Finite Group Databasebeta

← #315730

Subgroups — #315730

Order 512.

801 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33170C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#27961C211yesmaximal, characteristic, complemented
4#27960C211yesmaximal, characteristic, complemented
5#27909C211yesmaximal, characteristic
6C2×#1834C211yesmaximal, characteristic, complemented
7#10679C212yesmaximal, complemented
8#10678C212yesmaximal, complemented
9#9829C212yesmaximal, complemented
10#8821C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic, complemented
12C2×#492C2211yescharacteristic
13C2×#480—22no—
14C2×#465—22no—
15C22×C4×C8C2211yescharacteristic
16#2857C2211yescharacteristic
17#2856C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2100—22no—
21#2099—22no—
22#2093—22no—
23#2071—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1968—22no—
27#1957C2212yes—
28#1957C2212yes—
29#1957C2211yescharacteristic
30#1956C2211yescharacteristic
31#1874—22no—
32#1845C2212yes—
33#1844C2211yescharacteristic
34#1834—22no—
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38#1834C2212yes—
39#1834C2212yes—
40#1824—22no—
41#1823—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×#322C2212yes—
46#1273C2212yes—
47C22×SD16—22no—
48C23×C8D412yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×C22⋊Q8—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C42C2311yescharacteristic
62#492—24no—
63#480—48no—
64#465—48no—
65C2×#107—24no—
66C2×#106—22no—
67C2×#103—22no—
68C2×#102—24no—
69C2×#102—22no—
70C2×#102—24no—
71C2×C22⋊C8D412yes—
72C2×C22⋊C8D412yes—
73C2×C4×C8—24no—
74C2×C4×C8D412yes—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2312yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79#391—21no—
80#390—21no—
81#390—21no—
82#390—21no—
83#389—21no—
84#389—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#384C2312yes—
94#384C2312yes—
95#380—21no—
96#380—21no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#336D412yes—
105#336D412yes—
106#322—24no—
107C22×Q8—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—41no—
112C23×C4—41no—
113C23×C4—41no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C2×SD16—44no—
118C2×SD16—44no—
119C2×SD16—44no—
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—42no—
123C22×C8C2×D412yes—
124C22×C8C2×D414yes—
125C22×C8C2×D414yes—
126C22×C8C2×D412yes—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129#127—22no—
130#127—22no—
131C22⋊Q8—44no—
132#121—44no—
133#121—44no—
134#121—44no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—82no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2411yescharacteristic
172#107—48no—
173#106—44no—
174#103—44no—
175#102—44no—
176#102—48no—
177#102—84no—
178C22⋊C8—24no—
179C22⋊C8—24no—
180C4×C8—24no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8#4114yes—
187#95—41no—
188#95—41no—
189C24—41no—
190C24#12711yescharacteristic
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—82no—
206C2×D4—82no—
207C2×D4—82no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4#12711yescharacteristic
227C22×C4#12711yescharacteristic
228C22×C4#12711yescharacteristic
229C22×C4#13514yes—
230C22×C4AGL(1,ℤ/8)14yes—
231C22×C4AGL(1,ℤ/8)14yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235SD16—88no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—48no—
248C2×C8#12314yes—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C42—21no—
275C42—21no—
276C42—22no—
277C42—22no—
278C42C2×#4112yes—
279C422−1+412yes—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23#49512yes—
292C23#49612yes—
293C23#49612yes—
294C23C2×#12711yescharacteristic
295Q8—82no—
296Q8—82no—
297D4—84no—
298D4—84no—
299D4—82no—
300D4—82no—
301D4—164no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—164no—
328C2×C4—164no—
329C2×C4#49512yes—
330C2×C4#49612yes—
331C2×C4#49612yes—
332C2×C4C2×#12311yescharacteristic
333C2×C4#53611yescharacteristic
334C2×C4#54712yes—
335C2×C4#54712yes—
336C2×C4C2×#13512yes—
337C2×C4C2×AGL(1,ℤ/8)12yes—
338C2×C4C2×AGL(1,ℤ/8)12yes—
339C2×C4#57512yes—
340C2×C4#57712yes—
341C2×C4#57812yes—
342C8—24no—
343C8—44no—
344C8—44no—
345C8—42no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—161no—
353C22—161no—
354C22—161no—
355C22—161no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22#171714yes—
360C22#250111yescharacteristic
361C22C2×#49511yescharacteristic
362C22C2×#49611yescharacteristic
363C22C2×#49611yescharacteristic
364C22#315311yescharacteristic
365C22#315411yescharacteristic
366C22#315611yescharacteristic
367C4—22no—
368C4—22no—
369C4—22no—
370C4—161no—
371C4—162no—
372C4—161no—
373C4—162no—
374C4—161no—
375C4#309412yes—
376C4#323412yes—
377C4#333714yes—
378C4#334314yes—
379C2—24no—
380C2—321no—
381C2C2×#171711yescharacteristic
382C2#1962811yescharacteristic
383C2#1967711yescharacteristic
384C2#1970011yescharacteristic
385C2#2791811yescharacteristic
386C2#2792611yescharacteristic
387C2#2793911yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27961C2semidirect
4. #27960C2semidirect
6. C2×#1834C2semidirect
7. #10679C2semidirect
8. #10678C2semidirect
9. #9829C2semidirect
10. #8821C2semidirect
11. #8145C2semidirect

Maximal subgroups

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

subgroupindex
#331702
#279732
#279612
#279602
#279092
C2×#18342
#106792
#106782
#98292
#88212
#81452