Finite Group Databasebeta

← #206549

Subgroups — #206549

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22023C211yesmaximal, characteristic, complemented
2#22013C211yesmaximal, characteristic
3#21279C211yesmaximal, characteristic
4#9660C211yesmaximal, characteristic
5#9655C211yesmaximal, characteristic
6#9595C211yesmaximal, characteristic, complemented
7#9594C211yesmaximal, characteristic, complemented
8#9556C211yesmaximal, characteristic
9#9555C211yesmaximal, characteristic, complemented
10#9554C211yesmaximal, characteristic
11#9553C211yesmaximal, characteristic, complemented
12#8276C211yesmaximal, characteristic, complemented
13#8177C211yesmaximal, characteristic
14#8176C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×C4×SD16C2211yescharacteristic
17#2295C2211yescharacteristic
18#2293C2211yescharacteristic
19#1984C2211yescharacteristic
20#1984C2211yescharacteristic
21#1973C412yes—
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1944C412yes—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1936C412yes—
30#1935C412yes—
31#1924—21no—
32#1924—21no—
33#1917—21no—
34#1917—21no—
35#1840—21no—
36#1832—21no—
37#1818—21no—
38#1798—21no—
39C4×#107C2211yescharacteristic
40#1767C2211yescharacteristic
41#1767C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1744C2211yescharacteristic
45C4×#97C2211yescharacteristic
46C42×C8C2211yescharacteristic
47C2×#335C2211yescharacteristic
48C2×#334C2211yescharacteristic
49C2×Q8⋊C8C2211yescharacteristic
50C2×Q8⋊C8C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×#322C2211yescharacteristic
54C2×#322C2211yescharacteristic
55C2×C82C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63C22×SD16C2×C411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117—21no—
67C4×SD16—22no—
68C4×SD16—22no—
69C2×#107C2×C412yes—
70C2×#106C2×C411yescharacteristic
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×#103C2×C411yescharacteristic
80C2×#103C2×C412yes—
81C2×#102C2×C411yescharacteristic
82C2×#102C2×C412yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×#97—21no—
88C2×#97—21no—
89C2×#97C2×C412yes—
90C2×#97C2×C412yes—
91C2×C4×C8—22no—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391C2×C412yes—
99#390C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#388—22no—
103#387—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2311yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C4×C22⋊C4—21no—
108#376—21no—
109C43C2311yescharacteristic
110#340C2×C412yes—
111#340C2×C412yes—
112#340C2×C412yes—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#336—21no—
121#336—21no—
122#336—21no—
123#336D411yescharacteristic
124#336D411yescharacteristic
125#336D411yescharacteristic
126#336D411yescharacteristic
127#335—22no—
128#334—22no—
129Q8⋊C8—22no—
130Q8⋊C8—22no—
131D4⋊C8—22no—
132D4⋊C8—22no—
133#322—22no—
134#322—22no—
135C82—22no—
136C82—22no—
137C22×Q8C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C2×SD16—22no—
141C2×SD16—24no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8C22×C411yescharacteristic
149C22×C8C22×C412yes—
150C22×C8C22×C411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163#117—81no—
164#117—81no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C22×C412yes—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4112yes—
182C2×C42C2411yescharacteristic
183#107—24no—
184#106—22no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—81no—
194C4⋊C8—81no—
195C4⋊C8—81no—
196C4⋊C8—81no—
197#103—22no—
198#103—24no—
199#102—22no—
200#102—24no—
201C22⋊C8—81no—
202C22⋊C8—81no—
203C22⋊C8—81no—
204C22⋊C8—81no—
205#97—24no—
206#97—24no—
207#97—81no—
208#97—81no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—44no—
218C4×C8#4112yes—
219C4×C8#4112yes—
220C4×C8#4112yes—
221C4×C8#4112yes—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228C24—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4C4×D412yes—
249C22×C4C4×D412yes—
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C2×#4111yescharacteristic
253SD16—44no—
254C2×C8—24no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8#11712yes—
286C2×C8#11712yes—
287C2×C8#13114yes—
288C2×C8#13114yes—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C42—22no—
306C42—22no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—42no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42C2×#4111yescharacteristic
319C42C2×#4111yescharacteristic
320C42C2×#4111yescharacteristic
321C42C2×#4111yescharacteristic
322C23—42no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23C2×C4×D411yescharacteristic
328Q8—41no—
329Q8—41no—
330Q8—42no—
331D4—41no—
332D4—41no—
333D4—42no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4C2×#11711yescharacteristic
359C2×C4C4×#4111yescharacteristic
360C2×C4#52911yescharacteristic
361C2×C4#52911yescharacteristic
362C2×C4#53211yescharacteristic
363C2×C4#53211yescharacteristic
364C2×C4C2×#13112yes—
365C2×C4C2×#13112yes—
366C2×C4#57611yescharacteristic
367C2×C4#57711yescharacteristic
368C2×C4#57711yescharacteristic
369C2×C4#57811yescharacteristic
370C8—24no—
371C8—22no—
372C8—24no—
373C8—42no—
374C8—41no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—81no—
379C8—81no—
380C8—81no—
381C8—81no—
382C8#43314yes—
383C22—81no—
384C22—81no—
385C22—82no—
386C22—82no—
387C22—81no—
388C22#229311yescharacteristic
389C22#293112yes—
390C22#294611yescharacteristic
391C22#294711yescharacteristic
392C22#295612yes—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—82no—
400C4—82no—
401C4—81no—
402C4C2×#43311yescharacteristic
403C4#292211yescharacteristic
404C4#296612yes—
405C4#317411yescharacteristic
406C4#317511yescharacteristic
407C4#317811yescharacteristic
408C4#317911yescharacteristic
409C2—161no—
410C2#1796812yes—
411C2#1796912yes—
412C2#2126611yescharacteristic
413C2#2129111yescharacteristic
414C2#2129711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22023C2semidirect
6. #9595C2semidirect
7. #9594C2semidirect
9. #9555C2semidirect
11. #9553C2semidirect
12. #8276C2semidirect
14. #8176C2semidirect
15. C8×#97C2semidirect

Maximal subgroups

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

subgroupindex
#220232
#220132
#212792
#96602
#96552
#95952
#95942
#95562
#95552
#95542
#95532
#82762
#81772
#81762
C8×#972