Finite Group Databasebeta

← #10710656

Subgroups — #10710656

Order 768.

1160 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C211yesmaximal, characteristic
2#72139C211yesmaximal, characteristic, complemented
3#72137C211yesmaximal, characteristic, complemented
4C2×#4539C212yesmaximal, complemented
5C2×#4536C212yesmaximal, complemented
6#12689—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C24×C3×C4C411yescharacteristic, complemented
9C3×#525—24no—
10C3×#525—24no—
11C2×C3×#123C411yescharacteristic, complemented
12C2×C3×#123C2211yescharacteristic
13C2×C3×C22⋊Q8—21no—
14C2×C3×#121—21no—
15C2×C3×#120C411yescharacteristic, complemented
16C2×C3×#120C2211yescharacteristic
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4—21no—
19C2×C3×#117C411yescharacteristic, complemented
20C2×#836—21no—
21C2×#834—21no—
22C2×#834C2212yes—
23C2×#833C2212yes—
24#5211—21no—
25#5207—21no—
26#4539—24no—
27#4536—24no—
28C2×#525D311yescharacteristic
29C2×#352—32no—
30C2×AΣL(1,𝔽16)—32no—
31#1785—31no—
32#1782—31no—
33C25×C3C2×C411yescharacteristic
34C2×C3×#41—22no—
35C2×C3×#41—22no—
36C2×C3×#41—42no—
37C2×C3×#41—42no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4C2×C411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C22×C3×D4C2311yescharacteristic
44C23×C3×C4—22no—
45C23×C3×C4—21no—
46C23×C3×C4—21no—
47C23×C3×C4—22no—
48C23×C3×C4—42no—
49C23×C3×C4—42no—
50C23×C3×C4C2×C411yescharacteristic
51C23×C3×C4C2×C411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C3×#123—22no—
55C3×#123—22no—
56C3×#123—22no—
57C3×#123D414yes—
58C3×C22⋊Q8—28no—
59C3×#121—28no—
60C3×#120—22no—
61C3×#120—22no—
62C3×#120—22no—
63C3×#120D414yes—
64C3×C4×D4—28no—
65C3×C4×D4—28no—
66C3×#117—22no—
67C3×#117—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4C2×C411yescharacteristic
76C2×C3×C22⋊C4C2×C411yescharacteristic
77C2×C3×C42—21no—
78C2×#217—22no—
79C2×#208—22no—
80#836—42no—
81#834—22no—
82#834—22no—
83#834—42no—
84#833—22no—
85#833—22no—
86C3⋊#95—42no—
87#830—42no—
88C22×#41C2×D311yescharacteristic
89C24×C4Dic311yescharacteristic, complemented
90#525—24no—
91#525—24no—
92C2×#123Dic311yescharacteristic, complemented
93C2×#123C2×D311yescharacteristic
94C2×C22⋊Q8—21no—
95C2×#121—21no—
96C2×#120Dic311yescharacteristic, complemented
97C2×#120C2×D311yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×#117Dic311yescharacteristic, complemented
101C2×#101—61no—
102C2×#100—32no—
103C2×#99—32no—
104C2×#99—61no—
105#407—61no—
106#381—61no—
107#352—64no—
108AΣL(1,𝔽16)—64no—
109C24×C3—21no—
110C24×C3—21no—
111C24×C3—44no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3C22⋊C411yescharacteristic
115C24×C3C22⋊C412yes—
116C24×C3C22×C411yescharacteristic
117C24×C3C2×D411yescharacteristic
118C3×#41—48no—
119C3×#41—48no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—41no—
123C2×C3×D4—24no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—24no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—44no—
135C2×C3×D4—44no—
136C2×C3×D4C2×D412yes—
137C2×C3×D4C2×D412yes—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—48no—
155C22×C3×C4—48no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—41no—
161C22×C3×C4—41no—
162C22×C3×C4C22⋊C411yescharacteristic
163C22×C3×C4C22⋊C411yescharacteristic
164C22×C3×C4C22⋊C412yes—
165C22×C3×C4C22⋊C411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×Dic3—42no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—24no—
171C3×C22⋊C4—24no—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—21no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4C22⋊C412yes—
178C3×C22⋊C4C22⋊C412yes—
179C3×C42—24no—
180#217—42no—
181#217—42no—
182#208—42no—
183#208—42no—
184C2×C3⋊C8—42no—
185C25C2×Dic311yescharacteristic
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—42no—
189C2×#41—42no—
190C22×Q8#6711yescharacteristic
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C2×Dic311yescharacteristic
194C22×D4#6711yescharacteristic
195C22×D4C22×D311yescharacteristic
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4C2×Dic311yescharacteristic
203C23×C4C2×Dic311yescharacteristic
204C23×C4#6711yescharacteristic
205C23×C4#6711yescharacteristic
206C2×M16—62no—
207#123—22no—
208#123—22no—
209#123—22no—
210#123#6714yes—
211C22⋊Q8—28no—
212#121—28no—
213#120—22no—
214#120—22no—
215#120—22no—
216#120#6714yes—
217C4×D4—28no—
218C4×D4—28no—
219#117—22no—
220#117—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4C2×Dic311yescharacteristic
230C2×C22⋊C4C2×Dic311yescharacteristic
231C2×C42—21no—
232#101—122no—
233#100—62no—
234#100—62no—
235#99—62no—
236#99—62no—
237#99—122no—
238C22⋊C8—122no—
239#95—122no—
240C23×C3—24no—
241C23×C3—22no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—42no—
250C23×C3—44no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—44no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—44no—
258C23×C3—48no—
259C23×C3—44no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3C2×C22⋊C411yescharacteristic
263C23×C3C2×C22⋊C411yescharacteristic
264C23×C3C2×C22⋊C411yescharacteristic
265C23×C3#12012yes—
266C3×Q8—44no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—44no—
270C3×D4—48no—
271C3×D4—48no—
272C3×D4—42no—
273C3×D4—42no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—48no—
291C2×C3×C4—48no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4#9912yes—
299C2×C3×C4#9912yes—
300C2×C3×C4#12012yes—
301C2×Dic3—82no—
302C2×Dic3—84no—
303C3⋊C8—82no—
304C24—21no—
305C24—21no—
306C24—41no—
307C24—41no—
308C24—44no—
309C24#21711yescharacteristic
310C24#21712yes—
311C24C22×Dic311yescharacteristic
312C24C2×#6711yescharacteristic
313#41—48no—
314#41—48no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×D4—24no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4C2×#6712yes—
332C2×D4C2×#6712yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—122no—
358C22×C4#21711yescharacteristic
359C22×C4#21711yescharacteristic
360C22×C4#21712yes—
361C22×C4#21711yescharacteristic
362C22×C4C2×#6711yescharacteristic
363M16—122no—
364M16—122no—
365C2×C8—122no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—122no—
376C22⋊C4—122no—
377C22⋊C4#21712yes—
378C22⋊C4#21712yes—
379C42—24no—
380C22×C3—22no—
381C22×C3—24no—
382C22×C3—22no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—42no—
389C22×C3—44no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—48no—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—41no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3#37911yescharacteristic
404C22×C3C2×#9911yescharacteristic
405C22×C3C2×#9911yescharacteristic
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—41no—
412C3×C4—42no—
413Dic3—162no—
414C23—24no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—48no—
433C23—44no—
434C23—41no—
435C23—41no—
436C23C2×#21711yescharacteristic
437C23C2×#21711yescharacteristic
438C23C2×#21711yescharacteristic
439C23#95312yes—
440Q8—44no—
441D4—42no—
442D4—42no—
443D4—44no—
444D4—42no—
445D4—42no—
446D4—48no—
447D4—48no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—48no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—242no—
473C2×C4—244no—
474C2×C4#83412yes—
475C2×C4#83412yes—
476C2×C4#95312yes—
477C8—242no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—42no—
481C2×C3—44no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—81no—
485C2×C3#178111yescharacteristic
486C2×C3#212012yes—
487C22—22no—
488C22—24no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—44no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—48no—
502C22—42no—
503C22—44no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—41no—
508C22—81no—
509C22—81no—
510C22C2×#83411yescharacteristic
511C22C2×#83411yescharacteristic
512C22#524711yescharacteristic
513C4—22no—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—482no—
521C3#1268911yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—42no—
525C2—44no—
526C2—41no—
527C2—41no—
528C2—81no—
529C2#7171612yes—
530C2#7213011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC24×C3C22×C4
FrattiniC22×D4C22×D3
Fitting, prime radicalC2×C3×#525C2
socleC22×C3C2×#99
cosocleC22×C3×D4C23
hypercenterC2×#525D3
hypocenterC3#12689
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
2. #72139C2semidirect
3. #72137C2semidirect
4. C2×#4539C2semidirect
5. C2×#4536C2semidirect
8. C24×C3×C4C4semidirect
11. C2×C3×#123C4semidirect
15. C2×C3×#120C4semidirect
19. C2×C3×#117C4semidirect
89. C24×C4Dic3semidirect
92. C2×#123Dic3semidirect
96. C2×#120Dic3semidirect
100. C2×#117Dic3semidirect
521. C3#12689semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5252
#721392
#721372
C2×#45392
C2×#45362
#126893