Finite Group Databasebeta

← #10828040

Subgroups — #10828040

Order 768.

1286 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3485C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic, complemented
3#82914C212yesmaximal, complemented
4#72140C212yesmaximal, complemented
5#72137C211yesmaximal, characteristic, complemented
6#71863C214yesmaximal, complemented
7#71712C212yesmaximal, complemented
8#71647C212yesmaximal, complemented
9#24025—31nomaximal, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C22×C3×#41C2211yescharacteristic
12C3×#547—22no—
13C3×#547—22no—
14C3×#546—22no—
15C3×#546—22no—
16C3×#536C2212yes—
17C3×#534—22no—
18C3×#525C2212yes—
19C2×C3×#123—22no—
20C3×#518C2211yescharacteristic
21C2×#926C2211yescharacteristic, complemented
22C22×C4⋊Dic3C2211yescharacteristic, complemented
23C2×#837C2214yes—
24#5232C2214yes—
25#5229C2212yescomplemented
26C2×#836C2212yescomplemented
27#5221C2212yes—
28#5220—22no—
29C2×#833C2212yes—
30#5212—22no—
31#5207C2212yescomplemented
32#5044—22no—
33#5043—22no—
34#5034—22no—
35#5033—22no—
36#4995C2214yescomplemented
37#4532C2214yescomplemented
38#3485D311yescharacteristic
39#3092—32no—
40#2311—31no—
41#2038—34no—
42#1897—32no—
43#1856—32no—
44#1792—32no—
45#1785—31no—
46C3×H(5,𝔽2)—24no—
47C3×H(5,𝔽2)—42no—
48C2×C3×#41—22no—
49C2×C3×#41—24no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—42no—
53C2×C3×#41C2312yes—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4D412yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—21no—
62C3×C4⋊Q8—22no—
63C3×#124—22no—
64C3×#123—24no—
65C3×#123—24no—
66C3×#123—24no—
67C3×#123—22no—
68C3×#123—24no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8D412yes—
72C3×C22⋊Q8D412yes—
73C3×#121—24no—
74C3×#121—24no—
75C3×#121D412yes—
76C3×#121D412yes—
77C3×#120—24no—
78C3×#120—22no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×#117C2312yes—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×#217—21no—
90C2×#217—24no—
91#926—42no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2312yes—
97C2×C4×Dic3C2312yes—
98C2×#208C2314yes—
99#837—44no—
100#836—42no—
101#835—24no—
102#833—24no—
103#832—24no—
104C3⋊#95—24no—
105#830—22no—
106#830—22no—
107#808—24no—
108#807—24no—
109C2×H(5,𝔽2)C2×D311yescharacteristic
110C22×#41C2×D311yescharacteristic
111#547—22no—
112#547—22no—
113#546—22no—
114#546—22no—
115#536C2×D312yes—
116#534—22no—
117#525C2×D312yes—
118C2×#123—22no—
119#518C2×D311yescharacteristic
120C2×#117—31no—
121C22×C4⋊C4—31no—
122#484—62no—
123#483—62no—
124#481—62no—
125#480—62no—
126#428—34no—
127C2×#104—34no—
128#417—34no—
129C2×#101—32no—
130C2×#100—32no—
131#407—32no—
132#391—32no—
133#390—32no—
134#386—62no—
135#380—62no—
136#339—34no—
137C24×C3—22no—
138C24×C3C2×D411yescharacteristic
139C24×C3C2×D412yes—
140C3×#41—24no—
141C3×#41—24no—
142C3×#41—48no—
143C3×#41—48no—
144C3×#41—48no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8C2×D412yes—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—42no—
158C2×C3×D4—48no—
159C2×C3×D4—44no—
160C2×C3×D4—44no—
161C2×C3×D4C2×D412yes—
162C2×C3×D4#4114yes—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—42no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—42no—
181C22×Dic3—82no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—22no—
196C3×C42—22no—
197#217—42no—
198#217—84no—
199C4⋊Dic3—24no—
200C4⋊Dic3—42no—
201C4⋊Dic3—41no—
202C4⋊Dic3—41no—
203C4×Dic3—24no—
204#208—24no—
205#208—24no—
206C2×C3⋊C8—24no—
207H(5,𝔽2)—24no—
208H(5,𝔽2)—42no—
209C2×#41—22no—
210C2×#41—24no—
211C2×#41—22no—
212C2×#41—22no—
213C2×#41—42no—
214C2×#41C22×D312yes—
215C22×Q8C22×D311yescharacteristic
216C22×D4—22no—
217C22×D4—22no—
218C22×D4#6712yes—
219C22×D4C22×D311yescharacteristic
220C23×C4—61no—
221C23×C4#6712yes—
222C23×C4C22×D311yescharacteristic
223C2×M16—34no—
224C4⋊Q8—22no—
225#124—22no—
226#123—24no—
227#123—24no—
228#123—24no—
229#123—22no—
230#123—24no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8#6712yes—
234C22⋊Q8#6712yes—
235#121—24no—
236#121—24no—
237#121#6712yes—
238#121#6712yes—
239#120—24no—
240#120—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245#117—122no—
246#117C22×D312yes—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—61no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—64no—
259C2×C42—32no—
260#107—64no—
261#106—64no—
262#104—124no—
263#103—64no—
264#102—64no—
265#101—122no—
266#100—64no—
267C22⋊C8—62no—
268C22⋊C8—62no—
269#95—64no—
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—44no—
280C23×C3—82no—
281C23×C3—82no—
282C23×C3#12012yes—
283C23×C3#12314yes—
284C23×C3C22×D411yescharacteristic
285C3×Q8—24no—
286C3×Q8—44no—
287C3×D4—28no—
288C3×D4—24no—
289C3×D4—48no—
290C3×D4—48no—
291C3×D4—42no—
292C3×D4—42no—
293C3×D4—82no—
294C3×D4—82no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—44no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4—44no—
309C2×C3×C4—82no—
310C2×C3×C4—82no—
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4#12011yescharacteristic
313C2×C3×C4C22×D412yes—
314C2×C3×C4C2×#4112yes—
315C2×C3×C4H(5,𝔽2)12yes—
316C2×Dic3—42no—
317C2×Dic3—44no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—84no—
322C3⋊C8—44no—
323C24—22no—
324C24C2×#6712yes—
325C24C2×#6711yescharacteristic
326#41—24no—
327#41—24no—
328#41—48no—
329#41—48no—
330#41—48no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8C2×#6712yes—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4#23714yes—
348C2×D4C2×#6712yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—122no—
362C22×C4—242no—
363C22×C4C2×#6712yes—
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6712yes—
366C22×C4C2×#6712yes—
367C22×C4C23×D311yescharacteristic
368M16—64no—
369M16—64no—
370C2×C8—64no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—64no—
376C4⋊C4—122no—
377C4⋊C4—121no—
378C4⋊C4—121no—
379C4⋊C4C2×#6712yes—
380C4⋊C4C2×#6712yes—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—122no—
390C22⋊C4—244no—
391C42—22no—
392C42—64no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—24no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—44no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—82no—
408C22×C3—164no—
409C22×C3C2×#12011yescharacteristic
410C22×C3C2×#12311yescharacteristic
411C22×C3#53411yescharacteristic
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4#54614yes—
419Dic3—82no—
420Dic3—82no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—82no—
432C23—82no—
433C23#93514yes—
434C23#95312yes—
435C23C22×#6711yescharacteristic
436Q8—24no—
437Q8—44no—
438D4—28no—
439D4—24no—
440D4—48no—
441D4—48no—
442D4—42no—
443D4—42no—
444D4—82no—
445D4—82no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—122no—
463C2×C4—124no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—244no—
468C2×C4#95311yescharacteristic
469C2×C4#95311yescharacteristic
470C2×C4C2×#23712yes—
471C2×C4#100412yes—
472C2×C4C22×#6712yes—
473C8—124no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—42no—
477C2×C3—44no—
478C2×C3—82no—
479C2×C3#240511yescharacteristic
480C2×C3#302111yescharacteristic
481C2×C3#302211yescharacteristic
482C22—21no—
483C22—21no—
484C22—22no—
485C22—24no—
486C22—22no—
487C22—22no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—82no—
497C22—164no—
498C22C2×#93511yescharacteristic
499C22#580411yescharacteristic
500C22C2×#95311yescharacteristic
501C4—21no—
502C4—21no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—242no—
508C4—242no—
509C4#580614yes—
510C3#2402511yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—42no—
514C2—44no—
515C2—82no—
516C2#8404711yescharacteristic
517C2#8412211yescharacteristic
518C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3485C2
socleC22×C3C2×#120
hypercenter#3485D3
hypocenterC3#24025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
2. #84087C2semidirect
3. #82914C2semidirect
4. #72140C2semidirect
5. #72137C2semidirect
6. #71863C2semidirect
7. #71712C2semidirect
8. #71647C2semidirect
21. C2×#926C22semidirect
22. C22×C4⋊Dic3C22semidirect
25. #5229C22semidirect
26. C2×#836C22semidirect
31. #5207C22semidirect
36. #4995C22semidirect
37. #4532C22semidirect
510. C3#24025semidirect

Maximal subgroups

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

subgroupindex
C3×#34852
#840872
#829142
#721402
#721372
#718632
#717122
#716472
#240253