Finite Group Databasebeta

← #10901286

Subgroups — #10901286

Order 768.

1106 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic, complemented
2#82526C212yesmaximal, complemented
3#82476C211yesmaximal, characteristic, complemented
4#71903C212yesmaximal, complemented
5#71879C212yesmaximal, complemented
6#71867C214yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71637C211yesmaximal, characteristic, complemented
9#29679—31nomaximal, Sylow
10C3×#544C2211yescharacteristic
11C3×#542—22no—
12C3×#541—22no—
13C3×#539—22no—
14C3×#537C2211yescharacteristic
15C3×#536C2212yes—
16C3×#533—22no—
17C3×#529—22no—
18C3×#529—22no—
19C3×C4×#41C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#949C2214yes—
22C2×#947C2212yes—
23C2×#874C2211yescharacteristic
24C2×#837C2214yes—
25C2×#836C2212yes—
26#5027—22no—
27#5027—22no—
28#5015—22no—
29#5015—22no—
30#5001C2214yes—
31#4991—22no—
32#4988C2212yes—
33#4939—22no—
34C4×C4⋊Dic3C2211yescharacteristic
35#4924C2212yes—
36#4532C2214yes—
37#4530C2212yes—
38#3461D311yescharacteristic, complemented
39#3038—32no—
40#2548—31no—
41#2027—32no—
42#2006—32no—
43#2003—34no—
44#1806—32no—
45#1801—31no—
46C2×C3×#41C2312yes—
47C22×C3×Q8C2311yescharacteristic
48C22×D12—21no—
49C22×D12—21no—
50C22×D12C2312yes—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—22no—
53C3×#126—24no—
54C3×#126—24no—
55C3×#125—22no—
56C3×#125—24no—
57C3×#125—22no—
58C3×#124—22no—
59C3×#124—22no—
60C3×#123—24no—
61C3×#123—24no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—22no—
67C3×#121—22no—
68C3×#121—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8D412yes—
73C3×C4×Q8D412yes—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4D412yes—
78C3×C4×D4D412yes—
79C3×#117—24no—
80C3×#117—22no—
81C3×#117C2312yes—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—22no—
85C2×C3×C42D412yes—
86C2×C3×C42C2311yescharacteristic
87#949—44no—
88#949—44no—
89#947—44no—
90C2×#215—24no—
91C2×#213—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—24no—
94C2×C4⋊Dic3—21no—
95C2×C4×Dic3C2312yes—
96C2×#208C2314yes—
97#874—22no—
98#874—22no—
99#837—44no—
100#836—42no—
101C3⋊#95—21no—
102#809—24no—
103#809—24no—
104C4⋊(C3⋊C8)—22no—
105C4⋊(C3⋊C8)—22no—
106C2×AGL(1,ℤ/8)—34no—
107#544C2×D311yescharacteristic
108#542—22no—
109#541—22no—
110#539—22no—
111#537C2×D311yescharacteristic
112#536C2×D312yes—
113#533—22no—
114C2×#127—31no—
115#529—22no—
116#529—22no—
117C2×#124—32no—
118C4×#41C2×D312yes—
119C2×C4×Q8C2×D311yescharacteristic
120#460—62no—
121#460—62no—
122#459—62no—
123#459—62no—
124#423—32no—
125C2×#104—34no—
126#418—34no—
127C2×#101—32no—
128#394—62no—
129#390—32no—
130#390—62no—
131C4×C4⋊C4—31no—
132#343—32no—
133#339—34no—
134C23×D3—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8C2×D412yes—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4C2×D412yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—22no—
157C2×D12—24no—
158C2×D12—24no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—44no—
162C2×D12—41no—
163C2×D12—42no—
164C2×D12—41no—
165C2×D12—82no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4C2×D412yes—
176C3×C4⋊C4C2×D412yes—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—22no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190#215—48no—
191#213—48no—
192(C2×D3)⋊C4—48no—
193(C2×D3)⋊C4—84no—
194C4⋊Dic3—42no—
195C4×Dic3—24no—
196#208—24no—
197#208—24no—
198C2×C3⋊C8—24no—
199C2×#41C22×D312yes—
200C22×Q8C22×D311yescharacteristic
201C22×D4—32no—
202C22×D4—61no—
203C22×D4—61no—
204AGL(1,ℤ/8)—124no—
205AGL(1,ℤ/8)—124no—
206C2×SD16—64no—
207C2×D8—64no—
208C2×M16—34no—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211#127—62no—
212#127—62no—
213#126—24no—
214#126—24no—
215#125—22no—
216#125—24no—
217#125—22no—
218#124—22no—
219#124—22no—
220#124—124no—
221#123—24no—
222#123—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228#121—22no—
229#121—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8#6712yes—
234C4×Q8#6712yes—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4#6712yes—
239C4×D4#6712yes—
240#117—24no—
241#117—22no—
242#117C22×D312yes—
243C2×C4⋊C4—61no—
244C2×C4⋊C4#6712yes—
245C2×C4⋊C4C22×D311yescharacteristic
246C2×C22⋊C4—64no—
247C2×C22⋊C4—64no—
248C2×C42—22no—
249C2×C42—32no—
250C2×C42#6712yes—
251C2×C42C22×D311yescharacteristic
252C4⋊C8—62no—
253C4⋊C8—62no—
254#104—124no—
255#102—64no—
256#102—64no—
257#101—122no—
258#95—61no—
259C23×C3—22no—
260C23×C3C22×D411yescharacteristic
261C22×D3—44no—
262C22×D3—44no—
263C22×D3—42no—
264C22×D3—42no—
265C22×D3—42no—
266C3×Q8—22no—
267C3×Q8—22no—
268C3×Q8—42no—
269C3×Q8—42no—
270C3×Q8—82no—
271C3×D4—22no—
272C3×D4—22no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—82no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—44no—
288C2×C3×C4—42no—
289C2×C3×C4—82no—
290C2×C3×C4#12011yescharacteristic
291C2×C3×C4#12012yes—
292C2×C3×C4#12011yescharacteristic
293C2×C3×C4C22×D412yes—
294C2×C3×C4H(5,𝔽2)11yescharacteristic
295C2×C3×C4H(5,𝔽2)12yes—
296C2×C3×C42−1+411yescharacteristic
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300D12—44no—
301D12—44no—
302D12—82no—
303D12—84no—
304D12—82no—
305C3⋊C8—44no—
306C24—62no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311C2×Q8—22no—
312C2×Q8—21no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—22no—
316C2×Q8C2×#6712yes—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—64no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—64no—
323C2×D4—124no—
324C2×D4—121no—
325C2×D4—122no—
326C2×D4—121no—
327C2×D4—242no—
328C2×D4C2×#6712yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—62no—
333C22×C4C2×#6712yes—
334C22×C4C2×#6711yescharacteristic
335C22×C4C2×#6712yes—
336C22×C4C2×#6711yescharacteristic
337C22×C4C23×D311yescharacteristic
338SD16—128no—
339D8—128no—
340M16—64no—
341M16—64no—
342C2×C8—64no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—122no—
353C4⋊C4C2×#6712yes—
354C4⋊C4C2×#6712yes—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—128no—
362C22⋊C4—244no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—64no—
369C42C2×#6712yes—
370C42C2×#6712yes—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3C2×#12011yescharacteristic
377C22×C3#53511yescharacteristic
378C22×C3#53711yescharacteristic
379C2×D3—84no—
380C2×D3—84no—
381C2×D3—82no—
382C2×D3—82no—
383C2×D3—82no—
384C3×C4—21no—
385C3×C4—22no—
386C3×C4—42no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4#45412yes—
392C3×C4#53512yes—
393C3×C4#53712yes—
394Dic3—82no—
395C23—22no—
396C23—124no—
397C23—124no—
398C23—122no—
399C23—122no—
400C23—122no—
401C23C22×#6711yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—42no—
405Q8—42no—
406Q8—82no—
407D4—22no—
408D4—22no—
409D4—42no—
410D4—42no—
411D4—82no—
412D4—124no—
413D4—124no—
414D4—242no—
415D4—244no—
416D4—242no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4#95311yescharacteristic
435C2×C4#95312yes—
436C2×C4#95311yescharacteristic
437C2×C4#100911yescharacteristic
438C2×C4#100912yes—
439C2×C4#101011yescharacteristic
440C2×C4C22×#6712yes—
441C8—124no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—82no—
445C2×C3#261511yescharacteristic
446C2×C3C2×#45411yescharacteristic
447C2×C3#302111yescharacteristic
448D3—162no—
449C22—21no—
450C22—21no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—244no—
455C22—244no—
456C22—242no—
457C22—242no—
458C22—242no—
459C22#583011yescharacteristic
460C22#583311yescharacteristic
461C22C2×#95311yescharacteristic
462C4—21no—
463C4—22no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—242no—
470C4#524512yes—
471C4#583012yes—
472C4#583312yes—
473C3#2967911yescharacteristic, complemented, Sylow
474C2—21no—
475C2—21no—
476C2—82no—
477C2—482no—
478C2#8409811yescharacteristic
479C2C2×#524511yescharacteristic
480C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3461C2
socleC22×C3C2×#120
hypercenter#3461D3
hypocenterC3#29679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. C3×#3461C2semidirect
2. #82526C2semidirect
3. #82476C2semidirect
4. #71903C2semidirect
5. #71879C2semidirect
6. #71867C2semidirect
7. #71646C2semidirect
8. #71637C2semidirect
38. #3461D3semidirect
473. C3#29679semidirect

Maximal subgroups

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

subgroupindex
C3×#34612
#825262
#824762
#719032
#718792
#718672
#716462
#716372
#296793