Finite Group Databasebeta

← #407022

Subgroups — #407022

Order 512.

1337 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20909C212yesmaximal
2#20895C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic, complemented
4#11303C214yesmaximal, complemented
5#11192C214yesmaximal, complemented
6#10727C212yesmaximal, complemented
7#10723C211yesmaximal, characteristic, complemented
8C24×C8C2211yescharacteristic
9C2×#525C2211yescharacteristic
10C23×C4⋊C4C2212yes—
11C2×#417C2214yes—
12C2×#408C2212yes—
13C2×#407C2211yescharacteristic
14C2×#407C2212yes—
15#1874—22no—
16#1874—22no—
17#1872—22no—
18#1872—22no—
19#1869—22no—
20#1869—22no—
21C2×#343C2214yes—
22C2×#343C2214yes—
23C2×#341C2214yes—
24#1736C2214yes—
25C2×#340—22no—
26C2×#340—22no—
27C2×#340C2214yes—
28#1729C2212yes—
29C22×#41C2311yescharacteristic
30C24×C4—22no—
31C24×C4C2311yescharacteristic
32C22×M16C2312yes—
33C22×M16C2314yes—
34C23×C8—24no—
35C23×C8—44no—
36C23×C8D412yes—
37C23×C8D412yes—
38C23×C8C2312yes—
39#525—24no—
40#525—24no—
41C2×#123—21no—
42C2×C22⋊Q8D411yescharacteristic
43C2×C22⋊Q8D411yescharacteristic
44C2×#121D411yescharacteristic
45C2×#121D411yescharacteristic
46C2×#120—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×#117C2311yescharacteristic
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—28no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—44no—
57C22×C4⋊C4C2314yes—
58#417—28no—
59#417—44no—
60C2×#103—24no—
61C2×#102—24no—
62#408—44no—
63#407—24no—
64#407—24no—
65#407—24no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—24no—
71#343—28no—
72#343—44no—
73#341—44no—
74#340—28no—
75#340—24no—
76#340—24no—
77#336—24no—
78C25#4111yescharacteristic
79C2×#41—22no—
80C2×#41—22no—
81C2×#41—24no—
82C2×#41—24no—
83C22×Q8C2×D411yescharacteristic
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4C2×D411yescharacteristic
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—21no—
92C23×C4—48no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—84no—
96C23×C4—84no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4114yes—
103C23×C4#4114yes—
104C23×C4C2411yescharacteristic
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—24no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—24no—
115C22×C8—24no—
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—24no—
120C22×C8—22no—
121C22×C8—24no—
122C22×C8—44no—
123C22×C8—44no—
124C22×C8—416no—
125C22×C8C2×D414yes—
126#123—44no—
127C22⋊Q8—24no—
128C22⋊Q8—24no—
129#121—24no—
130#121—24no—
131#120—44no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136#117—22no—
137#117—22no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4#4118yes—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C42—21no—
163#103—48no—
164#102—48no—
165C22⋊C8—48no—
166C22⋊C8—44no—
167C22⋊C8—44no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170C24—24no—
171C24—24no—
172C24—22no—
173C24—22no—
174C24—21no—
175C24#12111yescharacteristic
176C24#12111yescharacteristic
177C24#12614yes—
178C24C2×#4111yescharacteristic
179#41—44no—
180#41—44no—
181#41—44no—
182#41—44no—
183C2×Q8—22no—
184C2×Q8—24no—
185C2×D4—22no—
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—44no—
192C2×D4—42no—
193C2×D4—42no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—28no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—28no—
209C22×C4—48no—
210C22×C4—48no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—832no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12314yes—
227C22×C4#12314yes—
228C22×C4#12312yes—
229C22×C4#12312yes—
230C22×C4#12412yes—
231C22×C4#12412yes—
232C22×C4#12614yes—
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4C2×#4112yes—
237C22×C4C2×#4114yes—
238C22×C4H(5,𝔽2)14yes—
239M16—44no—
240M16—44no—
241M16—44no—
242M16—44no—
243M16—44no—
244M16—44no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—48no—
249C2×C8—48no—
250C2×C8—48no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—44no—
256C2×C8—44no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—41no—
273C42—41no—
274C23—216no—
275C23—216no—
276C23—28no—
277C23—28no—
278C23—28no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—42no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23#39712yes—
293C23#39712yes—
294C23#47114yes—
295C23#47212yes—
296C23#47312yes—
297C23C2×#12111yescharacteristic
298C23C2×#12612yes—
299C23#55114yes—
300Q8—44no—
301D4—44no—
302D4—84no—
303D4—84no—
304D4—84no—
305C2×C4—28no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—48no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—88no—
325C2×C4—816no—
326C2×C4—816no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—88no—
333C2×C4#39712yes—
334C2×C4#39712yes—
335C2×C4C2×#12311yescharacteristic
336C2×C4C2×#12312yes—
337C2×C4#52512yes—
338C2×C4C2×#12411yescharacteristic
339C2×C4#52914yes—
340C2×C4#53412yes—
341C2×C4#53814yes—
342C2×C4#54614yes—
343C2×C4#55314yes—
344C2×C4#55314yes—
345C8—44no—
346C8—82no—
347C8—84no—
348C22—232no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—24no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—82no—
365C22—82no—
366C22#207814yes—
367C22#207914yes—
368C22C2×#39711yescharacteristic
369C22#251014yes—
370C22#253412yes—
371C22C2×#47112yes—
372C22C2×#47211yescharacteristic
373C22C2×#47311yescharacteristic
374C4—22no—
375C4—22no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—88no—
380C4#244414yes—
381C4#259112yes—
382C4#262211yescharacteristic
383C4#285311yescharacteristic
384C2—24no—
385C2—24no—
386C2—24no—
387C2—161no—
388C2#1494511yescharacteristic
389C2C2×#207811yescharacteristic
390C2C2×#207911yescharacteristic
391C2#2813412yes—
392C2#2813512yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#340C2semidirect
4. #11303C2semidirect
5. #11192C2semidirect
6. #10727C2semidirect
7. #10723C2semidirect

Maximal subgroups

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

subgroupindex
#209092
#208952
C22×#3402
#113032
#111922
#107272
#107232