Finite Group Databasebeta

← #428412

Subgroups — #428412

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2035C212yesmaximal
4C2×#1854C212yesmaximal
5C2×#1847C211yesmaximal, characteristic, complemented
6#15195C211yesmaximal, characteristic
7#11372C211yesmaximal, characteristic, complemented
8#11323C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#10928C211yesmaximal, characteristic, complemented
11#10922C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21#2499—24no—
22#2499—24no—
23C2×#397—22no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#387—22no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2096—24no—
33#2035—28no—
34#1854—28no—
35#1847—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1442—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—42no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—42no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D412yes—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—24no—
73C2×#107—24no—
74C2×#107—42no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—42no—
80C2×#106—42no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8D418yes—
90C2×#97—24no—
91C2×#97—22no—
92#397—216no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391D414yes—
97#387—216no—
98#384—22no—
99#384—22no—
100#384—22no—
101#384—22no—
102#382—22no—
103#382D414yes—
104#380—24no—
105#380—24no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—24no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—24no—
118#336—24no—
119#336—22no—
120C25C2×D411yescharacteristic
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8C2×D412yes—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2×D412yes—
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×D414yes—
158C22×C8C2×D414yes—
159C22×C8#4114yes—
160C22×C8#4114yes—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163#117—48no—
164#117—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
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—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—84no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219#107—48no—
220#106—48no—
221#106—48no—
222#103—48no—
223#103—48no—
224C22⋊C8—216no—
225#97—28no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—24no—
230#95—44no—
231#95—44no—
232#95—84no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—24no—
240C24#12712yes—
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4#12014yes—
316C22×C4#12712yes—
317C22×C4#13714yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C42−1+412yes—
325C22×C42−1+412yes—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8#12318yes—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—416no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—88no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—88no—
361C42—22no—
362C42—22no—
363C42—24no—
364C42—48no—
365C42—41no—
366C42—41no—
367C42—42no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—22no—
372C23—24no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23C2×#12011yescharacteristic
385C23C2×#12711yescharacteristic
386C23#53511yescharacteristic
387C23#54712yes—
388C23#54712yes—
389C23C2×#13712yes—
390C23#57512yes—
391C23#57714yes—
392Q8—44no—
393Q8—44no—
394Q8—82no—
395Q8—82no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4C2×#12312yes—
445C2×C4#52511yescharacteristic
446C2×C4#53211yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53612yes—
449C2×C4#53611yescharacteristic
450C2×C4#53912yes—
451C2×C4#54212yes—
452C2×C4D4212yes—
453C2×C4D4212yes—
454C2×C4#54712yes—
455C2×C4#54712yes—
456C2×C4#54712yes—
457C2×C4#54812yes—
458C2×C4#55312yes—
459C2×C4#56312yes—
460C8—28no—
461C8—44no—
462C22—42no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—84no—
468C22—82no—
469C22—82no—
470C22#240611yescharacteristic
471C22#243211yescharacteristic
472C22#243311yescharacteristic
473C22#250112yes—
474C22#271812yes—
475C22#300712yes—
476C22#301012yes—
477C22#315011yescharacteristic
478C22#315211yescharacteristic
479C22#315412yes—
480C22#321612yes—
481C22#322112yes—
482C22#334414yes—
483C22#334814yes—
484C22#335014yes—
485C22#335614yes—
486C4—42no—
487C4—44no—
488C4—44no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—162no—
494C4—162no—
495C4#242212yes—
496C4#244312yes—
497C4#267412yes—
498C4#268012yes—
499C2—82no—
500C2#1538411yescharacteristic
501C2#2410812yes—
502C2#2571512yes—
503C2#2572811yescharacteristic
504C2#2573311yescharacteristic
505C2#2793712yes—
506C2#2794012yes—
507C2#3229714yes—
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×#2096C2semidirect
5. C2×#1847C2semidirect
7. #11372C2semidirect
8. #11323C2semidirect
9. #11124C2semidirect
10. #10928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#20962
C2×#20352
C2×#18542
C2×#18472
#151952
#113722
#113232
#111242
#109282
#109222