Finite Group Databasebeta

← #10849279

Subgroups — #10849279

Order 768.

1324 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×#5071C211yesmaximal, characteristic, complemented
3#82483C211yesmaximal, characteristic, complemented
4C2×#5009C211yesmaximal, characteristic, complemented
5#71762C212yesmaximal, complemented
6#71762C212yesmaximal, complemented
7#71761C211yesmaximal, characteristic, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71609C211yesmaximal, characteristic, complemented
10#71607C212yesmaximal, complemented
11#71606C212yesmaximal, complemented
12#25655—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#550—24no—
15C3×#550—24no—
16C2×C3×#127C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121—22no—
19C2×C3×#121—21no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×#870C2211yescharacteristic
24#5230—21no—
25C2×#832C2212yes—
26C2×#832C2212yes—
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5071—24no—
33#5009—24no—
34#4983—21no—
35#4976—22no—
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39#4928C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550D311yescharacteristic
47C2×#490—31no—
48C2×#478—31no—
49#2584—31no—
50#2041—32no—
51#2041—32no—
52#1985—31no—
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1762—31no—
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—22no—
67C23×C3×C4—21no—
68C3×#127—24no—
69C3×#127—24no—
70C3×#127—22no—
71C3×#127D414yes—
72C3×#121—28no—
73C3×#121—216no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C3×C4×D4—22no—
79C3×C4×D4D414yes—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2312yes—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98#870—22no—
99#870—22no—
100#832—22no—
101#832—22no—
102#832—24no—
103#832—24no—
104#832—24no—
105#832—24no—
106#832—24no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—21no—
110#808—22no—
111#807—22no—
112C4⋊(C3⋊C8)—22no—
113#803—22no—
114C22×#41—21no—
115#550—24no—
116#550—24no—
117C2×#127C2×D312yes—
118C2×#127C2×D311yescharacteristic
119C2×#125—31no—
120C2×#121—22no—
121C2×#121—21no—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×D4C2×D312yes—
124C2×C4×D4C2×D311yescharacteristic
125#490—64no—
126#478—64no—
127C2×#107—31no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#102—31no—
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#97—31no—
138#398—61no—
139#395—61no—
140#389—32no—
141#389—32no—
142#389—62no—
143C4×C4⋊C4—31no—
144#340—32no—
145#336—32no—
146#336—32no—
147C24×C3—22no—
148C24×C3—21no—
149C3×#41—432no—
150C2×C3×Q8—22no—
151C2×C3×Q8—24no—
152C2×C3×D4—28no—
153C2×C3×D4—28no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—44no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4—81no—
168C2×C3×D4—81no—
169C2×C3×D4—82no—
170C2×C3×D4—82no—
171C2×C3×D4C2×D414yes—
172C2×C3×D4C2×D412yes—
173C22×C3×C4—28no—
174C22×C3×C4—21no—
175C22×C3×C4—24no—
176C22×C3×C4—44no—
177C22×C3×C4—44no—
178C22×C3×C4—44no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197C4⋊Dic3—22no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3#4114yescomplemented
203Dic3⋊C4—42no—
204Dic3⋊C4—42no—
205Dic3⋊C4—44no—
206Dic3⋊C4—82no—
207C4×Dic3—81no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×#41—28no—
216C2×#41—48no—
217C22×Q8#6711yescharacteristic
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4C22×D312yes—
223C22×D4C22×D311yescharacteristic
224C23×C4—22no—
225C23×C4—21no—
226C22×C8—32no—
227C22×C8—31no—
228C22×C8—31no—
229#127—24no—
230#127—24no—
231#127—22no—
232#127#6714yes—
233#125—62no—
234#125—62no—
235#121—28no—
236#121—216no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4#6714yes—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4#6712yes—
251C2×C4⋊C4#6711yescharacteristic
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C42—61no—
256C2×C42C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#107—62no—
259#106—62no—
260C4⋊C8—62no—
261#102—64no—
262#102—64no—
263#102—64no—
264#102—62no—
265#102—62no—
266#102—64no—
267#102—64no—
268#97—62no—
269#95—61no—
270#95—62no—
271#95—62no—
272C23×C3—24no—
273C23×C3—22no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—41no—
284C23×C3C22×D411yescharacteristic
285C3×Q8—28no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4—24no—
289C3×D4—24no—
290C3×D4—48no—
291C3×D4—416no—
292C3×D4—42no—
293C3×D4—42no—
294C3×D4—42no—
295C3×D4—84no—
296C3×D4—88no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—48no—
307C2×C3×C4—416no—
308C2×C3×C4—42no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4AGL(1,ℤ/8)14yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C42−1+412yes—
316C2×Dic3—24no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C3⋊C8—41no—
325C3⋊C8—41no—
326C3⋊C8—42no—
327C24—22no—
328C24—21no—
329#41—432no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4C2×#6714yes—
352C2×D4C2×#6712yes—
353C22×C4—28no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4C2×#6712yes—
363C22×C4C2×#6712yes—
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6711yescharacteristic
366C22×C4C23×D311yescharacteristic
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—61no—
373C2×C8—61no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—34no—
378C4⋊C4—62no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—124no—
384C4⋊C4—122no—
385C4⋊C4—122no—
386C4⋊C4—242no—
387C4⋊C4C2×#6712yes—
388C22⋊C4—28no—
389C22⋊C4—24no—
390C42—22no—
391C42—22no—
392C42—241no—
393C42C2×#6712yes—
394C42C2×#6712yes—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—42no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3C2×#12011yescharacteristic
408C22×C3#54812yes—
409C22×C3C2×AGL(1,ℤ/8)12yes—
410C22×C3#57611yescharacteristic
411C22×C3#57711yescharacteristic
412C3×C4—22no—
413C3×C4—24no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4#53211yescharacteristic
418C3×C4#53611yescharacteristic
419C3×C4#54712yes—
420Dic3—44no—
421Dic3—81no—
422Dic3—81no—
423C23—24no—
424C23—22no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—42no—
434C23—41no—
435C23C22×#6711yescharacteristic
436Q8—28no—
437D4—24no—
438D4—24no—
439D4—22no—
440D4—22no—
441D4—416no—
442D4—48no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—84no—
447D4—88no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—416no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—64no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4#93214yes—
469C2×C4#95312yes—
470C2×C4#95312yes—
471C2×C4D3×#4112yes—
472C2×C4#101012yes—
473C2×C4C22×#6711yescharacteristic
474C2×C4C22×#6711yescharacteristic
475C8—122no—
476C8—121no—
477C8—121no—
478C2×C3—82no—
479C2×C3—84no—
480C2×C3#243211yescharacteristic
481C2×C3#300311yescharacteristic
482C2×C3#301211yescharacteristic
483C2×C3#336612yes—
484C2×C3#336812yes—
485C22—44no—
486C22—42no—
487C22—44no—
488C22—42no—
489C22—44no—
490C22—42no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22C2×#93212yes—
498C22#583112yes—
499C22#583511yescharacteristic
500C22#583611yescharacteristic
501C22C2×#95311yescharacteristic
502C4—24no—
503C4—22no—
504C4—22no—
505C4—42no—
506C4—42no—
507C4—124no—
508C4—241no—
509C4—241no—
510C4#582412yes—
511C4#582511yescharacteristic
512C4#583211yescharacteristic
513C3#2565511yescharacteristic, complemented, Sylow
514C2—84no—
515C2—82no—
516C2#8358112yes—
517C2#8358712yes—
518C2#8403811yescharacteristic
519C2#8409311yescharacteristic
520C2#8410211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#550C2
socleC23×C3C22×D4
hypercenterC2×#550D3
hypocenterC3#25655
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C2×#5071C2semidirect
3. #82483C2semidirect
4. C2×#5009C2semidirect
5. #71762C2semidirect
6. #71762C2semidirect
7. #71761C2semidirect
8. #71635C2semidirect
9. #71609C2semidirect
10. #71607C2semidirect
11. #71606C2semidirect
202. C4⋊Dic3#41semidirect
513. C3#25655semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×#50712
#824832
C2×#50092
#717622
#717612
#716352
#716092
#716072
#716062
#256553