Finite Group Databasebeta

← #314742

Subgroups — #314742

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27627C212yesmaximal
2#21487C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic, complemented
4#10651C211yesmaximal, characteristic, complemented
5#10650C211yesmaximal, characteristic, complemented
6#10616C211yesmaximal, characteristic, complemented
7#10615C211yesmaximal, characteristic, complemented
8#9619C212yesmaximal, complemented
9#8729C212yesmaximal, complemented
10#8654C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×#436C2211yescharacteristic
13C2×#435—21no—
14C22×C4×C8C2211yescharacteristic
15C4×#121C2211yescharacteristic
16#2285C2212yes—
17#2077—22no—
18#2033—22no—
19#1954C2211yescharacteristic
20#1948C2212yes—
21#1947C2212yes—
22#1939C2211yescharacteristic
23#1926—22no—
24#1923—21no—
25#1921—22no—
26#1869—21no—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1834C2212yes—
30#1808—21no—
31#1808—21no—
32#1802C2212yes—
33#1802C2212yes—
34#1797C2211yescharacteristic
35#1797C2212yes—
36C4×#107—21no—
37C4×#107—21no—
38#1770C2211yescharacteristic
39#1753—21no—
40C2×#335C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×M16—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4C2311yescharacteristic
53C2×#117—22no—
54C22×C42C2311yescharacteristic
55#436—24no—
56#435—44no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—22no—
62C2×#106—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×#102—22no—
69C2×#102—22no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×#97—21no—
76C2×C4×C8—24no—
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#390D411yescharacteristic
82#389D411yescharacteristic
83#388—21no—
84#386—21no—
85#384C2312yes—
86#384C2312yes—
87#382—22no—
88#380—22no—
89#380—22no—
90#380D411yescharacteristic
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2311yescharacteristic
94C4×C22⋊C4—21no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—21no—
102#336—21no—
103#336—22no—
104#336—21no—
105#335—24no—
106C22×D4—21no—
107C22×D4C2×D411yescharacteristic
108C23×C4—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129#121—42no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—42no—
133C4×D4—42no—
134#117—84no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C4⋊C4C2×D411yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—42no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42C2411yescharacteristic
159#107—44no—
160#107—42no—
161#107—42no—
162#107—42no—
163#107—42no—
164#106—82no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—42no—
172#102—44no—
173#102—82no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—82no—
179#97—42no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8#4114yes—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—42no—
188#95—42no—
189C24—21no—
190C24#12111yescharacteristic
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4C2×#4111yescharacteristic
228M16—84no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8#12312yes—
252C2×C8#12312yes—
253C2×C8#13512yes—
254C2×C8AGL(1,ℤ/8)12yes—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—84no—
276C42—21no—
277C42—21no—
278C42—22no—
279C42—22no—
280C42—41no—
281C42—41no—
282C42—84no—
283C42C2×#4112yes—
284C422−1+412yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#47212yes—
299C23C2×#12111yescharacteristic
300D4—42no—
301D4—42no—
302D4—84no—
303C2×C4—28no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4#47312yes—
340C2×C4C2×#12311yescharacteristic
341C2×C4#53611yescharacteristic
342C2×C4#54712yes—
343C2×C4#55212yes—
344C2×C4C2×#13511yescharacteristic
345C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
346C2×C4#57511yescharacteristic
347C2×C4#57512yes—
348C2×C4#57511yescharacteristic
349C2×C4#57611yescharacteristic
350C2×C4#57811yescharacteristic
351C8—22no—
352C8—22no—
353C8—22no—
354C8—22no—
355C8—82no—
356C8—82no—
357C8—81no—
358C8—81no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#248611yescharacteristic
374C22#303511yescharacteristic
375C22#303611yescharacteristic
376C22#305411yescharacteristic
377C22#306111yescharacteristic
378C22C2×#47211yescharacteristic
379C22C2×#47311yescharacteristic
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—162no—
388C4—162no—
389C4#309111yescharacteristic
390C4#309411yescharacteristic
391C4#322811yescharacteristic
392C4#324011yescharacteristic
393C4#334712yes—
394C4#334912yes—
395C2—42no—
396C2—161no—
397C2#1916111yescharacteristic
398C2#1922211yescharacteristic
399C2#1928311yescharacteristic
400C2#1933311yescharacteristic
401C2#2765011yescharacteristic
402C2#2766411yescharacteristic
403C2#2767511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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. #20817C2semidirect
4. #10651C2semidirect
5. #10650C2semidirect
6. #10616C2semidirect
7. #10615C2semidirect
8. #9619C2semidirect
9. #8729C2semidirect
10. #8654C2semidirect
11. #8596C2semidirect

Maximal subgroups

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

subgroupindex
#276272
#214872
#208172
#106512
#106502
#106162
#106152
#96192
#87292
#86542
#85962