Finite Group Databasebeta

← #10900897

Subgroups — #10900897

Order 768.

1596 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 191 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×#534C211yesmaximal, characteristic
2#84047C212yesmaximal, complemented
3#82114C211yesmaximal, characteristic, complemented
4#72130C212yesmaximal, complemented
5#71850C214yesmaximal, complemented
6#71849C214yesmaximal, complemented
7#71542C211yesmaximal, characteristic, complemented
8#29635—31nomaximal, Sylow
9C23×C3×D4C2212yes—
10C3×#534—24no—
11C3×#534—22no—
12C3×#534C2214yes—
13C2×C3×#123—22no—
14C2×C3×#121—21no—
15C2×C3×#121—21no—
16C2×C3×#120—22no—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C2×#935C2214yescomplemented
20#5247—22no—
21#5247C2212yes—
22#5247C2211yescharacteristic
23#5222—24no—
24#5220C2214yescomplemented
25C2×#834C2218yescomplemented
26#5211—21no—
27#5210—22no—
28#5210—21no—
29#4952—22no—
30#4946C2212yescomplemented
31#4943C2212yescomplemented
32#4526C2214yescomplemented
33C2×#534D311yescharacteristic
34#2483—31no—
35#2405—32no—
36#2044—34no—
37#2022—34no—
38#1819—31no—
39#1781—32no—
40C25×C3D412yes—
41C25×C3C2311yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4—22no—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—48no—
49C22×C3×D4D414yes—
50C22×C3×D4C2314yes—
51C23×C3×C4D412yes—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×Dic3C2311yescharacteristic
55C23×Dic3C2311yescharacteristic
56C3×#123—28no—
57C3×#123—28no—
58C3×#121—24no—
59C3×#121—24no—
60C3×#121—24no—
61C3×#121—24no—
62C3×#120—28no—
63C3×#120—28no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4D412yes—
71C2×C3×C22⋊C4C2312yes—
72C2×#217—21no—
73C2×#217—21no—
74C2×#217—21no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—22no—
78C2×#217—24no—
79C2×#217—21no—
80C2×#217—24no—
81C2×#217—22no—
82C2×#217C2312yes—
83C2×#217C2314yes—
84#935—48no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—22no—
87#834—48no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—24no—
92C23×D4C2×D312yes—
93#534—24no—
94#534—22no—
95#534C2×D314yes—
96C2×#123—22no—
97C2×#123—34no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#120—22no—
101C22×C22⋊C4—31no—
102C22×C22⋊C4C2×D311yescharacteristic
103C2×#99—38no—
104#397—62no—
105#394—64no—
106#386—34no—
107#385—32no—
108#385—62no—
109#385—61no—
110#381—32no—
111#381—61no—
112#379—32no—
113#379—31no—
114#379—62no—
115#342—34no—
116C24×C3—24no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—44no—
120C24×C3—42no—
121C24×C3—44no—
122C24×C3—81no—
123C24×C3—82no—
124C24×C3C2×D412yes—
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D414yes—
127C24×C3C2×D412yes—
128C24×C3C2×D411yescharacteristic
129C24×C3C2411yescharacteristic
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—24no—
134C2×C3×D4—28no—
135C2×C3×D4—28no—
136C2×C3×D4—24no—
137C2×C3×D4—24no—
138C2×C3×D4—44no—
139C2×C3×D4—44no—
140C2×C3×D4—48no—
141C2×C3×D4—48no—
142C2×C3×D4—48no—
143C2×C3×D4—48no—
144C2×C3×D4—82no—
145C2×C3×D4—82no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—44no—
153C22×C3×C4—42no—
154C22×C3×C4—81no—
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—24no—
163C22×Dic3—22no—
164C22×Dic3—24no—
165C22×Dic3—81no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—24no—
168C3×C22⋊C4—28no—
169C3×C22⋊C4—28no—
170C3×C22⋊C4—24no—
171C3×C22⋊C4—24no—
172#217—24no—
173#217—28no—
174#217—48no—
175#217—42no—
176#217—44no—
177#217—42no—
178#217—41no—
179#217—41no—
180#217—81no—
181#217—82no—
182#217—84no—
183Dic3⋊C4—48no—
184Dic3⋊C4—82no—
185C25#6712yes—
186C25C22×D311yescharacteristic
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—48no—
194C22×D4#6714yes—
195C22×D4C22×D314yes—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4#6711yescharacteristic
199C23×C4#6712yes—
200C23×C4#6711yescharacteristic
201#123—28no—
202#123—28no—
203#123—128no—
204#121—24no—
205#121—24no—
206#121—24no—
207#121—24no—
208#120—28no—
209#120—28no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—62no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—34no—
219C2×C22⋊C4—32no—
220C2×C22⋊C4—62no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—62no—
223C2×C22⋊C4—61no—
224C2×C22⋊C4—64no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—64no—
230C2×C22⋊C4#6712yes—
231C2×C22⋊C4C22×D312yes—
232#99—128no—
233#95—62no—
234#95—62no—
235#95—62no—
236#95—64no—
237C23×C3—24no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—22no—
243C23×C3—24no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3—44no—
255C23×C3—44no—
256C23×C3—44no—
257C23×C3—44no—
258C23×C3—44no—
259C23×C3—44no—
260C23×C3—82no—
261C23×C3—82no—
262C23×C3—88no—
263C23×C3—816no—
264C23×C3#12012yes—
265C23×C3#12011yescharacteristic
266C23×C3#12011yescharacteristic
267C23×C3C22×D412yes—
268C23×C3C22×D411yescharacteristic
269C23×C3H(5,𝔽2)14yes—
270C3×D4—48no—
271C3×D4—48no—
272C3×D4—48no—
273C3×D4—48no—
274C3×D4—84no—
275C3×D4—84no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—81no—
288C2×C3×C4—81no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—44no—
293C2×Dic3—44no—
294C2×Dic3—44no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—44no—
298C2×Dic3—44no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—44no—
305C24—42no—
306C24—44no—
307C24—81no—
308C24—82no—
309C24C2×#6712yes—
310C24C2×#6711yescharacteristic
311C24C2×#6714yes—
312C24C2×#6712yes—
313C24C2×#6711yescharacteristic
314C24C23×D311yescharacteristic
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—28no—
318C2×D4—28no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—82no—
330C2×D4—82no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—64no—
340C22×C4—61no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—61no—
344C22×C4—64no—
345C22×C4—62no—
346C22×C4—81no—
347C22×C4—241no—
348C22×C4C2×#6714yes—
349C22×C4C2×#6712yes—
350C22×C4C2×#6712yes—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—128no—
354C4⋊C4—242no—
355C22⋊C4—28no—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—64no—
360C22⋊C4—68no—
361C22⋊C4—128no—
362C22⋊C4—122no—
363C22⋊C4—124no—
364C22⋊C4—122no—
365C22⋊C4—121no—
366C22⋊C4—121no—
367C22⋊C4—242no—
368C22⋊C4—244no—
369C22⋊C4—241no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—48no—
385C22×C3—48no—
386C22×C3—44no—
387C22×C3—44no—
388C22×C3—44no—
389C22×C3—84no—
390C22×C3—82no—
391C22×C3—84no—
392C22×C3—88no—
393C22×C3—88no—
394C22×C3#45812yes—
395C22×C3C2×#12011yescharacteristic
396C22×C3#53414yes—
397C22×C3#53512yes—
398C3×C4—44no—
399C3×C4—42no—
400C3×C4—42no—
401Dic3—84no—
402Dic3—82no—
403Dic3—81no—
404Dic3—81no—
405C23—22no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—82no—
429C23—82no—
430C23—88no—
431C23—816no—
432C23#95312yes—
433C23#95311yescharacteristic
434C23#95311yescharacteristic
435C23#100414yes—
436C23C22×#6712yes—
437C23C22×#6711yescharacteristic
438D4—48no—
439D4—48no—
440D4—48no—
441D4—48no—
442D4—84no—
443D4—84no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—122no—
458C2×C4—124no—
459C2×C4—124no—
460C2×C4—122no—
461C2×C4—124no—
462C2×C4—122no—
463C2×C4—124no—
464C2×C4—124no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C3—24no—
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—41no—
473C2×C3—42no—
474C2×C3—84no—
475C2×C3#261311yescharacteristic
476C2×C3C2×#45811yescharacteristic
477C2×C3#302611yescharacteristic
478C22—21no—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—21no—
489C22—22no—
490C22—44no—
491C22—48no—
492C22—44no—
493C22—48no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—84no—
498C22—82no—
499C22—84no—
500C22—88no—
501C22—88no—
502C22#524312yes—
503C22#580414yes—
504C22#580512yes—
505C22C2×#95311yescharacteristic
506C4—44no—
507C4—42no—
508C4—42no—
509C4—244no—
510C4—242no—
511C4—241no—
512C4—241no—
513C3#2963511yescharacteristic, complemented, Sylow
514C2—24no—
515C2—22no—
516C2—22no—
517C2—42no—
518C2—41no—
519C2—84no—
520C2#8405411yescharacteristic
521C2C2×#524311yescharacteristic
522C2#8411011yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
Fitting, prime radicalC2×C3×#534C2
socleC22×C3C2×#120
hypercenterC2×#534D3
hypocenterC3#29635
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
2. #84047C2semidirect
3. #82114C2semidirect
4. #72130C2semidirect
5. #71850C2semidirect
6. #71849C2semidirect
7. #71542C2semidirect
19. C2×#935C22semidirect
24. #5220C22semidirect
25. C2×#834C22semidirect
30. #4946C22semidirect
31. #4943C22semidirect
32. #4526C22semidirect
513. C3#29635semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5342
#840472
#821142
#721302
#718502
#718492
#715422
#296353