Finite Group Databasebeta

← #413949

Subgroups — #413949

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2035C212yesmaximal
2C2×#1956C211yesmaximal, characteristic, complemented
3C2×#1894C212yesmaximal
4C2×#1753C211yesmaximal, characteristic
5#15195C211yesmaximal, characteristic
6#15178C211yesmaximal, characteristic
7#11371C211yesmaximal, characteristic, complemented
8#11122C212yesmaximal, complemented
9#11010C211yesmaximal, characteristic, complemented
10#10928C212yesmaximal, complemented
11#10861C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#391C2212yes—
24C2×#384C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2035—28no—
28#1956—24no—
29#1894—28no—
30#1753—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1444—22no—
37#1442—22no—
38#1439—22no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×Q8C2312yes—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—42no—
48C22×M16—42no—
49C22×M16D414yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8D414yes—
93C2×#97—22no—
94C2×#97—42no—
95C2×C4×C8—42no—
96C2×C4×C8D414yes—
97#391—24no—
98#391—42no—
99#384—22no—
100#384—22no—
101#380—24no—
102#380—24no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95—24no—
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—22no—
115#336—22no—
116C25C2×D411yescharacteristic
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8#4114yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C2×M16—28no—
150C2×M16—48no—
151C2×M16—44no—
152C2×M16—44no—
153C2×M16—48no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8#4112yes—
173C22×C8#4112yes—
174C4⋊Q8—28no—
175C4⋊Q8—48no—
176C4⋊Q8—44no—
177C22⋊Q8—216no—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—84no—
205C2×C4⋊C4#4114yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—84no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D412yes—
221C2×C42#4112yes—
222#106—48no—
223#103—48no—
224#103—48no—
225#103—48no—
226#103—48no—
227C22⋊C8—28no—
228#97—44no—
229C4×C8—44no—
230#95—24no—
231#95—48no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—84no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—24no—
243C24#12712yes—
244C24C22×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4#12114yes—
325C22×C4#12712yes—
326C22×C4#13714yes—
327C22×C4#13714yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)12yes—
337C22×C4H(5,𝔽2)12yes—
338M16—416no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8#12114yes—
358C2×C8#12114yes—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—416no—
362C4⋊C4—416no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—88no—
374C22⋊C4—416no—
375C22⋊C4—48no—
376C22⋊C4—88no—
377C42—22no—
378C42—22no—
379C42—24no—
380C42—41no—
381C42—41no—
382C42—42no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—24no—
387C23—24no—
388C23—44no—
389C23—44no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—42no—
396C23—44no—
397C23—44no—
398C23—42no—
399C23C2×#12111yescharacteristic
400C23C2×#12711yescharacteristic
401C23#53211yescharacteristic
402C23#54612yes—
403C23#54612yes—
404C23C2×#13712yes—
405C23C2×#13712yes—
406C23#57714yes—
407Q8—44no—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—82no—
412Q8—82no—
413Q8—82no—
414Q8—82no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—84no—
454C2×C4—82no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4#47018yes—
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12111yescharacteristic
460C2×C4#52512yes—
461C2×C4#52511yescharacteristic
462C2×C4#52912yes—
463C2×C4#53211yescharacteristic
464C2×C4#53211yescharacteristic
465C2×C4#53211yescharacteristic
466C2×C4#53412yes—
467C2×C4#53411yescharacteristic
468C2×C4#53511yescharacteristic
469C2×C4#53812yes—
470C2×C4#54612yes—
471C2×C4#54612yes—
472C2×C4#55112yes—
473C2×C4#55312yes—
474C2×C4#55312yes—
475C2×C4#55912yes—
476C8—24no—
477C8—24no—
478C8—42no—
479C8—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—44no—
485C22—84no—
486C22—82no—
487C22—82no—
488C22#253411yescharacteristic
489C22#253912yes—
490C22#254411yescharacteristic
491C22#255912yes—
492C22#285611yescharacteristic
493C22C2×#47012yes—
494C22#305712yes—
495C22#315211yescharacteristic
496C22#315211yescharacteristic
497C22#315412yes—
498C22#317412yes—
499C22#318512yes—
500C22#331614yes—
501C22#332014yes—
502C22#332914yes—
503C22#333114yes—
504C4—28no—
505C4—42no—
506C4—84no—
507C4—82no—
508C4—84no—
509C4—162no—
510C4—162no—
511C4#239211yescharacteristic
512C4#253511yescharacteristic
513C4#254311yescharacteristic
514C4#255712yes—
515C4#260212yes—
516C4#261711yescharacteristic
517C2—82no—
518C2#1516011yescharacteristic
519C2#2810812yes—
520C2#2836212yes—
521C2#2836712yes—
522C2#2860812yes—
523C2#2907014yes—
524C2#3317611yescharacteristic
525C2#3317811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1956C2semidirect
7. #11371C2semidirect
8. #11122C2semidirect
9. #11010C2semidirect
10. #10928C2semidirect
11. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20352
C2×#19562
C2×#18942
C2×#17532
#151952
#151782
#113712
#111222
#110102
#109282
#108612