Finite Group Databasebeta

← #10826911

Subgroups — #10826911

Order 768.

1694 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3487C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C2×#5246C212yesmaximal, complemented
4#82933C212yesmaximal, complemented
5#72137C211yesmaximal, characteristic, complemented
6#71857C214yesmaximal, complemented
7#71712C212yesmaximal, complemented
8#71660C212yesmaximal, complemented
9#23916—31nomaximal, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C22×C3×#41C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C3×#550—24no—
14C3×#548—24no—
15C3×#535—22no—
16C3×#535C2212yes—
17C3×#525—22no—
18C3×#525C2212yes—
19C3×#519C2211yescharacteristic
20#5825—28no—
21C2×#951C2214yes—
22C2×#944C2212yes—
23C2×#936C2212yes—
24C2×#935—22no—
25C2×#926C2211yescharacteristic
26C2×#924—22no—
27#5246—28no—
28C2×#837C2214yes—
29C2×#836C2212yes—
30C2×#833C2212yes—
31#5207C2212yes—
32#5050—24no—
33#5040—24no—
34#5004C2214yes—
35#4532C2214yes—
36#3487D311yescharacteristic
37C2×#536—31no—
38C2×#456—32no—
39#3004—32no—
40#2004—34no—
41#1897—32no—
42#1886—32no—
43#1785—31no—
44C3×2−1+4—24no—
45C3×2−1+4—42no—
46C2×C3×#41—24no—
47C2×C3×#41—22no—
48C2×C3×#41—24no—
49C2×C3×#41—22no—
50C2×C3×#41—42no—
51C2×C3×#41C2312yes—
52C22×C3×Q8—22no—
53C22×C3×Q8C2311yescharacteristic
54C22×C3×D4D412yes—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C22×Dic6—21no—
60C22×Dic6—24no—
61C22×Dic6—24no—
62C22×Dic6—42no—
63C22×Dic6C2312yes—
64C3×#127—24no—
65C3×#127—22no—
66C3×#124—24no—
67C3×#124—22no—
68C3×#123—22no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8D414yes—
71C3×#121—24no—
72C3×#121—28no—
73C3×#121—24no—
74C3×#121D414yes—
75C3×#120—24no—
76C3×#120—22no—
77C3×C4×Q8—24no—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×#117—22no—
81C3×#117C2312yes—
82C2×#217—24no—
83C2×#217—21no—
84#951—28no—
85#951—44no—
86#944—24no—
87#944D418yescomplemented
88C2×#216—24no—
89#936—24no—
90#936—24no—
91#935—216no—
92C2×#214—24no—
93#926—24no—
94C2×C4⋊Dic3—21no—
95#924—216no—
96C2×Dic3⋊C4—24no—
97C2×C4×Dic3C2312yes—
98C2×#208C2314yes—
99#837—28no—
100#836—24no—
101#833—42no—
102#830—24no—
103#810—28no—
104C2×2−1+4C2×D311yescharacteristic
105C22×#41C2×D311yescharacteristic
106C23×Q8—31no—
107C2×#137—34no—
108#550—24no—
109#548—24no—
110#536—68no—
111#535—22no—
112#535C2×D312yes—
113C2×C4⋊Q8—32no—
114C2×#124—32no—
115#525—22no—
116#525C2×D312yes—
117C2×#123—62no—
118C2×C22⋊Q8—62no—
119#519C2×D311yescharacteristic
120C2×#117—31no—
121#456—68no—
122#451—64no—
123#448—64no—
124C2×#104—34no—
125#419—34no—
126C2×#101—32no—
127C2×#100—32no—
128#407—32no—
129#339—34no—
130C24×C3C2×D412yes—
131C24×C3C2×D411yescharacteristic
132C3×#41—24no—
133C3×#41—24no—
134C3×#41—416no—
135C3×#41—44no—
136C3×#41—48no—
137C3×#41—44no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—48no—
142C2×C3×Q8C2×D414yes—
143C2×C3×Q8C2×D412yes—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—28no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—42no—
154C2×C3×D4C2×D412yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—42no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—24no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—42no—
170C22×Dic3—42no—
171C2×Dic6—24no—
172C2×Dic6—24no—
173C2×Dic6—24no—
174C2×Dic6—22no—
175C2×Dic6—416no—
176C2×Dic6—42no—
177C2×Dic6—42no—
178C2×Dic6C2×D414yes—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4C2×D414yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—24no—
186C3×C42—24no—
187C3×C42—22no—
188#217—216no—
189#217—24no—
190#216—48no—
191#214—48no—
192C4⋊Dic3—24no—
193Dic3⋊C4—216no—
194C4×Dic3—22no—
195C4×Dic3C2×D414yes—
196#208—24no—
197#208—24no—
198C2×C3⋊C8—24no—
1992−1+4—24no—
2002−1+4—42no—
201C2×#41—24no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—24no—
205C2×#41—42no—
206C2×#41C22×D312yes—
207C22×Q8—22no—
208C22×Q8—32no—
209C22×Q8—64no—
210C22×Q8—61no—
211C22×Q8—64no—
212C22×Q8—122no—
213C22×Q8C22×D311yescharacteristic
214C22×D4#6712yes—
215C22×D4C22×D311yescharacteristic
216C23×C4—61no—
217C23×C4#6712yes—
218C23×C4C22×D311yescharacteristic
219#137—68no—
220#137—124no—
221C2×Q16—64no—
222C2×SD16—64no—
223C2×M16—34no—
224C4⋊Q8—38no—
225C4⋊Q8—64no—
226#127—24no—
227#127—22no—
228#124—24no—
229#124—22no—
230#124—64no—
231#124—64no—
232#123—22no—
233#123—616no—
234C22⋊Q8—24no—
235C22⋊Q8—616no—
236C22⋊Q8#6714yes—
237#121—24no—
238#121—28no—
239#121—24no—
240#121#6714yes—
241#120—24no—
242#120—22no—
243C4×Q8—24no—
244C4×D4—28no—
245C4×D4—24no—
246#117—22no—
247#117—64no—
248#117C22×D312yes—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—64no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—64no—
253C2×C42—32no—
254#104—68no—
255#103—68no—
256#101—64no—
257#100—122no—
258C22⋊C8—64no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—42no—
266C23×C3—44no—
267C23×C3#12012yes—
268C23×C3C22×D411yescharacteristic
269C3×Q8—28no—
270C3×Q8—24no—
271C3×Q8—44no—
272C3×D4—24no—
273C3×D4—48no—
274C3×D4—44no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—44no—
278C3×D4—84no—
279C3×D4—88no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—28no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—48no—
288C2×C3×C4—48no—
289C2×C3×C4—44no—
290C2×C3×C4#12014yes—
291C2×C3×C4#12012yes—
292C2×C3×C4C22×D412yes—
293C2×C3×C4C22×D412yes—
294C2×C3×C4H(5,𝔽2)12yes—
295C2×Dic3—24no—
296C2×Dic3—24no—
297C2×Dic3—48no—
298C2×Dic3—44no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301Dic6—28no—
302Dic6—48no—
303Dic6—44no—
304C3⋊C8—44no—
305C24C2×#6712yes—
306C24C2×#6711yescharacteristic
307#41—24no—
308#41—24no—
309#41—416no—
310#41—44no—
311#41—48no—
312#41—44no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—34no—
317C2×Q8—48no—
318C2×Q8—64no—
319C2×Q8—62no—
320C2×Q8—64no—
321C2×Q8—64no—
322C2×Q8—1216no—
323C2×Q8—122no—
324C2×Q8—122no—
325C2×Q8D3×D414yes—
326C2×Q8C2×#6712yes—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—28no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4C2×#6712yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—64no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4C2×#6712yes—
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6714yes—
353C22×C4C23×D311yescharacteristic
354Q16—128no—
355SD16—128no—
356M16—64no—
357M16—64no—
358C2×C8—64no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—64no—
362C4⋊C4—616no—
363C4⋊C4C2×#6714yes—
364C22⋊C4—28no—
365C22⋊C4—24no—
366C22⋊C4—22no—
367C22⋊C4—24no—
368C22⋊C4—64no—
369C22⋊C4—616no—
370C42—24no—
371C42—22no—
372C42—34no—
373C42—62no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—24no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—82no—
385C22×C3C2×#12011yescharacteristic
386C22×C3C2×#12011yescharacteristic
387C22×C3#53411yescharacteristic
388C3×C4—22no—
389C3×C4—24no—
390C3×C4—42no—
391C3×C4—44no—
392C3×C4D4214yes—
393Dic3—44no—
394Dic3—44no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—44no—
403C23#95312yes—
404C23C22×#6711yescharacteristic
405Q8—28no—
406Q8—24no—
407Q8—44no—
408Q8—68no—
409Q8—128no—
410Q8—124no—
411D4—24no—
412D4—48no—
413D4—44no—
414D4—42no—
415D4—42no—
416D4—44no—
417D4—84no—
418D4—88no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—64no—
430C2×C4—64no—
431C2×C4—128no—
432C2×C4—124no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4#93714yes—
436C2×C4#95312yes—
437C2×C4C2×D3×D412yes—
438C2×C4#100412yes—
439C2×C4C22×#6712yes—
440C8—124no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—42no—
444C2×C3—82no—
445C2×C3—82no—
446C2×C3#240311yescharacteristic
447C2×C3#302212yes—
448C22—22no—
449C22—22no—
450C22—24no—
451C22—42no—
452C22—42no—
453C22—44no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22—82no—
459C22C2×#93711yescharacteristic
460C22#580411yescharacteristic
461C22C2×#95311yescharacteristic
462C4—22no—
463C4—24no—
464C4—42no—
465C4—44no—
466C4—124no—
467C4—124no—
468C4D4×#6714yes—
469C3#2391611yescharacteristic, complemented, Sylow
470C2—21no—
471C2—21no—
472C2—42no—
473C2—82no—
474C2—82no—
475C2#8405211yescharacteristic
476C2#8412312yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3487C2
socleC22×C3C2×#120
hypercenter#3487D3
hypocenterC3#23916
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3487

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C2×#5246C2semidirect
4. #82933C2semidirect
5. #72137C2semidirect
6. #71857C2semidirect
7. #71712C2semidirect
8. #71660C2semidirect
87. #944D4semidirect
469. C3#23916semidirect

Maximal subgroups

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

subgroupindex
C3×#34872
C2×#58252
C2×#52462
#829332
#721372
#718572
#717122
#716602
#239163