Finite Group Databasebeta

← #324420

Subgroups — #324420

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33290C211yesmaximal, characteristic
2#29123C211yesmaximal, characteristic
3#28373C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10483C212yesmaximal
6#10478C212yesmaximal
7#9900C211yesmaximal, characteristic
8#9697C211yesmaximal, characteristic
9#9218C212yesmaximal
10#9216C212yesmaximal
11C4×#335C211yesmaximal, characteristic
12C2×#477C2212yes—
13C2×#462C2212yes—
14C2×C8×Q8C2212yes—
15#2862C2211yescharacteristic
16#2859C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2100—21no—
19#2095—21no—
20#2043—21no—
21#2016—21no—
22#1991—21no—
23#1991—21no—
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1984C2212yes—
29#1969—22no—
30#1939—21no—
31#1939—21no—
32#1933—22no—
33#1842C2211yescharacteristic
34#1835—21no—
35#1835—21no—
36#1803C2212yes—
37#1803C2212yes—
38C8×C4⋊C4—21no—
39C8×C4⋊C4—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×#335C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×Q16—22no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#125—21no—
54C2×C4×Q8C2312yes—
55#477—24no—
56#477—42no—
57#462—24no—
58#462—42no—
59C8×Q8—42no—
60C8×Q8—42no—
61C2×#107—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×C4×C8—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391D412yes—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#384D411yescharacteristic
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#382—21no—
96#382—21no—
97#382—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C43C2311yescharacteristic
101#340—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#340—21no—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#336—21no—
114#336—21no—
115#335—24no—
116#333—42no—
117#332—42no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8C2×D412yes—
121C2×Q16—44no—
122C2×Q16—44no—
123C2×Q16—82no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C4⋊Q8—24no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136#125—42no—
137#125—42no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4C2×D412yes—
160C2×C42—22no—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42C2411yescharacteristic
168#107—44no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—44no—
175C4⋊C8—42no—
176C4⋊C8—42no—
177C4⋊C8#4114yes—
178#103—44no—
179#103—44no—
180#103—44no—
181#103—82no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—44no—
188C4×C8—81no—
189C4×C8—81no—
190C4×C8Q1614yescomplemented
191#95—22no—
192#95—22no—
193#95—41no—
194#95—41no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—81no—
210C2×Q8—81no—
211C2×Q8#13514yes—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225Q16—88no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—24no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8C2×Q1612yes—
246C2×C8C2×Q1612yes—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C42—22no—
273C42—21no—
274C42—21no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42—42no—
279C42C2×Q1612yes—
280C42#13512yes—
281C42#13712yes—
282C42#13712yes—
283C42C2×#4112yes—
284C42H(5,𝔽2)12yes—
285C23C2×#12111yescharacteristic
286Q8—24no—
287Q8—24no—
288Q8—42no—
289Q8—42no—
290Q8—82no—
291Q8—82no—
292Q8—162no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4#46212yes—
322C2×C4#46212yes—
323C2×C4#46412yes—
324C2×C4#46412yes—
325C2×C4#53211yescharacteristic
326C2×C4#53511yescharacteristic
327C2×C4#54612yes—
328C2×C4#55912yes—
329C2×C4C22×Q1611yescharacteristic
330C2×C4C2×#13511yescharacteristic
331C2×C4C2×#13512yes—
332C2×C4C2×#13711yescharacteristic
333C2×C4C2×#13711yescharacteristic
334C2×C4#57611yescharacteristic
335C2×C4#57811yescharacteristic
336C8—22no—
337C8—22no—
338C8—41no—
339C8—41no—
340C8—44no—
341C8—82no—
342C8—82no—
343C22#254411yescharacteristic
344C22C2×#46211yescharacteristic
345C22C2×#46411yescharacteristic
346C22#305011yescharacteristic
347C22#306111yescharacteristic
348C22#306711yescharacteristic
349C22#306711yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—44no—
356C4—82no—
357C4—161no—
358C4—161no—
359C4—161no—
360C4—161no—
361C4#317011yescharacteristic
362C4#318511yescharacteristic
363C4#320911yescharacteristic
364C4#321011yescharacteristic
365C4#329712yes—
366C4#330012yes—
367C2#1911811yescharacteristic
368C2#1912311yescharacteristic
369C2#1921111yescharacteristic
370C2#1921711yescharacteristic
371C2#2860111yescharacteristic
372C2#2861611yescharacteristic
373C2#2863311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
190. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332902
#291232
#283732
C8×C4⋊Q82
#104832
#104782
#99002
#96972
#92182
#92162
C4×#3352