Finite Group Databasebeta

← #416754

Subgroups — #416754

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4#14687C211yesmaximal, characteristic
5#11392C214yesmaximal
6#11217C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10839C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C2×#544C2212yes—
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2868—21no—
15#2859—21no—
16#2660—22no—
17#2552—21no—
18#2552—21no—
19#2551—21no—
20#2551—21no—
21#2440—22no—
22#2440—22no—
23#2317C2212yes—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382—22no—
28C4×#117C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1948—21no—
31#1948—21no—
32#1947—21no—
33#1947—21no—
34#1836—22no—
35#1834—21no—
36#1834—21no—
37C2×#343C2214yes—
38#1736C2214yes—
39#1736C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2214yes—
43C4⋊#95—22no—
44C4⋊#95C2214yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47#544—28no—
48#544—44no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×C22⋊Q8—24no—
56C2×#117C2312yes—
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#428—28no—
70C2×#107—22no—
71C2×#106—22no—
72#423—22no—
73#423Q814yescomplemented
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C4×M16—22no—
77C4×M16Q814yescomplemented
78C2×#97—21no—
79C2×C4×C8—21no—
80#398—24no—
81#398—24no—
82#395—24no—
83#395—24no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#385—24no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D411yescharacteristic
104#382—22no—
105#382—48no—
106#382D412yes—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117#376—21no—
118C2×#95—28no—
119C43—21no—
120C43—22no—
121C43Q814yescomplemented
122#343—44no—
123#340—24no—
124#340—24no—
125#340—22no—
126#340—22no—
127#339—24no—
128#339—24no—
129#339—44no—
130C22×Q8—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D414yes—
136C23×C4C2411yescharacteristic
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16C2×Q814yes—
146C2×M16C2×Q814yes—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—48no—
150C4⋊Q8—44no—
151C4⋊Q8—44no—
152C4⋊Q8—84no—
153C4⋊Q8—84no—
154#125—48no—
155#125—44no—
156#125—44no—
157C22⋊Q8—416no—
158#117—22no—
159#117—22no—
160#117—24no—
161#117—24no—
162#117#4114yes—
163#117#4114yes—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×Q814yes—
225C2×C42C2×Q812yes—
226C2×C42C2×Q812yes—
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229#107—44no—
230#106—44no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233#97—24no—
234C4×C8—24no—
235#95—24no—
236#95—24no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—416no—
244C24C22×D411yescharacteristic
245C2×Q8—48no—
246C2×Q8—84no—
247C2×Q8—84no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4C22⋊Q812yes—
278C22×C4C22⋊Q812yes—
279C22×C4C4⋊Q812yes—
280C22×C4C4⋊Q812yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×Q811yescharacteristic
284C22×C4C22×Q811yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4112yes—
288C22×C4C2×#4112yes—
289C22×C4H(5,𝔽2)12yes—
290C22×C42−1+412yes—
291M16—24no—
292M16—24no—
293M16—28no—
294M16—44no—
295M16—42no—
296M16—42no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4#12412yes—
333C4⋊C4#12412yes—
334C4⋊C4#12412yes—
335C4⋊C4#12412yes—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—44no—
339C42—216no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—48no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42C22⋊Q812yes—
354C42C22⋊Q812yes—
355C42C22⋊Q812yes—
356C42C22⋊Q812yes—
357C42C4⋊Q812yes—
358C42C4⋊Q812yes—
359C42C4⋊Q812yes—
360C42C4⋊Q812yes—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23C2×#12011yescharacteristic
366C23C2×C22⋊Q811yescharacteristic
367C23C2×C4⋊Q811yescharacteristic
368C23#54612yes—
369C23#54812yes—
370Q8—168no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—88no—
408C2×C4—88no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#39512yes—
413C2×C4#39512yes—
414C2×C4#39512yes—
415C2×C4#39512yes—
416C2×C4#45312yes—
417C2×C4#45412yes—
418C2×C4#45512yes—
419C2×C4#45612yes—
420C2×C4C2×C22⋊Q811yescharacteristic
421C2×C4C2×C22⋊Q811yescharacteristic
422C2×C4C2×#12411yescharacteristic
423C2×C4C2×#12411yescharacteristic
424C2×C4C2×C4⋊Q811yescharacteristic
425C2×C4C2×C4⋊Q811yescharacteristic
426C2×C4#53312yes—
427C2×C4#53312yes—
428C2×C4#53511yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#54412yes—
431C2×C4#54412yes—
432C2×C4#54612yes—
433C2×C4#54812yes—
434C2×C4#55412yes—
435C2×C4#55512yes—
436C2×C4#55612yes—
437C2×C4#55712yes—
438C8—44no—
439C8—44no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22C2×#39511yescharacteristic
449C22#243511yescharacteristic
450C22#243611yescharacteristic
451C22#243912yes—
452C22#255012yes—
453C22C2×#45311yescharacteristic
454C22C2×#45511yescharacteristic
455C22C2×#45411yescharacteristic
456C22C2×#45611yescharacteristic
457C4—22no—
458C4—22no—
459C4—48no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—84no—
465C4—82no—
466C4—82no—
467C4—162no—
468C4#206014yes—
469C4#206114yes—
470C4#243912yes—
471C4#255012yes—
472C4#256311yescharacteristic
473C4#256811yescharacteristic
474C4#266011yescharacteristic
475C4#286211yescharacteristic
476C2—22no—
477C2—22no—
478C2#1523011yescharacteristic
479C2C2×#206011yescharacteristic
480C2C2×#206111yescharacteristic
481C2#2587511yescharacteristic
482C2#2587611yescharacteristic
483C2#2592311yescharacteristic
484C2#2592411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
73. #423Q8semidirect
77. C4×M16Q8semidirect
121. C43Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#211712
#207642
#146872
#113922
#112172
#108572
#108392
C4×#3392