Finite Group Databasebeta

← #416680

Subgroups — #416680

Order 512.

1094 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#20800C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic
5#11340C214yesmaximal
6#11217C214yesmaximal
7#10851C211yesmaximal, characteristic
8#10851C211yesmaximal, characteristic
9C4×#343C211yesmaximal, characteristic
10C2×#428C2214yes—
11C2×C4×M16C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2851—22no—
14#2646—22no—
15#2571—21no—
16#2571—21no—
17#2570—22no—
18#2570—22no—
19#2552—21no—
20#2552—21no—
21#2317C2212yes—
22#2317C2212yes—
23C2×#389—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1847—22no—
31#1845—21no—
32#1844—21no—
33#1836—22no—
34#1834—21no—
35#1834—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1736C2214yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41#1444—22no—
42C4⋊#95C2212yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C2×#125—22no—
46C2×#117C2312yes—
47C2×#117C2312yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42D412yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58#428—28no—
59#428—44no—
60C2×#107—24no—
61C2×#106—24no—
62C4×M16—22no—
63C4×M16—22no—
64C4×M16Q814yes—
65C4×M16Q814yes—
66C2×#97—21no—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70#400—24no—
71#400—24no—
72#398—24no—
73#398—24no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—48no—
81#387—24no—
82#387—24no—
83#385—24no—
84#385—24no—
85#384—22no—
86#384—22no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—44no—
92#384D411yescharacteristic
93#384D411yescharacteristic
94#384D411yescharacteristic
95#384D411yescharacteristic
96#382—22no—
97#382—22no—
98#382—22no—
99#382D412yes—
100#382D412yes—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4Q814yes—
115C2×#95—21no—
116C2×#95—24no—
117C2×#95D412yes—
118#343—24no—
119#343—24no—
120#340—24no—
121#340—24no—
122#339—28no—
123#339—44no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16C2×Q814yes—
140C2×M16C2×Q814yes—
141C2×M16#4114yes—
142C2×M16#4114yes—
143C22×C8—22no—
144C22×C8—22no—
145#125—84no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—84no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—44no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×Q814yes—
205C2×C42C2×Q812yes—
206C2×C42C2×Q812yes—
207C2×C42C2×Q812yes—
208C2×C42#4112yes—
209#107—48no—
210#106—48no—
211#97—24no—
212#97—24no—
213C4×C8—24no—
214C4×C8—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—24no—
226#95—48no—
227C24C22×D411yescharacteristic
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4C22⋊Q812yes—
263C22×C4C22⋊Q812yes—
264C22×C4C4⋊Q812yes—
265C22×C4C4⋊Q812yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4112yes—
273C22×C4C2×#4112yes—
274C22×C4H(5,𝔽2)12yes—
275C22×C42−1+412yes—
276M16—24no—
277M16—24no—
278M16—24no—
279M16—24no—
280M16—24no—
281M16—42no—
282M16—42no—
283M16#12518yescomplemented
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—216no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—816no—
317C4⋊C4—82no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C42—28no—
323C42—44no—
324C42—44no—
325C42—44no—
326C42—44no—
327C42—82no—
328C42C22⋊Q812yes—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42C22⋊Q812yes—
332C42C4⋊Q812yes—
333C42C4⋊Q812yes—
334C42C4⋊Q812yes—
335C42C4⋊Q812yes—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23C2×#12011yescharacteristic
341C23C2×C22⋊Q811yescharacteristic
342C23C2×C4⋊Q811yescharacteristic
343C23#53211yescharacteristic
344C23#53211yescharacteristic
345C23#53511yescharacteristic
346C23#53611yescharacteristic
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#39512yes—
392C2×C4#39512yes—
393C2×C4#39512yes—
394C2×C4#39512yes—
395C2×C4#45312yes—
396C2×C4#45612yes—
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4C2×#12512yes—
400C2×C4C2×C4⋊Q811yescharacteristic
401C2×C4C2×C4⋊Q811yescharacteristic
402C2×C4#53312yes—
403C2×C4#54412yes—
404C2×C4#54412yes—
405C2×C4#54612yes—
406C2×C4#54712yes—
407C2×C4#54812yes—
408C2×C4#55012yes—
409C2×C4#55212yes—
410C2×C4#55312yes—
411C2×C4#55412yes—
412C2×C4#55712yes—
413C8—28no—
414C8—44no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22C2×#39511yescharacteristic
428C22#243111yescharacteristic
429C22#243211yescharacteristic
430C22#256511yescharacteristic
431C22#256611yescharacteristic
432C22#266011yescharacteristic
433C22#285911yescharacteristic
434C22C2×#45311yescharacteristic
435C22C2×#45611yescharacteristic
436C22#302111yescharacteristic
437C22#302211yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—48no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—162no—
449C4#206014yes—
450C4#244112yes—
451C4#244212yes—
452C4#255012yes—
453C4#255212yes—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1522811yescharacteristic
459C2C2×#206011yescharacteristic
460C2#2356211yescharacteristic
461C2#2562211yescharacteristic
462C2#2562311yescharacteristic
463C2#2569411yescharacteristic
464C2#2569511yescharacteristic
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
283. M16#125semidirect

Maximal subgroups

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

subgroupindex
#351812
#208002
#207642
#139102
#113402
#112172
#108512
C4×#3432