Finite Group Databasebeta

← #10769310

Subgroups — #10769310

Order 768.

1172 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5483C211yesmaximal, characteristic, complemented
2#82115C211yesmaximal, characteristic
3#78642C212yesmaximal, complemented
4C3×#1462C211yesmaximal, characteristic, complemented
5#69980C212yesmaximal, complemented
6#68117C214yesmaximal, complemented
7#68114C214yesmaximal, complemented
8#18318—31nomaximal, Sylow
9C22×C3×C22⋊C4C2211yescharacteristic
10C2×#953—21no—
11C22×#217—21no—
12C2×#930C2211yescharacteristic
13C2×#929—21no—
14C2×#924—21no—
15C2×#924C2212yes—
16#5483—22no—
17#5483—24no—
18#5483C2214yes—
19C2×#878—22no—
20C2×#878—22no—
21C3×#407C2212yes—
22C4×#217C2211yescharacteristic
23#5134—24no—
24#4960C2212yes—
25#4957—22no—
26#4949C2212yes—
27#4738—22no—
28#4736—22no—
29#4731C2214yescomplemented
30#4722C2214yescomplemented
31C3×#323C2214yes—
32#4467C2218yescomplemented
33C2×#543—31no—
34#3193—32no—
35#2539—31no—
36#1614—34no—
37#1596—34no—
38#1508—32no—
39#1462D311yescharacteristic, complemented
40C25×C3D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—22no—
45C23×Dic3—21no—
46C23×Dic3—21no—
47C22×D12C2311yescharacteristic
48C22×Dic6—21no—
49C22×Dic6D412yescomplemented
50C2×C3×M16—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4D412yes—
54C2×C3×C22⋊C4C2312yes—
55#953—44no—
56C2×#217—22no—
57C2×#217—24no—
58C2×#217—21no—
59C2×#217—24no—
60C2×#217—22no—
61C2×#217—22no—
62#930—22no—
63#930—22no—
64#929—44no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D412yescomplemented
69C2×C4⋊Dic3C2311yescharacteristic
70#924—24no—
71#924—22no—
72#924—22no—
73#924—24no—
74#924D414yescomplemented
75#924C2314yes—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3—21no—
80C2×#204—24no—
81#878—28no—
82#878—28no—
83#878—28no—
84#878—28no—
85C3×C22⋊C8—22no—
86C3×C22⋊C8C2314yes—
87C3⋊#95C2312yes—
88#821—24no—
89#818—24no—
90#816—24no—
91#543—34no—
92#543—64no—
93#543—62no—
94C2×#123—61no—
95C2×C22⋊Q8—32no—
96C2×C22⋊Q8—62no—
97C2×C22⋊Q8—62no—
98C2×C22⋊Q8—61no—
99C2×#121—31no—
100C2×#120—61no—
101C22×C22⋊C4—61no—
102C22×C22⋊C4C2×D311yescharacteristic
103#484—34no—
104#480—62no—
105#468—64no—
106#450—34no—
107#448—62no—
108#407C2×D312yes—
109#399—32no—
110#392—32no—
111#390—62no—
112C4×C22⋊C4—31no—
113#328—38no—
114#323C2×D314yes—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—24no—
118C24×C3—42no—
119C24×C3—81no—
120C24×C3C2×D411yescharacteristic
121C24×C3C2×D412yes—
122C23×D3—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—42no—
126C22×C3×C4—81no—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2411yescharacteristic
132C2×#67—44no—
133C2×#67—42no—
134C2×#67—42no—
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—44no—
138C2×#67—82no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—24no—
143C22×Dic3—24no—
144C22×Dic3—21no—
145C22×Dic3—82no—
146C22×Dic3—82no—
147C2×D12—22no—
148C2×D12—81no—
149C2×Dic6—24no—
150C2×Dic6—22no—
151C2×Dic6—82no—
152C2×Dic6—82no—
153C2×Dic6C2×D414yes—
154C3×M16—82no—
155C2×C3×C8—24no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158#217—28no—
159#217—24no—
160#217—24no—
161#217—24no—
162#217—44no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—42no—
165C4⋊Dic3—24no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3C2×D414yes—
169Dic3⋊C4—28no—
170Dic3⋊C4—24no—
171Dic3⋊C4—24no—
172Dic3⋊C4—44no—
173C4×Dic3—81no—
174#204—84no—
175C25D1211yescharacteristic
176C22×Q8—32no—
177C22×Q8—61no—
178C22×D4—31no—
179C22×D4—61no—
180C22×D4—62no—
181C23×C4—61no—
182C23×C4—61no—
183C23×C4D1211yescharacteristic
184C23×C4C22×D311yescharacteristic
185C2×SD16—64no—
186C2×M16—22no—
187#123—124no—
188C22⋊Q8—34no—
189C22⋊Q8—34no—
190C22⋊Q8—68no—
191C22⋊Q8—68no—
192C22⋊Q8—68no—
193C22⋊Q8—68no—
194C22⋊Q8—64no—
195C22⋊Q8—62no—
196C22⋊Q8—64no—
197C22⋊Q8—62no—
198#121—62no—
199#121—62no—
200#120—124no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4—62no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—64no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—62no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4D1212yes—
218C2×C22⋊C4C22×D312yes—
219C2×C42—61no—
220#107—64no—
221#103—64no—
222#102—64no—
223C22⋊C8—22no—
224C22⋊C8C22×D314yes—
225#95—32no—
226C23×C3—24no—
227C23×C3—24no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—81no—
238C23×C3—81no—
239C23×C3—88no—
240C23×C3#12011yescharacteristic
241C23×C3#12011yescharacteristic
242C23×C3#13714yes—
243C23×C3C22×D411yescharacteristic
244C22×D3—41no—
245C22×D3—42no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—42no—
249C2×C3×C4—22no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—81no—
257C2×C3×C4#12012yes—
258C2×C3×C4C22×D412yes—
259#67—88no—
260#67—82no—
261#67—82no—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—42no—
266C2×Dic3—44no—
267C2×Dic3—42no—
268C2×Dic3—44no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—84no—
273C2×Dic3—84no—
274D12—44no—
275Dic6—44no—
276Dic6—82no—
277Dic6—84no—
278C3×C8—44no—
279C24—21no—
280C24—21no—
281C24—24no—
282C24—42no—
283C24—61no—
284C24—81no—
285C24C2×D1211yescharacteristic
286C24C2×D1212yes—
287C2×Q8—34no—
288C2×Q8—64no—
289C2×Q8—62no—
290C2×Q8—242no—
291C2×Q8—242no—
292C2×D4—62no—
293C2×D4—124no—
294C2×D4—122no—
295C2×D4—122no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—124no—
299C2×D4—241no—
300C2×D4—242no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—64no—
308C22×C4—64no—
309C22×C4—61no—
310C22×C4—81no—
311C22×C4—242no—
312C22×C4—242no—
313C22×C4C2×D1212yes—
314C22×C4C2×D1211yescharacteristic
315C22×C4D3×D412yes—
316C22×C4D3×D412yes—
317C22×C4C23×D311yescharacteristic
318SD16—244no—
319M16—82no—
320C2×C8—24no—
321C4⋊C4—34no—
322C4⋊C4—68no—
323C4⋊C4—64no—
324C4⋊C4—62no—
325C4⋊C4—64no—
326C4⋊C4—64no—
327C4⋊C4—62no—
328C4⋊C4—124no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—68no—
332C22⋊C4—64no—
333C22⋊C4—64no—
334C22⋊C4—64no—
335C22⋊C4—124no—
336C22⋊C4—124no—
337C22⋊C4—122no—
338C42—241no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—24no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—48no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—82no—
353C22×C3—84no—
354C22×C3#45312yes—
355C22×C3C2×#12011yescharacteristic
356C22×C3C2×#13712yes—
357C2×D3—82no—
358C2×D3—82no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—42no—
365Dic3—44no—
366Dic3—44no—
367Dic3—44no—
368Dic3—81no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—81no—
381C23—81no—
382C23—88no—
383C23—121no—
384C23—121no—
385C23—122no—
386C23—121no—
387C23—122no—
388C23#88211yescharacteristic
389C23#88211yescharacteristic
390C23#90914yes—
391C23C22×D1211yescharacteristic
392Q8—124no—
393Q8—242no—
394Q8—244no—
395D4—124no—
396D4—248no—
397D4—242no—
398D4—242no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—64no—
407C2×C4—64no—
408C2×C4—64no—
409C2×C4—81no—
410C2×C4—122no—
411C2×C4—124no—
412C2×C4—122no—
413C2×C4—124no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—244no—
418C2×C4—244no—
419C2×C4#88212yes—
420C2×C4C2×D3×D412yes—
421C8—44no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—41no—
425C2×C3—44no—
426C2×C3#300411yescharacteristic
427C2×C3C2×#45311yescharacteristic
428C2×C3#302611yescharacteristic
429D3—161no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—21no—
437C22—21no—
438C22—44no—
439C22—44no—
440C22—48no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22—84no—
445C22—242no—
446C22—242no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22#474712yes—
451C22C2×#88211yescharacteristic
452C22C2×#90912yes—
453C4—22no—
454C4—22no—
455C4—42no—
456C4—124no—
457C4—124no—
458C4—124no—
459C4—241no—
460C3#1831811yescharacteristic, complemented, Sylow
461C2—22no—
462C2—22no—
463C2—41no—
464C2—44no—
465C2—481no—
466C2#7864511yescharacteristic
467C2#7868611yescharacteristic
468C2C2×#474711yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1462C2
socleC22×C3C2×#120
prime radical#82115C2
hypercenter#1462D3
hypocenterC3#18318
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
1. C2×#5483C2semidirect
3. #78642C2semidirect
4. C3×#1462C2semidirect
5. #69980C2semidirect
6. #68117C2semidirect
7. #68114C2semidirect
29. #4731C22semidirect
30. #4722C22semidirect
32. #4467C22semidirect
39. #1462D3semidirect
49. C22×Dic6D4semidirect
68. C2×C4⋊Dic3D4semidirect
74. #924D4semidirect
460. C3#18318semidirect

Maximal subgroups

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

subgroupindex
C2×#54832
#821152
#786422
C3×#14622
#699802
#681172
#681142
#183183