Finite Group Databasebeta

← #10849312

Subgroups — #10849312

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×#5071C211yesmaximal, characteristic
3#82483C211yesmaximal, characteristic, complemented
4C2×#5022C211yesmaximal, characteristic, complemented
5#71761C211yesmaximal, characteristic
6#71722C212yesmaximal, complemented
7#71722C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71631C211yesmaximal, characteristic, complemented
10#71607C212yesmaximal, complemented
11#71606C212yesmaximal, complemented
12#25658—31nomaximal, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124C2212yes—
18C2×C3×#123—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C2×#870C2211yescharacteristic
24#5230—21no—
25C2×#835C2212yes—
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5071—24no—
33#5022—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×#559D311yescharacteristic
47C2×#490—31no—
48C2×#479—31no—
49#2584—31no—
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1763—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4—22no—
63C3×#127—22no—
64C3×#127D414yes—
65C3×#124—28no—
66C3×#124—24no—
67C3×#124—24no—
68C3×#123—216no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yes—
73C3×C4×D4—28no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4D412yes—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—21no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97#870—22no—
98#870—22no—
99#835—22no—
100#835—22no—
101#835—24no—
102#832—24no—
103#832—24no—
104#832—24no—
105#832—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109#808—22no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113#559—24no—
114#559—24no—
115C2×#127C2×D311yescharacteristic
116C2×#125—31no—
117C2×#124—21no—
118C2×#124C2×D312yes—
119C2×#123—22no—
120C2×C4×Q8C2×D311yescharacteristic
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4C2×D312yes—
124#490—64no—
125#479—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8—31no—
129C2×#103—31no—
130C2×#103—31no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#97—31no—
137#398—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4—31no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—42no—
152C2×C3×Q8C2×D412yes—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—44no—
161C2×C3×D4—81no—
162C2×C3×D4—81no—
163C2×C3×D4C2×D414yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—28no—
167C22×C3×C4—82no—
168C22×C3×C4—82no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—22no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—28no—
184C3×C22⋊C4—28no—
185C3×C42—24no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190C4⋊Dic3—22no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3#4114yes—
196Dic3⋊C4—44no—
197Dic3⋊C4—42no—
198Dic3⋊C4—42no—
199Dic3⋊C4—82no—
200C4×Dic3—81no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C22×Q8#6711yescharacteristic
209C22×Q8C22×D311yescharacteristic
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×D312yes—
213C23×C4—22no—
214C22×C8—32no—
215C22×C8—31no—
216C22×C8—31no—
217#127—22no—
218#127#6714yes—
219#125—62no—
220#125—62no—
221#124—28no—
222#124—24no—
223#124—24no—
224#123—216no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8#6714yes—
229C4×D4—28no—
230C4×D4—24no—
231C4×D4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4#6711yescharacteristic
241C2×C4⋊C4#6712yes—
242C2×C4⋊C4C22×D311yescharacteristic
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42—21no—
247C2×C42—61no—
248C2×C42C22×D312yes—
249C2×C42C22×D311yescharacteristic
250#107—62no—
251#106—62no—
252C4⋊C8—62no—
253#103—64no—
254#103—62no—
255#103—62no—
256#102—64no—
257#102—64no—
258#102—64no—
259#102—64no—
260#97—62no—
261#95—61no—
262#95—62no—
263#95—62no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3C22×D411yescharacteristic
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—44no—
275C3×Q8—42no—
276C3×D4—24no—
277C3×D4—24no—
278C3×D4—48no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—84no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—88no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C42−1+412yes—
303C2×Dic3—24no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C3⋊C8—41no—
312C3⋊C8—41no—
313C3⋊C8—42no—
314C24—22no—
315C2×Q8—22no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—42no—
320C2×Q8C2×#6712yes—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—44no—
329C2×D4—81no—
330C2×D4—81no—
331C2×D4C2×#6714yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—28no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4C2×#6712yes—
341C22×C4C2×#6712yes—
342C22×C4C2×#6711yescharacteristic
343C22×C4C2×#6711yescharacteristic
344C22×C4C23×D311yescharacteristic
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—62no—
350C2×C8—61no—
351C2×C8—61no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—34no—
357C4⋊C4—62no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—124no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C4⋊C4—242no—
366C4⋊C4C2×#6712yes—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—28no—
370C42—24no—
371C42—22no—
372C42—22no—
373C42—241no—
374C42C2×#6712yes—
375C42C2×#6712yes—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3C2×#12011yescharacteristic
384C22×C3#54812yes—
385C22×C3#57512yes—
386C22×C3#57611yescharacteristic
387C22×C3#57711yescharacteristic
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4#53211yescharacteristic
395C3×C4#53611yescharacteristic
396C3×C4#54712yes—
397Dic3—44no—
398Dic3—81no—
399Dic3—81no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23C22×#6711yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—44no—
411Q8—42no—
412D4—24no—
413D4—24no—
414D4—48no—
415D4—42no—
416D4—42no—
417D4—84no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—64no—
433C2×C4—88no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4#95312yes—
442C2×C4#95312yes—
443C2×C4D3×#4112yes—
444C2×C4#101012yes—
445C2×C4C22×#6711yescharacteristic
446C2×C4C22×#6711yescharacteristic
447C8—122no—
448C8—121no—
449C8—121no—
450C2×C3—84no—
451C2×C3#243211yescharacteristic
452C2×C3#300511yescharacteristic
453C2×C3#301211yescharacteristic
454C2×C3#337012yes—
455C2×C3#337112yes—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22#582212yes—
464C22#583112yes—
465C22#583511yescharacteristic
466C22#583611yescharacteristic
467C22C2×#95311yescharacteristic
468C4—22no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—124no—
475C4—241no—
476C4—241no—
477C4#582412yes—
478C4#582511yescharacteristic
479C4#583211yescharacteristic
480C3#2565811yescharacteristic, complemented, Sylow
481C2—84no—
482C2#8371412yes—
483C2#8371812yes—
484C2#8408111yescharacteristic
485C2#8409311yescharacteristic
486C2#8410211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#559D3
hypocenterC3#25658
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
3. #82483C2semidirect
4. C2×#5022C2semidirect
6. #71722C2semidirect
7. #71722C2semidirect
8. #71635C2semidirect
9. #71631C2semidirect
10. #71607C2semidirect
11. #71606C2semidirect
480. C3#25658semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50712
#824832
C2×#50222
#717612
#717222
#716352
#716312
#716072
#716062
#256583