Finite Group Databasebeta

← #496959

Subgroups — #496959

Order 512.

724 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
1C2×#2206C211yesmaximal, characteristic
2C4×#493C211yesmaximal, characteristic, complemented
3#21275C211yesmaximal, characteristic, complemented
4#13252C211yesmaximal, characteristic, complemented
5#13239C212yesmaximal, complemented
6#13177C211yesmaximal, characteristic, complemented
7#13149C212yesmaximal, complemented
8#13012C212yesmaximal, complemented
9#12937C212yesmaximal, complemented
10#12811C211yesmaximal, characteristic, complemented
11#12653C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C2×#493C2211yescharacteristic
14C2×C4×D8—21no—
15C2×C4×D8C2211yescharacteristic
16C2×C4×D8C2212yes—
17C4×#127—21no—
18#2293—21no—
19#2206C418yescomplemented
20C2×#367C2212yes—
21C2×C4⋊C16C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×#357C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#346C2212yes—
28#1970—21no—
29#1970C2211yescharacteristic
30#1970C2211yescharacteristic
31#1970C2212yes—
32#1957—21no—
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1934—21no—
36#1934—21no—
37#1845C2211yescharacteristic
38C4×#107C2211yescharacteristic
39C4×#102—21no—
40C42×C8C2211yescharacteristic
41#1380C2212yes—
42#1380C2212yes—
43#1374C2212yes—
44C22×D8—21no—
45C22×D8C2312yes—
46C22×D8C2311yescharacteristic
47C2×#127—21no—
48C2×C4×D4—21no—
49C2×C4×D4—22no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×#117—41no—
53C2×D16—42no—
54C2×D16—42no—
55C2×D16C2×C418yescomplemented
56C22×C16C2312yes—
57C22×C16C2312yes—
58#493—22no—
59#493C2×C414yes—
60C4×D8—22no—
61C4×D8—42no—
62C4×D8—42no—
63C4×D8—44no—
64C4×D8C2×C414yescomplemented
65C2×#107—21no—
66C2×#107C2312yes—
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#102—21no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—21no—
74C2×#102—21no—
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#390—22no—
82#390—21no—
83#390—21no—
84#390—41no—
85#390—41no—
86#389—21no—
87#389—21no—
88#388—41no—
89#388—41no—
90#384—21no—
91C4×C4⋊C4—21no—
92C4×C22⋊C4—41no—
93C43D411yescharacteristic
94#367—42no—
95C4⋊C16—22no—
96C4⋊C16C2×C414yescomplemented
97#357—42no—
98#357—42no—
99#357—42no—
100#357C2×C418yescomplemented
101#346—42no—
102#340—21no—
103#340—21no—
104#340—21no—
105C22×D4—22no—
106C22×D4—21no—
107C22×D4—41no—
108C22×D4—41no—
109C23×C4—41no—
110C23×C4—42no—
111C2×D8—24no—
112C2×D8—21no—
113C2×D8—22no—
114C2×D8—21no—
115C2×D8—22no—
116C2×D8—22no—
117C2×D8—22no—
118C2×D8—21no—
119C2×D8—21no—
120C2×D8—22no—
121C2×D8—21no—
122C2×D8—42no—
123C2×D8—42no—
124C2×D8C22×C414yes—
125C2×D8C22×C412yes—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2411yescharacteristic
130#127—24no—
131#127—42no—
132C4×D4—24no—
133C4×D4—44no—
134C4×D4—44no—
135C4×D4—42no—
136C4×D4—84no—
137C4×D4—84no—
138#117—84no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C42—41no—
156C2×C42D811yescharacteristic
157C2×C42D811yescharacteristic
158C2×C42D811yescharacteristic
159C2×C42D811yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163D16—48no—
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16C22×C414yes—
170#107—21no—
171#107—21no—
172#107—21no—
173#107—22no—
174#107—21no—
175#107—22no—
176#107—21no—
177#107—42no—
178#107C22×C412yes—
179#102—24no—
180#102—42no—
181#102—44no—
182#102—42no—
183#102—44no—
184C4×C8—22no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—21no—
189C4×C8—22no—
190C4×C8—41no—
191C4×C8C22×C412yes—
192C4×C8#4112yes—
193#95—41no—
194#95—41no—
195#95—41no—
196C24—41no—
197C24—42no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—42no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—82no—
213C2×D4—82no—
214C2×D4—82no—
215C2×D4—82no—
216C22×C4—41no—
217C22×C4—44no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—84no—
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C22×D411yescharacteristic
233D8—22no—
234D8—24no—
235D8—22no—
236D8—24no—
237D8—48no—
238D8—41no—
239D8—42no—
240D8—41no—
241D8—42no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8C4×D412yes—
250C2×C8AGL(1,ℤ/8)12yes—
251C2×C8C23×C411yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8H(5,𝔽2)11yescharacteristic
254C2×C8H(5,𝔽2)11yescharacteristic
255C2×C8H(5,𝔽2)11yescharacteristic
256C2×C82−1+411yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C22⋊C4—44no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—84no—
274C22⋊C4—84no—
275C22⋊C4—84no—
276C22⋊C4—82no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—82no—
286C42C4×D412yes—
287C42#13512yes—
288C16—24no—
289C16—42no—
290C23—42no—
291C23—44no—
292C23—81no—
293C23—82no—
294C23—82no—
295C23—81no—
296C23—81no—
297C23—82no—
298C23—81no—
299C23—82no—
300C23—82no—
301C23—81no—
302C23—81no—
303C23—82no—
304C23C22×D811yescharacteristic
305D4—42no—
306D4—44no—
307D4—44no—
308D4—42no—
309D4—88no—
310D4—88no—
311D4—82no—
312D4—81no—
313D4—81no—
314D4—82no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—168no—
342C2×C4C4×D812yes—
343C2×C4C4×D812yes—
344C2×C4#50912yes—
345C2×C4#50912yes—
346C2×C4#50912yes—
347C2×C4C2×C4×D411yescharacteristic
348C2×C4#53511yescharacteristic
349C2×C4#53711yescharacteristic
350C2×C4C2×#13511yescharacteristic
351C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
352C2×C4#57511yescharacteristic
353C8—22no—
354C8—22no—
355C8—41no—
356C8#51811yescharacteristic
357C8#52011yescharacteristic
358C8#54011yescharacteristic
359C8#54211yescharacteristic
360C22—84no—
361C22—82no—
362C22—82no—
363C22—84no—
364C22—84no—
365C22—82no—
366C22—162no—
367C22—161no—
368C22—161no—
369C22—162no—
370C22—161no—
371C22—162no—
372C22—162no—
373C22—161no—
374C22C2×C4×D811yescharacteristic
375C22#320511yescharacteristic
376C22#324511yescharacteristic
377C22C2×#50911yescharacteristic
378C22C2×#50911yescharacteristic
379C22C2×#50911yescharacteristic
380C22#341411yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—82no—
387C4—161no—
388C4—161no—
389C4—162no—
390C4#217714yes—
391C4#232411yescharacteristic
392C4C4×AGL(1,ℤ/8)11yescharacteristic
393C4#321211yescharacteristic
394C4#324711yescharacteristic
395C2—162no—
396C2—164no—
397C2#2214511yescharacteristic
398C2C2×#217711yescharacteristic
399C2C4×#50911yescharacteristic
400C2#3378011yescharacteristic
401C2#3378311yescharacteristic
402C2#3381711yescharacteristic
403C2#3381911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2206C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#493C2semidirect
3. #21275C2semidirect
4. #13252C2semidirect
5. #13239C2semidirect
6. #13177C2semidirect
7. #13149C2semidirect
8. #13012C2semidirect
9. #12937C2semidirect
10. #12811C2semidirect
11. #12653C2semidirect
19. #2206C4semidirect
55. C2×D16C2×C4semidirect
64. C4×D8C2×C4semidirect
96. C4⋊C16C2×C4semidirect
100. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22062
C4×#4932
#212752
#132522
#132392
#131772
#131492
#130122
#129372
#128112
#126532