Finite Group Databasebeta

← #416739

Subgroups — #416739

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#20765C212yesmaximal
3#13909C211yesmaximal, characteristic
4#11392C212yesmaximal
5#11392C212yesmaximal
6#11211C212yesmaximal
7#11210C212yesmaximal
8#10851C212yesmaximal
9C4×#343C211yesmaximal, characteristic
10C2×#544C2212yes—
11C2×C4×M16C2212yes—
12#2857—22no—
13#2659—22no—
14#2552—22no—
15#2551—22no—
16#2550—24no—
17#2316C2212yes—
18#2285C2212yes—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1836—22no—
25#1836—22no—
26#1835—24no—
27C2×#343C2212yes—
28C2×#343C2212yes—
29#1736C2214yes—
30#1736C2214yes—
31C2×#339C2214yes—
32C2×#339C2214yes—
33#1441—22no—
34#1436—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C22×M16C2314yes—
38#544—24no—
39#544—24no—
40#544—44no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—24no—
43C2×#125—24no—
44C2×C22⋊Q8—24no—
45C2×#117C2314yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42D412yes—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C4×M16—24no—
57C4×M16Q818yescomplemented
58C2×#97—22no—
59C2×C4×C8—22no—
60#398—28no—
61#395—28no—
62#391—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#384—22no—
68#384—24no—
69#384D412yes—
70#384D412yes—
71#382—24no—
72#382—22no—
73#382—22no—
74#382D414yes—
75#380—24no—
76#380—22no—
77#380—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4Q814yescomplemented
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95D412yes—
91#343—24no—
92#343—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#339—44no—
98#339—44no—
99C22×Q8—24no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16C2×Q818yes—
115C22×C8—24no—
116C4⋊Q8—48no—
117C4⋊Q8—48no—
118C4⋊Q8—84no—
119#125—48no—
120#125—48no—
121C22⋊Q8—48no—
122C22⋊Q8—48no—
123#117—24no—
124#117—24no—
125#117—24no—
126#117—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4C2×D414yes—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4#4114yes—
154C2×C4⋊C4#4114yes—
155C2×C22⋊C4—24no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—24no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—82no—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×Q812yes—
169C2×C42C2×Q812yes—
170C2×C42C2×Q814yes—
171C2×C42C2×Q812yes—
172C2×C42#4112yes—
173#97—28no—
174C4×C8—28no—
175#95—28no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—44no—
181#95—41no—
182#95—42no—
183#95—41no—
184#95—44no—
185#95—82no—
186C24C22×D411yescharacteristic
187C2×Q8—48no—
188C2×Q8—84no—
189C2×Q8—84no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4#12011yescharacteristic
229C22×C4#12012yes—
230C22×C4#12011yescharacteristic
231C22×C4C22⋊Q812yes—
232C22×C4C22⋊Q812yes—
233C22×C4#12414yes—
234C22×C4#12414yes—
235C22×C4C4⋊Q812yes—
236C22×C4C4⋊Q812yes—
237C22×C4C22×D412yes—
238C22×C4C22×Q812yes—
239C22×C4C22×Q811yescharacteristic
240C22×C4C2×#4111yescharacteristic
241C22×C4C2×#4112yes—
242C22×C4C2×#4112yes—
243C22×C4H(5,𝔽2)12yes—
244C22×C42−1+412yes—
245M16—28no—
246M16—28no—
247M16—44no—
248M16—44no—
249C2×C8—28no—
250C2×C8—44no—
251C2×C8—44no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—416no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4—88no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C42—24no—
273C42—24no—
274C42—44no—
275C42—44no—
276C42—44no—
277C42—42no—
278C42—42no—
279C42—82no—
280C42C22⋊Q814yes—
281C42C22⋊Q814yes—
282C42C4⋊Q814yes—
283C42C4⋊Q814yes—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23C2×#12011yescharacteristic
289C23C2×C22⋊Q811yescharacteristic
290C23C2×#12411yescharacteristic
291C23C2×#12411yescharacteristic
292C23C2×C4⋊Q811yescharacteristic
293C23#53511yescharacteristic
294C23#53611yescharacteristic
295Q8—168no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4#39514yes—
347C2×C4#39512yes—
348C2×C4#39512yes—
349C2×C4C2×C22⋊Q812yes—
350C2×C4C2×C4⋊Q812yes—
351C2×C4#53314yes—
352C2×C4#54414yes—
353C2×C4#54614yes—
354C2×C4#54814yes—
355C2×C4#55412yes—
356C2×C4#55512yes—
357C2×C4#55612yes—
358C2×C4#55712yes—
359C8—48no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22C2×#39511yescharacteristic
373C22#243511yescharacteristic
374C22#243611yescharacteristic
375C22#256311yescharacteristic
376C22#256811yescharacteristic
377C22#266011yescharacteristic
378C22#286211yescharacteristic
379C22#301811yescharacteristic
380C22#301811yescharacteristic
381C22#301911yescharacteristic
382C22#302011yescharacteristic
383C4—24no—
384C4—22no—
385C4—22no—
386C4—44no—
387C4—44no—
388C4—44no—
389C4—82no—
390C4—84no—
391C4—84no—
392C4—161no—
393C4—161no—
394C4#243914yes—
395C4#255014yes—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2#1523011yescharacteristic
401C2#2356011yescharacteristic
402C2#2356111yescharacteristic
403C2#2587311yescharacteristic
404C2#2587411yescharacteristic
405C2#2592111yescharacteristic
406C2#2592211yescharacteristic
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
57. C4×M16Q8semidirect
83. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#207652
#139092
#113922
#112112
#112102
#108512
C4×#3432