Finite Group Databasebeta

← #10849354

Subgroups — #10849354

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×#5671C211yesmaximal, characteristic
2C2×C3×#489C211yesmaximal, characteristic
3#83239C211yesmaximal, characteristic, complemented
4C2×#5091C211yesmaximal, characteristic, complemented
5#71721C211yesmaximal, characteristic
6#70284C212yesmaximal, complemented
7#70235C212yesmaximal, complemented
8#70216C212yesmaximal
9#70167C212yesmaximal
10#69838C211yesmaximal, characteristic, complemented
11#69813C211yesmaximal, characteristic, complemented
12#25661—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C2×#944—21no—
17C2×#944C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20#5671—24no—
21#5671—24no—
22C2×#879—21no—
23C2×#879—21no—
24C2×#878—21no—
25C2×#877—21no—
26C2×C4×Dic6C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#489—24no—
29C2×C3×#103C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#97C2211yescharacteristic
32#5229—21no—
33C2×#835C2212yes—
34C2×#832C2212yes—
35C2×#816C2212yes—
36#5091—24no—
37#4983—21no—
38#4983—21no—
39#4972C2212yes—
40#4972C2212yes—
41C2×#810C2212yes—
42C2×#807C2212yes—
43C4×C4⋊Dic3C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4551C2212yes—
46#4550C2212yes—
47#4550C2212yes—
48C2×#556—31no—
49C2×#489D311yescharacteristic
50C2×#477—31no—
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×C3×D4C2311yescharacteristic
62C23×Dic3—21no—
63C22×Dic6D411yescharacteristic
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×#124—22no—
69C3×#124D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—21no—
76C2×#217—21no—
77C2×#217—21no—
78Q8×Dic3—22no—
79Q8×Dic3—22no—
80#944—22no—
81#944—22no—
82#944—44no—
83#936—22no—
84#936—22no—
85D4×Dic3—22no—
86D4×Dic3—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3D411yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4D411yescharacteristic
97C2×Dic3⋊C4D411yescharacteristic
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104#879—28no—
105#879—28no—
106#878—28no—
107#877—28no—
108C4×Dic6—22no—
109C4×Dic6—22no—
110C4×Dic6—22no—
111C4×Dic6D414yes—
112C3×#103—24no—
113C3×#102—24no—
114C3×#97—22no—
115#835—24no—
116#832—24no—
117C3⋊#95—21no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#816—24no—
121#810—24no—
122#807—24no—
123C4⋊(C3⋊C8)—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×#124C2×D311yescharacteristic
133C2×C22⋊Q8—61no—
134C2×C4×Q8—31no—
135C2×C4×Q8—31no—
136C2×C4×Q8—31no—
137C2×C4×D4—31no—
138C2×#117—61no—
139#489—24no—
140#477—64no—
141C2×#107—32no—
142C2×C4⋊C8—31no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#103C2×D312yes—
147C2×#102—32no—
148C2×#102C2×D312yes—
149C2×#97C2×D311yescharacteristic
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—
159C24×C3—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—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4C2×D412yes—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—24no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—42no—
181C22×Dic3—42no—
182C22×Dic3C2×D411yescharacteristic
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
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—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—24no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203#217—24no—
204#217—24no—
205#217—24no—
206C4⋊Dic3—24no—
207C4⋊Dic3—22no—
208C4⋊Dic3—21no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D412yes—
211C4⋊Dic3#4114yes—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—22no—
215Dic3⋊C4—22no—
216Dic3⋊C4—24no—
217Dic3⋊C4—44no—
218Dic3⋊C4—44no—
219C4×Dic3—24no—
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×D4C22×D311yescharacteristic
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—22no—
245#124—62no—
246#124—62no—
247#124#6714yes—
248C22⋊Q8—68no—
249C4×Q8—34no—
250C4×Q8—62no—
251C4×Q8—62no—
252C4×Q8—62no—
253C4×Q8—62no—
254C4×Q8—62no—
255C4×D4—62no—
256C4×D4—62no—
257#117—68no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—31no—
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—61no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C42—31no—
277C2×C42—31no—
278C2×C42—61no—
279C2×C42—61no—
280C2×C42—61no—
281C2×C42C22×D311yescharacteristic
282#107—64no—
283C4⋊C8—62no—
284#103—24no—
285#103—64no—
286#103—64no—
287#103—64no—
288#102—24no—
289#102—64no—
290#97—22no—
291#95—62no—
292#95—62no—
293#95—61no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C3×Q8—24no—
301C3×Q8—41no—
302C3×Q8—41no—
303C3×Q8—84no—
304C3×D4—24no—
305C3×D4—41no—
306C3×D4—41no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—42no—
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—22no—
325C2×Dic3—21no—
326C2×Dic3—22no—
327C2×Dic3—21no—
328C2×Dic3—24no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—48no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—42no—
339C2×Dic3AGL(1,ℤ/8)12yes—
340Dic6—22no—
341Dic6—22no—
342Dic6—44no—
343Dic6—42no—
344C3×C8—42no—
345C3⋊C8—42no—
346C24—21no—
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—21no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4C2×#6712yes—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—64no—
372C22×C4—62no—
373C22×C4—61no—
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—22no—
387C4⋊C4—22no—
388C4⋊C4—32no—
389C4⋊C4—34no—
390C4⋊C4—42no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—64no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—64no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—124no—
401C4⋊C4—124no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C42—64no—
408C42—64no—
409C42—61no—
410C42—61no—
411C42—61no—
412C42—61no—
413C42—122no—
414C42D3×D412yes—
415C42C2×#6712yes—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54812yes—
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3#57511yescharacteristic
425C22×C3#57812yes—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—42no—
429C3×C4—82no—
430C3×C4#53211yescharacteristic
431C3×C4#53611yescharacteristic
432C3×C4#54712yes—
433Dic3—22no—
434Dic3—44no—
435Dic3—42no—
436Dic3—41no—
437Dic3—42no—
438Dic3—41no—
439Dic3—41no—
440C23—22no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23C2×D3×D411yescharacteristic
446Q8—24no—
447Q8—41no—
448Q8—41no—
449Q8—62no—
450Q8—62no—
451Q8—84no—
452Q8—124no—
453Q8—122no—
454D4—24no—
455D4—41no—
456D4—41no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—32no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—64no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—61no—
474C2×C4—62no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—62no—
478C2×C4—128no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4#93712yes—
485C2×C4#93712yes—
486C2×C4C2×D3×D411yescharacteristic
487C2×C4D3×#4112yes—
488C2×C4#101012yes—
489C2×C4C22×#6711yescharacteristic
490C8—42no—
491C8—122no—
492C2×C3—82no—
493C2×C3#243211yescharacteristic
494C2×C3#300611yescharacteristic
495C2×C3#301111yescharacteristic
496C2×C3#336712yes—
497C2×C3#337712yes—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22#567712yes—
504C22D3×AGL(1,ℤ/8)11yescharacteristic
505C22#577311yescharacteristic
506C22#577912yes—
507C22C2×#93711yescharacteristic
508C4—22no—
509C4—22no—
510C4—42no—
511C4—62no—
512C4—82no—
513C4—124no—
514C4—122no—
515C4—121no—
516C4—121no—
517C4—121no—
518C4—122no—
519C4#566811yescharacteristic
520C4#567511yescharacteristic
521C4#582412yes—
522C3#2566111yescharacteristic, complemented, Sylow
523C2—82no—
524C2#8167012yes—
525C2#8167712yes—
526C2#8295911yescharacteristic
527C2#8394611yescharacteristic
528C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#489D3
hypocenterC3#25661
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
3. #83239C2semidirect
4. C2×#5091C2semidirect
6. #70284C2semidirect
7. #70235C2semidirect
10. #69838C2semidirect
11. #69813C2semidirect
522. C3#25661semidirect

Maximal subgroups

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

subgroupindex
C2×#56712
C2×C3×#4892
#832392
C2×#50912
#717212
#702842
#702352
#702162
#701672
#698382
#698132
#256613