Finite Group Databasebeta

← #10945077

Subgroups — #10945077

Order 768.

1528 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1528 conjugacy classes of subgroups — too many to draw. Only its 223 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×C3×#518C211yesmaximal, characteristic
2#84047C211yesmaximal, characteristic
3#82114C212yesmaximal, complemented
4#72130C211yesmaximal, characteristic
5#71657C214yesmaximal, complemented
6#71655C214yesmaximal, complemented
7#71542C212yesmaximal, complemented
8#33022—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C3×#518—22no—
11C3×#518—24no—
12C3×#518C2214yes—
13C2×C3×C4×D4—22no—
14C2×C3×C4×D4—22no—
15C2×C3×#117—21no—
16C22×C3×C22⋊C4—21no—
17C22×C3×C22⋊C4C2212yes—
18C22×#217C2211yescharacteristic
19C2×#935C2212yes—
20#5247—21no—
21#5247C2211yescharacteristic
22#5247C2212yes—
23#5222—22no—
24#5220C2212yes—
25C2×#834C2214yes—
26#5211—22no—
27#5210—21no—
28#5210—22no—
29#4952—24no—
30#4946C2214yescomplemented
31#4943C2214yescomplemented
32#4526C2218yescomplemented
33C2×#518D311yescharacteristic
34#2483—32no—
35#2405—31no—
36#1890—34no—
37#1867—34no—
38#1819—32no—
39#1781—31no—
40C25×C3D411yescharacteristic
41C25×C3C2311yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4—22no—
44C22×C3×D4—28no—
45C22×C3×D4D412yes—
46C22×C3×D4C2312yes—
47C23×C3×C4—21no—
48C23×C3×C4—21no—
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D412yes—
51C23×C3×C4D412yes—
52C23×Dic3C2311yescharacteristic
53C23×Dic3C2311yescharacteristic
54C3×C4×D4—28no—
55C3×C4×D4—28no—
56C3×C4×D4—28no—
57C3×C4×D4—28no—
58C3×#117—24no—
59C3×#117—24no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—24no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4D414yes—
70C2×C3×C22⋊C4C2314yes—
71C2×C3×C42—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—21no—
76C2×#217—21no—
77C2×#217—21no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81C2×#217—21no—
82C2×#217C2314yes—
83C2×#217C2312yes—
84#935—28no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—24no—
87#834—28no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—22no—
92C23×D4C2×D311yescharacteristic
93C2×#123—32no—
94#518—22no—
95#518—24no—
96#518C2×D314yes—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×#117—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—31no—
102C22×C22⋊C4C2×D312yes—
103C2×#99—34no—
104#397—64no—
105#394—62no—
106#386—32no—
107#385—34no—
108#385—61no—
109#385—62no—
110#381—34no—
111#381—62no—
112#379—31no—
113#379—32no—
114#379—61no—
115#342—38no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—24no—
119C24×C3—24no—
120C24×C3—42no—
121C24×C3—42no—
122C24×C3—81no—
123C24×C3—81no—
124C24×C3C2×D411yescharacteristic
125C24×C3C2×D412yes—
126C24×C3C2×D411yescharacteristic
127C24×C3C2×D412yes—
128C24×C3C2×D412yes—
129C24×C3C2411yescharacteristic
130C2×C3×D4—216no—
131C2×C3×D4—216no—
132C2×C3×D4—24no—
133C2×C3×D4—24no—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4#4118yes—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—44no—
148C22×C3×C4—82no—
149C22×C3×C4—82no—
150C22×C3×C4—82no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D414yes—
153C22×C3×C4C2×D414yes—
154C22×Dic3—22no—
155C22×Dic3—24no—
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—24no—
161C22×Dic3—81no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—24no—
164C3×C22⋊C4—28no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—24no—
169C3×C42—28no—
170#217—28no—
171#217—24no—
172#217—44no—
173#217—42no—
174#217—41no—
175#217—42no—
176#217—41no—
177#217—44no—
178#217—82no—
179#217—81no—
180#217—82no—
181Dic3⋊C4—44no—
182Dic3⋊C4—84no—
183C25#6711yescharacteristic
184C25C22×D311yescharacteristic
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—28no—
188C22×D4#6712yes—
189C22×D4C22×D312yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—31no—
193C23×C4—31no—
194C23×C4#6711yescharacteristic
195C23×C4#6712yes—
196C23×C4#6712yes—
197#123—68no—
198C4×D4—28no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202#117—24no—
203#117—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4—64no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—32no—
216C2×C22⋊C4—34no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—62no—
221C2×C22⋊C4—62no—
222C2×C22⋊C4—61no—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4#6714yes—
228C2×C22⋊C4C22×D314yes—
229C2×C42—22no—
230#99—68no—
231#95—64no—
232#95—64no—
233#95—64no—
234#95—62no—
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—24no—
240C23×C3—24no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—24no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—81no—
258C23×C3—81no—
259C23×C3—88no—
260C23×C3—88no—
261C23×C3#12011yescharacteristic
262C23×C3#12012yes—
263C23×C3#12011yescharacteristic
264C23×C3#12314yes—
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D412yes—
267C23×C3C2×#4114yes—
268C3×D4—216no—
269C3×D4—216no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—82no—
281C2×C3×C4—82no—
282C2×C3×C4—84no—
283C2×C3×C4—84no—
284C2×C3×C4#12314yes—
285C2×Dic3—44no—
286C2×Dic3—44no—
287C2×Dic3—44no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—44no—
296C2×Dic3—44no—
297C24—21no—
298C24—24no—
299C24—21no—
300C24—24no—
301C24—42no—
302C24—42no—
303C24—81no—
304C24—81no—
305C24C2×#6711yescharacteristic
306C24C2×#6712yes—
307C24C2×#6712yes—
308C24C2×#6711yescharacteristic
309C24C2×#6712yes—
310C24C23×D311yescharacteristic
311C2×D4—216no—
312C2×D4—216no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4#23518yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—44no—
329C22×C4—64no—
330C22×C4—62no—
331C22×C4—64no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—61no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—241no—
340C22×C4C2×#6712yes—
341C22×C4C2×#6714yes—
342C22×C4C2×#6714yes—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—124no—
346C4⋊C4—244no—
347C22⋊C4—28no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—68no—
353C22⋊C4—64no—
354C22⋊C4—124no—
355C22⋊C4—122no—
356C22⋊C4—124no—
357C22⋊C4—121no—
358C22⋊C4—122no—
359C22⋊C4—121no—
360C22⋊C4—241no—
361C22⋊C4—242no—
362C22⋊C4—242no—
363C42—28no—
364C22×C3—21no—
365C22×C3—22no—
366C22×C3—21no—
367C22×C3—22no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—24no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—21no—
377C22×C3—44no—
378C22×C3—48no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—48no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—82no—
385C22×C3—82no—
386C22×C3—84no—
387C22×C3—84no—
388C22×C3#45812yes—
389C22×C3C2×#12011yescharacteristic
390C22×C3C2×#12312yes—
391C22×C3#52514yes—
392C3×C4—24no—
393C3×C4—44no—
394C3×C4—44no—
395Dic3—82no—
396Dic3—84no—
397Dic3—81no—
398Dic3—81no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—41no—
415C23—42no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—81no—
422C23—81no—
423C23—88no—
424C23—88no—
425C23#92914yes—
426C23#95311yescharacteristic
427C23#95312yes—
428C23#95311yescharacteristic
429C23C2×#23514yes—
430C23C22×#6711yescharacteristic
431C23C22×#6712yes—
432D4—216no—
433D4—216no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—124no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—124no—
452C2×C4—122no—
453C2×C4—124no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—124no—
457C2×C4—124no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4#92914yes—
461C2×C3—22no—
462C2×C3—24no—
463C2×C3—22no—
464C2×C3—42no—
465C2×C3—41no—
466C2×C3—44no—
467C2×C3#284911yescharacteristic
468C2×C3C2×#45811yescharacteristic
469C2×C3#302611yescharacteristic
470C22—21no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22—22no—
475C22—21no—
476C22—21no—
477C22—24no—
478C22—22no—
479C22—22no—
480C22—21no—
481C22—22no—
482C22—21no—
483C22—44no—
484C22—44no—
485C22—48no—
486C22—44no—
487C22—48no—
488C22—42no—
489C22—42no—
490C22—82no—
491C22—82no—
492C22—84no—
493C22—84no—
494C22#524312yes—
495C22C2×#92912yes—
496C22#579114yes—
497C22C2×#95311yescharacteristic
498C4—24no—
499C4—44no—
500C4—44no—
501C4—242no—
502C4—244no—
503C4—241no—
504C4—241no—
505C3#3302211yescharacteristic, complemented, Sylow
506C2—22no—
507C2—24no—
508C2—22no—
509C2—41no—
510C2—42no—
511C2—44no—
512C2#8403311yescharacteristic
513C2C2×#524311yescharacteristic
514C2#8411011yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#518C2
socleC22×C3C2×#120
prime radicalC23×C3×D4C22
hypercenterC2×#518D3
hypocenterC3#33022
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
3. #82114C2semidirect
5. #71657C2semidirect
6. #71655C2semidirect
7. #71542C2semidirect
30. #4946C22semidirect
31. #4943C22semidirect
32. #4526C22semidirect
505. C3#33022semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5182
#840472
#821142
#721302
#716572
#716552
#715422
#330223