Finite Group Databasebeta

← #287232

Subgroups — #287232

Order 512.

609 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#29206C211yesmaximal, characteristic
2#29141C211yesmaximal, characteristic
3#28255C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5#10620C212yesmaximal
6#10479C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9602C211yesmaximal, characteristic
9#9218C212yesmaximal
10#8643C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#462—21no—
13C2×#462C2212yes—
14C2×C8×Q8C2212yes—
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2058—22no—
19#2056—22no—
20#2024—22no—
21#1990C2212yes—
22#1984C2212yes—
23#1956—21no—
24#1948—21no—
25#1948—21no—
26#1939—21no—
27#1916—22no—
28#1844C2212yes—
29#1842C2211yescharacteristic
30#1834—21no—
31#1803—21no—
32#1803C2212yes—
33#1803C2212yes—
34#1803C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42#1296C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1277C2212yes—
47C22×Q16—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8D411yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8C2312yes—
55#462—24no—
56#462—42no—
57#462—44no—
58C8×Q8—24no—
59C8×Q8—42no—
60C2×#106—22no—
61C2×#106—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×C4×C8—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#391—21no—
81#391D412yes—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384D411yescharacteristic
90#384C2311yescharacteristic
91#384C2312yes—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C43C2311yescharacteristic
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#336—22no—
103#336—22no—
104#336—21no—
105#336—21no—
106#336D412yes—
107#334—42no—
108#332—22no—
109#332—22no—
110C22×Q8—21no—
111C22×Q8—41no—
112C22×Q8C2×D412yes—
113C2×Q16—44no—
114C2×Q16—44no—
115C2×Q16—82no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—81no—
128C4⋊Q8—81no—
129C4⋊Q8—81no—
130C4⋊Q8—81no—
131C4⋊Q8—82no—
132C4⋊Q8SD1614yes—
133C4⋊Q8#4112yes—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—42no—
139C4×Q8—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4C2×D412yes—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42C2411yescharacteristic
167#106—82no—
168#106—82no—
169C4⋊C8—22no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—44no—
180C4⋊C8—42no—
181C4⋊C8—81no—
182C4⋊C8Q1614yescomplemented
183C4⋊C8#4112yes—
184#103—44no—
185#103—44no—
186#103—82no—
187#103—82no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—44no—
191C4×C8—42no—
192C4×C8—42no—
193#95—22no—
194#95—22no—
195#95—41no—
196#95—41no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8C2×SD1614yes—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C2×#4111yescharacteristic
224Q16—88no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8C2×Q1614yes—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—22no—
277C42—41no—
278C42—42no—
279C42—42no—
280C42—82no—
281C42C2×SD1612yes—
282C42C2×Q1612yes—
283C42#13512yes—
284C42#13712yes—
285C42C2×#4111yescharacteristic
286C42C2×#4111yescharacteristic
287C42H(5,𝔽2)11yescharacteristic
288C422−1+411yescharacteristic
289C23C2×#12111yescharacteristic
290Q8—24no—
291Q8—24no—
292Q8—42no—
293Q8—42no—
294Q8—42no—
295Q8—42no—
296Q8—81no—
297Q8—81no—
298Q8—164no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4#46212yes—
324C2×C4#46212yes—
325C2×C4#46512yes—
326C2×C4#46512yes—
327C2×C4#47112yes—
328C2×C4#52911yescharacteristic
329C2×C4#54211yescharacteristic
330C2×C4#54611yescharacteristic
331C2×C4#54611yescharacteristic
332C2×C4#54711yescharacteristic
333C2×C4#54811yescharacteristic
334C2×C4C22×SD1611yescharacteristic
335C2×C4C22×Q1611yescharacteristic
336C2×C4C2×#13511yescharacteristic
337C2×C4C2×#13711yescharacteristic
338C2×C4#57611yescharacteristic
339C2×C4#57711yescharacteristic
340C2×C4#57711yescharacteristic
341C2×C4#57711yescharacteristic
342C8—24no—
343C8—42no—
344C8—42no—
345C8—81no—
346C8—81no—
347C8—82no—
348C8—81no—
349C22#242711yescharacteristic
350C22C2×#46211yescharacteristic
351C22#304311yescharacteristic
352C22C2×#46511yescharacteristic
353C22#306211yescharacteristic
354C22C2×#47111yescharacteristic
355C22#306911yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—22no—
360C4—41no—
361C4—44no—
362C4—82no—
363C4—82no—
364C4—162no—
365C4—161no—
366C4#167214yes—
367C4#329511yescharacteristic
368C4#329911yescharacteristic
369C4#330311yescharacteristic
370C4#333111yescharacteristic
371C4#333511yescharacteristic
372C4#334111yescharacteristic
373C4#336311yescharacteristic
374C4#336911yescharacteristic
375C2C2×#167211yescharacteristic
376C2#1924011yescharacteristic
377C2#1939711yescharacteristic
378C2#1956411yescharacteristic
379C2#2532411yescharacteristic
380C2#2533911yescharacteristic
381C2#2536111yescharacteristic
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
182. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#292062
#291412
#282552
#231492
#106202
#104792
#98962
#96022
#92182
#86432
C4×#3322