Finite Group Databasebeta

← #237835

Subgroups — #237835

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#21759C212yesmaximal
3#14159C211yesmaximal, characteristic
4#10343C214yesmaximal, complemented
5#10341C214yesmaximal, complemented
6C2×#1294C212yesmaximal
7#7907C211yesmaximal, characteristic
8C2×#543—21no—
9C22×C22⋊Q8C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2857—22no—
12#2659C2214yes—
13#2549—24no—
14#2440—24no—
15#2315—21no—
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C22×#95C2211yescharacteristic
19#1892—24no—
20#1872—24no—
21#1824C2214yes—
22#1787C2214yes—
23#1446C2212yes—
24#1445C2212yes—
25#1436—22no—
26C22⋊#95—21no—
27C22⋊#95—21no—
28#1294C2218yes—
29#1280C2214yes—
30C23×Q8C2311yescharacteristic
31C24×C4—21no—
32C24×C4C2311yescharacteristic
33C23×C8—22no—
34#543—44no—
35#543—44no—
36C2×C4⋊Q8—22no—
37C2×C22⋊Q8—28no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8Q814yescomplemented
44C22×C4⋊C4—21no—
45C22×C4⋊C4D414yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4D411yescharacteristic
50C22×C22⋊C4D411yescharacteristic
51C22×C42D412yes—
52C2×#103—28no—
53C2×C22⋊C8—24no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8C2314yes—
57#398—28no—
58#395—28no—
59#391—24no—
60#389—24no—
61#389—24no—
62#385—28no—
63#384—22no—
64#380—24no—
65#380—22no—
66#380—22no—
67#380D414yes—
68#380C2314yes—
69C4×C22⋊C4—22no—
70C4×C22⋊C4—22no—
71C2×#95—22no—
72C2×#95—24no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95C2312yes—
79C2×#95C2312yes—
80#340—28no—
81C25C2×D411yescharacteristic
82C22×Q8—24no—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—82no—
87C22×Q8C2×Q812yes—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—44no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—82no—
97C23×C4—82no—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×Q812yes—
104C23×C4#4114yes—
105C23×C4#4112yes—
106C23×C4C2411yescharacteristic
107C22×C8—24no—
108C22×C8—44no—
109C22×C8—84no—
110C4⋊Q8—84no—
111C22⋊Q8—28no—
112C22⋊Q8—28no—
113C22⋊Q8—432no—
114C22⋊Q8—48no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117C22⋊Q8—48no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4C2×D418yes—
132C2×C4⋊C4C2×Q812yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4SD1614yes—
145C2×C22⋊C4Q1614yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—82no—
149#103—88no—
150C22⋊C8—24no—
151C22⋊C8—24no—
152C22⋊C8—44no—
153#95—28no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—44no—
159#95—44no—
160#95—42no—
161#95—82no—
162#95—82no—
163C24—42no—
164C24—42no—
165C24#12012yes—
166C24C22⋊Q812yes—
167C24#12412yes—
168C24C2×SD1612yes—
169C24C2×Q1612yes—
170C24AGL(1,ℤ/8)12yes—
171C24#13712yes—
172C24C22×D411yescharacteristic
173C2×Q8—24no—
174C2×Q8—48no—
175C2×Q8—48no—
176C2×Q8—42no—
177C2×Q8—82no—
178C2×Q8—82no—
179C2×Q8—84no—
180C2×Q8—84no—
181C2×Q8—84no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—48no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—48no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—88no—
220C22×C4—88no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4#12011yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4C22⋊Q814yes—
226C22×C4C22⋊Q814yes—
227C22×C4C22⋊Q812yes—
228C22×C4#12414yes—
229C22×C4#12414yes—
230C22×C4#12412yes—
231C22×C4C2×SD1614yes—
232C22×C4C2×Q1614yes—
233C22×C4C22×D412yes—
234C22×C4C22×Q811yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4C2×#4111yescharacteristic
237C2×C8—48no—
238C2×C8—44no—
239C2×C8—88no—
240C4⋊C4—24no—
241C4⋊C4—48no—
242C4⋊C4—48no—
243C4⋊C4—48no—
244C4⋊C4—416no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—84no—
249C4⋊C4—84no—
250C22⋊C4—28no—
251C22⋊C4—28no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—48no—
255C22⋊C4—48no—
256C42—44no—
257C42—82no—
258C23—22no—
259C23—24no—
260C23—22no—
261C23—24no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—44no—
266C23—44no—
267C23—42no—
268C23—44no—
269C23—44no—
270C23#39314yes—
271C23#45014yes—
272C23#45114yes—
273C23#47514yes—
274C23#47714yes—
275C23#48714yes—
276C23#48914yes—
277C23C2×#12011yescharacteristic
278C23C2×C22⋊Q811yescharacteristic
279C23C2×#12411yescharacteristic
280C23C22×SD1611yescharacteristic
281C23C22×Q1611yescharacteristic
282C23C2×AGL(1,ℤ/8)11yescharacteristic
283C23C2×#13711yescharacteristic
284Q8—48no—
285Q8—84no—
286Q8—168no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—816no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—168no—
331C2×C4#39312yes—
332C2×C4#39312yes—
333C2×C4C2×C22⋊Q812yes—
334C2×C4C2×#12412yes—
335C8—84no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—21no—
340C22—21no—
341C22—24no—
342C22—22no—
343C22—22no—
344C22—28no—
345C22—22no—
346C22—22no—
347C22—28no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22#202418yes—
353C22C2×#39311yescharacteristic
354C22C2×#45011yescharacteristic
355C22C2×#45111yescharacteristic
356C22#301812yes—
357C22#302612yes—
358C22C2×#47511yescharacteristic
359C22C2×#47711yescharacteristic
360C22C2×#48711yescharacteristic
361C22C2×#48911yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—44no—
367C4—44no—
368C4—84no—
369C4—82no—
370C4—82no—
371C4—88no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—28no—
377C2#1838112yes—
378C2#1838312yes—
379C2C2×#202411yescharacteristic
380C2#2346111yescharacteristic
381C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10343C2semidirect
5. #10341C2semidirect
43. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#217592
#141592
#103432
#103412
C2×#12942
#79072