Finite Group Databasebeta

← #10945051

Subgroups — #10945051

Order 768.

1216 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×#5516C211yesmaximal, characteristic
2C3×#2625C211yesmaximal, characteristic
3C2×#5050C211yesmaximal, characteristic
4C2×#5045C211yesmaximal, characteristic
5#71584C211yesmaximal, characteristic
6#69690C212yesmaximal, complemented
7#69690C212yesmaximal, complemented
8#69632C212yesmaximal, complemented
9#69631C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic
11#69503C211yesmaximal, characteristic
12#33019—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#216C2212yes—
17C2×#926C2211yescharacteristic
18#5516—24no—
19#5516—24no—
20C2×#887—21no—
21C2×#887—21no—
22C2×#877—22no—
23C2×C4×Dic6C2212yes—
24C2×C4×Dic6C2212yes—
25C3×#400—21no—
26C3×#398—21no—
27C3×#397—22no—
28C3×#391C2212yes—
29C3×#389C2212yes—
30C3×#380C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5050—24no—
36#5045—24no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#519—31no—
47C2×#485—31no—
48C2×#451—31no—
49#2625D311yescharacteristic
50#1878—32no—
51#1878—32no—
52#1865—32no—
53#1863—32no—
54#1825—31no—
55#1818—31no—
56#1787—31no—
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—21no—
60C22×Dic6—21no—
61C22×Dic6—22no—
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6D414yes—
65C22×Dic6C2312yes—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×#117—22no—
69C3×#117—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4D412yes—
79C2×C3×C42C2312yes—
80C2×#217—21no—
81C2×#216—24no—
82C2×#216—24no—
83C2×#216—24no—
84#926—22no—
85#926D414yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—21no—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96#887—28no—
97#887—28no—
98#877—216no—
99C4×Dic6—24no—
100C4×Dic6—24no—
101C4×Dic6—24no—
102C4×Dic6—24no—
103C3×#95—21no—
104C3×#95—22no—
105C3×#95—22no—
106C3×#95—22no—
107#835—22no—
108#835—22no—
109#830—22no—
110#830—22no—
111#810—24no—
112#810—24no—
113#810—24no—
114#808—22no—
115#807—22no—
116C23×Q8—31no—
117C22×Q16—32no—
118C2×C22⋊Q8C2×D311yescharacteristic
119#519—64no—
120#519—64no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—61no—
124C2×C4×Q8—61no—
125C2×#117—31no—
126C2×#117—62no—
127C2×#117C2×D311yescharacteristic
128#485—64no—
129#451—64no—
130C2×#107—31no—
131C2×#106—31no—
132C2×#103—31no—
133C2×#103—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8—31no—
139#400—21no—
140#398—21no—
141#397—22no—
142#391C2×D312yes—
143#389C2×D312yes—
144#380C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—28no—
166C22×Dic3—24no—
167C22×Dic3—22no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C2×Dic6—28no—
171C2×Dic6—28no—
172C2×Dic6—28no—
173C2×Dic6—24no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6—216no—
178C2×Dic6C2×D414yes—
179C2×Dic6#4114yes—
180C2×Dic6#4114yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—82no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—42no—
191C3×C22⋊C4—81no—
192C3×C42—22no—
193C3×C42—22no—
194#217—24no—
195#216—48no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3C2×D412yes—
199C4⋊Dic3C2×D412yes—
200Dic3⋊C4—28no—
201Dic3⋊C4—28no—
202C4×Dic3—28no—
203C4×Dic3—28no—
204C4×Dic3—24no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C22×Q8—34no—
213C22×Q8—32no—
214C22×Q8—64no—
215C22×Q8—64no—
216C22×Q8—62no—
217C22×Q8—61no—
218C22×Q8C22×D311yescharacteristic
219C23×C4—61no—
220C23×C4C22×D311yescharacteristic
221C2×Q16—64no—
222C2×Q16—64no—
223C2×Q16—64no—
224C22×C8—32no—
225C22×C8—31no—
226C22×C8—31no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C4×Q8—64no—
230C4×Q8—64no—
231C4×Q8—64no—
232C4×Q8—64no—
233C4×Q8—68no—
234C4×Q8—68no—
235#117—22no—
236#117—22no—
237#117—34no—
238#117—616no—
239#117—62no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4#6711yescharacteristic
247C2×C4⋊C4#6711yescharacteristic
248C2×C4⋊C4C22×D311yescharacteristic
249C2×C4⋊C4C22×D312yes—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4#6712yes—
254C2×C42—61no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42C22×D312yes—
258#107—62no—
259#106—62no—
260#103—64no—
261#103—64no—
262#103—64no—
263#103—62no—
264#103—62no—
265C22⋊C8—62no—
266C22⋊C8—62no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271C23×C3—22no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—21no—
275C23×C3#12012yes—
276C23×C3C22×D411yescharacteristic
277C3×Q8—42no—
278C3×Q8—41no—
279C3×Q8—41no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4#12012yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4112yes—
301C2×C3×C4C2×#4112yes—
302C2×Dic3—216no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311Dic6—216no—
312Dic6—24no—
313Dic6—24no—
314Dic6—24no—
315Dic6#12318yes—
316C3⋊C8—42no—
317C3⋊C8—41no—
318C3⋊C8—41no—
319C24C2×#6711yescharacteristic
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—34no—
325C2×Q8—34no—
326C2×Q8—34no—
327C2×Q8—616no—
328C2×Q8—68no—
329C2×Q8—68no—
330C2×Q8—68no—
331C2×Q8—64no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—68no—
341C22×C4—64no—
342C22×C4—62no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4D3×D412yes—
346C22×C4D3×D412yes—
347C22×C4C2×#6712yes—
348C22×C4C2×#6712yes—
349C22×C4C2×#6711yescharacteristic
350C22×C4C23×D311yescharacteristic
351Q16—128no—
352C2×C8—61no—
353C2×C8—61no—
354C2×C8—62no—
355C2×C8—62no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—62no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—32no—
366C4⋊C4—32no—
367C4⋊C4—42no—
368C4⋊C4—68no—
369C4⋊C4—68no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—82no—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—64no—
376C22⋊C4—81no—
377C42—22no—
378C42—22no—
379C42—68no—
380C42—68no—
381C42—64no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3C2×#12011yescharacteristic
388C22×C3#52512yes—
389C22×C3#57711yescharacteristic
390C22×C3#57812yes—
391C22×C3#57811yescharacteristic
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—81no—
396C3×C4—82no—
397C3×C4—81no—
398C3×C4C2×#12312yes—
399C3×C4#52511yescharacteristic
400C3×C4#52511yescharacteristic
401Dic3—28no—
402Dic3—42no—
403Dic3—42no—
404C23—22no—
405C23—21no—
406C23—22no—
407C23—21no—
408C23#93712yes—
409C23C22×#6711yescharacteristic
410Q8—38no—
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—616no—
415Q8—64no—
416Q8—64no—
417Q8—64no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—616no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—121no—
442C2×C4—122no—
443C2×C4—121no—
444C2×C4#93712yes—
445C2×C4C2×D3×D411yescharacteristic
446C2×C4C2×D3×D411yescharacteristic
447C2×C4D3×#4112yes—
448C2×C4D3×#4112yes—
449C8—122no—
450C8—121no—
451C8—121no—
452C2×C3—42no—
453C2×C3#284911yescharacteristic
454C2×C3#301111yescharacteristic
455C2×C3#301311yescharacteristic
456C2×C3#312112yes—
457C2×C3#312712yes—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—42no—
462C22—42no—
463C22C2×#93711yescharacteristic
464C22#582812yes—
465C22#583611yescharacteristic
466C22#583812yes—
467C22#583811yescharacteristic
468C4—22no—
469C4—42no—
470C4—42no—
471C4—68no—
472C4—82no—
473C4—81no—
474C4—81no—
475C4—122no—
476C4—122no—
477C4D3×#12312yes—
478C4#565611yescharacteristic
479C4#565611yescharacteristic
480C3#3301911yescharacteristic, complemented, Sylow
481C2—42no—
482C2#8291211yescharacteristic
483C2#8341111yescharacteristic
484C2#8342711yescharacteristic
485C2#8345112yes—
486C2#8346712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2625C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2625D3
hypocenterC3#33019
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
6. #69690C2semidirect
7. #69690C2semidirect
8. #69632C2semidirect
9. #69631C2semidirect
480. C3#33019semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
C3×#26252
C2×#50502
C2×#50452
#715842
#696902
#696322
#696312
#695252
#695032
#330193