—122no—
442C23—122no—
443C23—121no—
444C23—121no—
445C23C2×D3×D411yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—42no—
449Q8—64no—
450Q8—84no—
451Q8—124no—
452Q8—121no—
453Q8—121no—
454D4—64no—
455D4—121no—
456D4—121no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—32no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—64no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—62no—
480C2×C4—128no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4#93712yes—
485C2×C4#93712yes—
486C2×C4C2×D3×D411yescharacteristic
487C2×C4#100712yes—
488C2×C4D3×#4112yes—
489C2×C4C22×#6711yescharacteristic
490C8—42no—
491C8—122no—
492C2×C3#243211yescharacteristic
493C2×C3#300611yescharacteristic
494C2×C3#301111yescharacteristic
495C2×C3#336712yes—
496C2×C3#337712yes—
497D3—82no—
498C22—122no—
499C22—122no—
500C22—122no—
501C22—242no—
502C22—242no—
503C22#563312yes—
504C22#576612yes—
505C22D3×AGL(1,ℤ/8)11yescharacteristic
506C22#577311yescharacteristic
507C22C2×#93711yescharacteristic
508C4—22no—
509C4—22no—
510C4—42no—
511C4—42no—
512C4—62no—
513C4—82no—
514C4—122no—
515C4—121no—
516C4—121no—
517C4—124no—
518C4—121no—
519C4#563112yes—
520C4#581611yescharacteristic
521C4#583211yescharacteristic
522C3#2566111yescharacteristic, complemented, Sylow
523C2—242no—
524C2#8110012yes—
525C2#8110112yes—
526C2#8275911yescharacteristic
527C2#8388411yescharacteristic
528C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#477D3
hypocenterC3#25661
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×C3×#477C2semidirect
3. #83239C2semidirect
5. #71631C2semidirect
6. #70642C2semidirect
8. #70473C2semidirect
10. #70284C2semidirect
50. C2×#477D3semidirect
522. C3#25661semidirect

Maximal subgroups

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

subgroupindex
C2×#55682
C2×C3×#4772
#832392
C2×#50892
#716312
#706422
#706382
#704732
#704582
#702842
#702792
#256613