Finite Group Databasebeta

← #411938

Subgroups — #411938

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2027C212yesmaximal
3#20646C212yesmaximal
4#13785C211yesmaximal, characteristic
5#11392C212yesmaximal, complemented
6#11296C212yesmaximal, complemented
7#11054C212yesmaximal
8#10847C211yesmaximal, characteristic, complemented
9#10840C212yesmaximal, complemented
10C2×#544C2212yes—
11C22×C4×D4C2211yescharacteristic
12C2×#423C2212yes—
13C2×#418C2214yes—
14#2857—22no—
15#2659—22no—
16#2552—22no—
17#2551—22no—
18#2550—24no—
19#2316C2212yes—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22#2285C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#2027—28no—
26#1804—24no—
27#1802—22no—
28#1802—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1437—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C22⋊#95—22no—
38C23×D4C2311yescharacteristic
39C24×C4—21no—
40C22×M16C2314yes—
41#544—24no—
42#544—24no—
43#544—44no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—24no—
46C2×#125—24no—
47C2×C22⋊Q8—24no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—24no—
52C2×C4×D4Q814yescomplemented
53C2×#117C2314yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#423—24no—
65#423—24no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68#418—28no—
69#418—44no—
70C2×#102—24no—
71C2×#102—24no—
72#398—28no—
73#395—28no—
74#391—24no—
75#390—22no—
76#390—22no—
77#390D414yes—
78#390Q814yescomplemented
79#389—24no—
80#389—24no—
81#389—24no—
82#389—24no—
83#384—22no—
84#384D412yes—
85#384D412yes—
86#382D414yes—
87#380—24no—
88#380—22no—
89#380—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94#376—24no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100#343—24no—
101#343—24no—
102#340—24no—
103#340—24no—
104#339—28no—
105#339—44no—
106C25—21no—
107C22×Q8—24no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C2×D412yes—
116C22×D4C2×Q812yes—
117C22×D4C2×Q812yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—24no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—82no—
125C23×C4—82no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C22×C8—24no—
138C4⋊Q8—48no—
139C4⋊Q8—48no—
140C4⋊Q8—84no—
141#125—48no—
142#125—48no—
143C22⋊Q8—48no—
144C22⋊Q8—48no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—44no—
148C4×D4—44no—
149#117—24no—
150#117—24no—
151#117—24no—
152#117—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4C2×Q812yes—
173C2×C4⋊C4#4112yes—
174C2×C4⋊C4#4112yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—82no—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×Q812yes—
200C2×C42C2×Q812yes—
201C2×C42#4114yes—
202C2×C42#4112yes—
203C4⋊C8—44no—
204C4⋊C8—44no—
205#102—48no—
206#102—48no—
207#95—28no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—42no—
212#95—42no—
213#95—82no—
214#95—82no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24C22×D411yescharacteristic
227C2×Q8—48no—
228C2×Q8—84no—
229C2×Q8—84no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4C4⋊Q814yescomplemented
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—48no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4—88no—
283C22×C4—88no—
284C22×C4#12011yescharacteristic
285C22×C4#12012yes—
286C22×C4#12011yescharacteristic
287C22×C4#12114yes—
288C22×C4C22⋊Q814yes—
289C22×C4C22⋊Q812yes—
290C22×C4C22⋊Q812yes—
291C22×C4#12412yes—
292C22×C4#12412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×Q811yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4H(5,𝔽2)12yes—
302C22×C42−1+412yes—
303M16—44no—
304M16—44no—
305M16—44no—
306M16—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—416no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—88no—
333C22⋊C4—88no—
334C42—24no—
335C42—24no—
336C42—44no—
337C42—44no—
338C42—44no—
339C42—42no—
340C42—42no—
341C42—82no—
342C42C22⋊Q814yes—
343C42C22⋊Q814yes—
344C42#12414yes—
345C42#12414yes—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—24no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23C2×#12011yescharacteristic
374C23C2×#12111yescharacteristic
375C23C2×C22⋊Q811yescharacteristic
376C23C2×C22⋊Q811yescharacteristic
377C23C2×#12411yescharacteristic
378C23#53511yescharacteristic
379C23#53611yescharacteristic
380Q8—168no—
381D4—28no—
382D4—44no—
383D4—44no—
384D4—44no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—816no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#39312yes—
432C2×C4#39312yes—
433C2×C4#39314yes—
434C2×C4C2×C22⋊Q812yes—
435C2×C4C2×#12412yes—
436C2×C4C2×C4⋊Q811yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4#53311yescharacteristic
439C2×C4#53311yescharacteristic
440C2×C4#54212yes—
441C2×C4#54412yes—
442C2×C4D4212yes—
443C2×C4#54612yes—
444C2×C4#54712yes—
445C2×C4#54812yes—
446C2×C4D4×Q812yes—
447C2×C4#55412yes—
448C2×C4#55512yes—
449C2×C4#55612yes—
450C8—84no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—48no—
464C22—44no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22C2×#39311yescharacteristic
475C22#240811yescharacteristic
476C22#240911yescharacteristic
477C22#254711yescharacteristic
478C22#256311yescharacteristic
479C22#262811yescharacteristic
480C22#285711yescharacteristic
481C22#301911yescharacteristic
482C22#302011yescharacteristic
483C22#302111yescharacteristic
484C22#302111yescharacteristic
485C4—22no—
486C4—22no—
487C4—24no—
488C4—44no—
489C4—44no—
490C4—82no—
491C4—82no—
492C4—84no—
493C4—84no—
494C4—82no—
495C4#243912yes—
496C4#244012yes—
497C4#249112yes—
498C4#249212yes—
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—84no—
504C2#1512511yescharacteristic
505C2#2346211yescharacteristic
506C2#2346311yescharacteristic
507C2#2422611yescharacteristic
508C2#2422711yescharacteristic
509C2#2431611yescharacteristic
510C2#2431711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35167C2semidirect
5. #11392C2semidirect
6. #11296C2semidirect
8. #10847C2semidirect
9. #10840C2semidirect
52. C2×C4×D4Q8semidirect
78. #390Q8semidirect
245. C2×D4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#20272
#206462
#137852
#113922
#112962
#110542
#108472
#108402