Finite Group Databasebeta

← #4463992

Subgroups — #4463992

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17292C214yesmaximal, complemented
2#17262C214yesmaximal
3#17260C214yesmaximal
4#17245C214yesmaximal, complemented
5#15238C212yesmaximal, complemented
6#15216C211yesmaximal, characteristic
7#15200C212yesmaximal, complemented
8#15186C212yesmaximal, complemented
9#15180C212yesmaximal, complemented
10#15179C212yesmaximal
11#15178C211yesmaximal, characteristic
12#14258C212yesmaximal
13#13844C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×#117C2211yescharacteristic
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#389C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1445C2214yes—
29#1445C2214yes—
30#1444C2212yes—
31#1444C2214yes—
32#1443C2214yes—
33#1443C2214yes—
34#1442C2212yes—
35#1442C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2212yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4×#95C2212yes—
67C23×Q8C2312yes—
68C24×C4C2311yescharacteristic
69C2×C4⋊Q8—42no—
70C2×C4⋊Q8—44no—
71C2×#125—42no—
72C2×#125—44no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8—44no—
76C2×#117—44no—
77C2×#117—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2311yescharacteristic
93C22×C22⋊C4C2314yes—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2312yes—
98#391—44no—
99#391—44no—
100#391—44no—
101#391—44no—
102#391—44no—
103#389—44no—
104#380—44no—
105C4×C4⋊C4—44no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—42no—
108C2×#95C2312yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2311yescharacteristic
112C2×#95C2311yescharacteristic
113C2×#95C2312yes—
114C2×#95C2312yes—
115C2×#95C2312yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C25C2411yescharacteristic
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4#4114yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2414yes—
156C23×C4C2412yes—
157C23×C4C2412yes—
158C23×C4C2412yes—
159C23×C4C2412yes—
160C23×C4C2411yescharacteristic
161C23×C4C2412yes—
162C23×C4C2412yes—
163C23×C4C2411yescharacteristic
164C4⋊Q8—416no—
165#125—416no—
166C22⋊Q8—432no—
167#117—416no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224#95—22no—
225#95—22no—
226#95—24no—
227#95—28no—
228#95—28no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24C2×#4112yes—
264C24C2511yescharacteristic
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—48no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—216no—
289C22×C4—28no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4H(5,𝔽2)14yes—
352C22×C4H(5,𝔽2)14yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C42−1+414yes—
355C22×C42−1+414yes—
356C22×C42−1+412yes—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—416no—
360C4⋊C4—48no—
361C4⋊C4—416no—
362C4⋊C4—416no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—88no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—416no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—88no—
373C42—48no—
374C42—44no—
375C42—48no—
376C42—44no—
377C23—24no—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23C22×#4111yescharacteristic
399C23C2×H(5,𝔽2)12yes—
400C23C2×H(5,𝔽2)12yes—
401C23C2×H(5,𝔽2)11yescharacteristic
402C23C2×2−1+411yescharacteristic
403C23C2×2−1+412yes—
404C23C2×2−1+412yes—
405C23#58512yes—
406C23#58512yes—
407Q8—816no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—416no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4#53818yes—
485C2×C4#53914yes—
486C2×C4#56014yes—
487C2×C4H(3,𝔽4)14yes—
488C2×C4#56218yes—
489C22—42no—
490C22—42no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22C2×#53812yes—
499C22C2×#53911yescharacteristic
500C22#349512yes—
501C22#350911yescharacteristic
502C22#351211yescharacteristic
503C22C2×#56011yescharacteristic
504C22C2×H(3,𝔽4)11yescharacteristic
505C22C2×#56212yes—
506C22#353812yes—
507C22#354112yes—
508C22#354112yes—
509C22#354712yes—
510C22#354712yes—
511C22#355112yes—
512C22#355212yes—
513C22#355412yes—
514C22#355412yes—
515C22#355512yes—
516C22#355611yescharacteristic
517C22#355711yescharacteristic
518C22#356212yes—
519C4—44no—
520C4—44no—
521C4—42no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—84no—
526C4—84no—
527C4—82no—
528C4—84no—
529C4—84no—
530C4#282418yes—
531C2—82no—
532C2C2×#282411yescharacteristic
533C2#3738211yescharacteristic
534C2#3739411yescharacteristic
535C2#3739812yes—
536C2#3740212yes—
537C2#5300512yes—
538C2#5301312yes—
539C2#5304712yes—
540C2#5313812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17292C2semidirect
4. #17245C2semidirect
5. #15238C2semidirect
7. #15200C2semidirect
8. #15186C2semidirect
9. #15180C2semidirect

Maximal subgroups

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

subgroupindex
#172922
#172622
#172602
#172452
#152382
#152162
#152002
#151862
#151802
#151792
#151782
#142582
#138442