Finite Group Databasebeta

← #319062

Subgroups — #319062

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33170C211yesmaximal, characteristic
2#27960C211yesmaximal, characteristic, complemented
3#27909C211yesmaximal, characteristic
4#25903C211yesmaximal, characteristic
5#25813C211yesmaximal, characteristic, complemented
6C2×#1956C211yesmaximal, characteristic, complemented
7#10675C212yesmaximal, complemented
8#10308C212yesmaximal, complemented
9#9823C212yesmaximal, complemented
10#9822C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#500C2212yes—
14C2×#492C2211yescharacteristic
15C2×#486C2212yes—
16C2×#480—22no—
17C2×#465—22no—
18C22×#103—22no—
19C22×C4×C8C2211yescharacteristic
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2099—22no—
24#2093—22no—
25#2057—22no—
26#2023—22no—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1968—22no—
31#1957C2212yes—
32#1957C2211yescharacteristic
33#1956—22no—
34#1956C2212yes—
35#1956C2212yes—
36#1956C2211yescharacteristic
37#1874—22no—
38#1852—22no—
39#1845C2212yes—
40#1844C2211yescharacteristic
41#1834C2212yes—
42#1823—22no—
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×C82C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C22×SD16—22no—
49C23×C8D412yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2312yes—
57C2×#127C2311yescharacteristic
58C2×C22⋊Q8—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C42C2311yescharacteristic
68#500—24no—
69#500—42no—
70#492—42no—
71#486—24no—
72#486—42no—
73#480—48no—
74#465—48no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#106—22no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—22no—
81C2×#103—48no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—22no—
85C2×C22⋊C8D412yes—
86C2×C22⋊C8D412yes—
87C2×C4×C8—42no—
88C2×C4×C8D412yes—
89C2×C4×C8C2312yes—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2312yes—
92C2×C4×C8C2311yescharacteristic
93#391—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2312yes—
105#380—21no—
106#380—21no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336D412yes—
112#336D412yes—
113C82—42no—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×Q8—41no—
120C22×Q8—44no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×SD16—44no—
131C2×SD16—44no—
132C2×SD16—44no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8D818yes—
136C22×C8C2×D414yes—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C22×C8C2×D414yes—
140C4⋊Q8—24no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C4⋊Q8—42no—
149C4⋊Q8#4114yes—
150#127—22no—
151#127—22no—
152C22⋊Q8—44no—
153#121—44no—
154#121—44no—
155#121—44no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2411yescharacteristic
190#107—48no—
191#107—48no—
192#106—44no—
193#103—48no—
194#103—48no—
195#103—44no—
196#102—48no—
197#102—44no—
198#102—48no—
199C22⋊C8—24no—
200C22⋊C8SD1618yescomplemented
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—42no—
208#95—41no—
209#95—41no—
210C24—41no—
211C24#12711yescharacteristic
212C2×Q8—24no—
213C2×Q8—44no—
214C2×Q8—416no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8—82no—
223C2×Q8—88no—
224C2×Q8—82no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—82no—
236C2×D4—82no—
237C2×D4—82no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4#12711yescharacteristic
257C22×C4#12711yescharacteristic
258C22×C4#12711yescharacteristic
259C22×C4C2×D814yes—
260C22×C4C2×SD1614yes—
261C22×C4#13514yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265SD16—88no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—48no—
276C2×C8C2×D814yes—
277C2×C8C2×D814yes—
278C2×C8C2×SD1618yes—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42C2×#4112yes—
309C42H(5,𝔽2)12yes—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23#49212yes—
322C23#49312yes—
323C23#49512yes—
324C23C2×#12711yescharacteristic
325Q8—48no—
326Q8—88no—
327Q8—84no—
328Q8—82no—
329Q8—82no—
330D4—82no—
331D4—82no—
332D4—84no—
333D4—84no—
334D4—164no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—44no—
346C2×C4—88no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—164no—
362C2×C4#49212yes—
363C2×C4#49312yes—
364C2×C4#49512yes—
365C2×C4#52511yescharacteristic
366C2×C4#53411yescharacteristic
367C2×C4#54612yes—
368C2×C4#54612yes—
369C2×C4C22×D812yes—
370C2×C4C22×SD1612yes—
371C2×C4C2×#13512yes—
372C2×C4#57512yes—
373C2×C4#57812yes—
374C2×C4#57812yes—
375C8—24no—
376C8—24no—
377C8—28no—
378C8—44no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—161no—
386C22—161no—
387C22—161no—
388C22—161no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22#170614yes—
393C22#253911yescharacteristic
394C22C2×#49211yescharacteristic
395C22C2×#49311yescharacteristic
396C22C2×#49511yescharacteristic
397C22#315311yescharacteristic
398C22#315611yescharacteristic
399C22#315611yescharacteristic
400C4—22no—
401C4—22no—
402C4—22no—
403C4—84no—
404C4—162no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4#310012yes—
409C4#318612yes—
410C4#329814yes—
411C4#330014yes—
412C2—24no—
413C2—321no—
414C2C2×#170611yescharacteristic
415C2#1962411yescharacteristic
416C2#1962811yescharacteristic
417C2#1966211yescharacteristic
418C2#2830111yescharacteristic
419C2#2833511yescharacteristic
420C2#2833611yescharacteristic
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
2. #27960C2semidirect
5. #25813C2semidirect
6. C2×#1956C2semidirect
7. #10675C2semidirect
8. #10308C2semidirect
9. #9823C2semidirect
10. #9822C2semidirect
11. C8×C22⋊C8C2semidirect
200. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#331702
#279602
#279092
#259032
#258132
C2×#19562
#106752
#103082
#98232
#98222
C8×C22⋊C82