Finite Group Databasebeta

← #288056

Subgroups — #288056

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28676C211yesmaximal, characteristic
2#25453C212yesmaximal
3#23184C211yesmaximal, characteristic
4C2×#1707C211yesmaximal, characteristic
5C2×#1670C211yesmaximal, characteristic
6#10544C212yesmaximal
7#10073C212yesmaximal
8#9730C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#9065C212yesmaximal
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#498C2212yes—
14C2×#494C2211yescharacteristic
15C2×#487C2212yes—
16C2×#475C2212yes—
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16—21no—
19#2567C2212yes—
20#2365C2212yes—
21#2087—22no—
22#2084—22no—
23#2066—22no—
24#1987C2212yes—
25#1987C2212yes—
26#1981C2212yes—
27#1969—21no—
28#1948—22no—
29#1942—21no—
30#1942—21no—
31#1939C2212yes—
32#1927—22no—
33#1878—22no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1707—24no—
38#1670—24no—
39C2×#335C2211yescharacteristic
40C2×#333C2212yes—
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×C82C2211yescharacteristic
44#1279C2211yescharacteristic
45#1279C2212yes—
46#1277C2211yescharacteristic
47C22×Q16—21no—
48C22×Q16—21no—
49C2×C4⋊Q8C2312yes—
50C2×C4⋊Q8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8C2311yescharacteristic
53#500—22no—
54#500—22no—
55#498—24no—
56#498—42no—
57#494—22no—
58#494D414yes—
59#487—24no—
60#487—24no—
61#475—24no—
62#475—42no—
63C8×Q8—22no—
64C8×Q8—22no—
65C4×Q16—44no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×#106—22no—
71C2×#106—22no—
72C2×#106—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—42no—
82C2×#103D411yescharacteristic
83C2×#103D411yescharacteristic
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#395—42no—
90#391—22no—
91#391—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#387—42no—
96#384—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2312yes—
99C4×C4⋊C4C2311yescharacteristic
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340D411yescharacteristic
105#340D412yes—
106#340D411yescharacteristic
107#336—22no—
108#336—21no—
109#335—22no—
110#333—42no—
111#332—24no—
112Q8⋊C8—22no—
113C82—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8C2×D411yescharacteristic
117C2×Q16—24no—
118C2×Q16—24no—
119C2×Q16—42no—
120C2×Q16—42no—
121C2×Q16—81no—
122C2×Q16—81no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8C2×D414yes—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—44no—
144C4×Q8—82no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—42no—
162C2×C42C2411yescharacteristic
163#107—22no—
164#107—42no—
165#107—42no—
166#107SD1614yes—
167#106—44no—
168#106—44no—
169#106—44no—
170C4⋊C8—21no—
171C4⋊C8—22no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—22no—
176C4⋊C8—42no—
177C4⋊C8—44no—
178#103—22no—
179#103—48no—
180#103—44no—
181#103—44no—
182#103—82no—
183#103Q1614yescomplemented
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—42no—
188C4×C8—42no—
189C4×C8C2×D412yes—
190C4×C8#4112yes—
191C4×C8#4112yes—
192#95—21no—
193#95—22no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—42no—
198C2×Q8—21no—
199C2×Q8—24no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8C2×Q1612yes—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4H(5,𝔽2)11yescharacteristic
225Q16—48no—
226Q16—84no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—42no—
245C2×C8#12112yes—
246C2×C8#12112yes—
247C2×C8C2×SD1612yes—
248C2×C8C2×Q1612yes—
249C2×C8#13512yes—
250C2×C8#13712yes—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—24no—
255C4⋊C4—21no—
256C4⋊C4—24no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—84no—
270C4⋊C4—162no—
271C4⋊C4—162no—
272C4⋊C4C2×SD1614yes—
273C4⋊C4C2×Q1612yes—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42—41no—
279C42—42no—
280C42—84no—
281C42C22×D411yescharacteristic
282C42C2×#4111yescharacteristic
283C42C2×#4111yescharacteristic
284C42H(5,𝔽2)11yescharacteristic
285C23D4211yescharacteristic
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—44no—
291Q8—44no—
292Q8—84no—
293Q8—82no—
294Q8—82no—
295Q8—162no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4C2×#12111yescharacteristic
323C2×C4#53211yescharacteristic
324C2×C4#54011yescharacteristic
325C2×C4D4211yescharacteristic
326C2×C4#54611yescharacteristic
327C2×C4#55011yescharacteristic
328C2×C4C22×SD1611yescharacteristic
329C2×C4C22×Q1611yescharacteristic
330C2×C4C2×#13511yescharacteristic
331C2×C4C2×#13711yescharacteristic
332C2×C4#57711yescharacteristic
333C2×C4#57711yescharacteristic
334C2×C4#57811yescharacteristic
335C2×C4#57811yescharacteristic
336C8—22no—
337C8—22no—
338C8—22no—
339C8—42no—
340C8—81no—
341C8—82no—
342C8—81no—
343C8#46114yes—
344C22#242811yescharacteristic
345C22#327811yescharacteristic
346C22D4×SD1611yescharacteristic
347C22#328311yescharacteristic
348C22D4×Q1611yescharacteristic
349C22#329011yescharacteristic
350C22#329011yescharacteristic
351C4—21no—
352C4—21no—
353C4—44no—
354C4—42no—
355C4—42no—
356C4—81no—
357C4—84no—
358C4—81no—
359C4—82no—
360C4—162no—
361C4C2×#46111yescharacteristic
362C4#304211yescharacteristic
363C4#317011yescharacteristic
364C4#317711yescharacteristic
365C4#328511yescharacteristic
366C4D4×Q1611yescharacteristic
367C4#339211yescharacteristic
368C4#340011yescharacteristic
369C2#1993211yescharacteristic
370C2#1995911yescharacteristic
371C2#1998011yescharacteristic
372C2SD16×Q1611yescharacteristic
373C2#2536811yescharacteristic
374C2#2541511yescharacteristic
375C2#2543211yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
183. #103Q16semidirect

Maximal subgroups

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

subgroupindex
#286762
#254532
#231842
C2×#17072
C2×#16702
#105442
#100732
#97302
#95942
#90652
C8×#1032