Finite Group Databasebeta

← #6378568

Subgroups — #6378568

Order 512.

1471 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36692C211yesmaximal, characteristic
2#22362C214yesmaximal, complemented
3#22334C214yesmaximal, complemented
4#22300C211yesmaximal, characteristic, complemented
5#22261C211yesmaximal, characteristic, complemented
6#22134C212yesmaximal, complemented
7#22015C212yesmaximal, complemented
8#21927C211yesmaximal, characteristic, complemented
9#21684C211yesmaximal, characteristic, complemented
10#21677C211yesmaximal, characteristic, complemented
11#21571C214yesmaximal, complemented
12#21496C214yesmaximal, complemented
13#21493C212yesmaximal, complemented
14#20485C212yesmaximal, complemented
15#20474C211yesmaximal, characteristic, complemented
16C2×#436C2212yes—
17C2×#436C2214yes—
18C2×#435C2214yes—
19C2×C8×D4C2212yes—
20C2×#433C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C2×#405C2214yes—
23#2837C414yes—
24#2836C414yes—
25#2829C414yes—
26#2828C414yes—
27#2402C2211yescharacteristic
28#2397C2211yescharacteristic
29#2395C2214yes—
30#2392C2214yes—
31#2392C2212yes—
32#2298C2211yescharacteristic
33#2293C2212yes—
34#1955C2214yes—
35#1955C2212yes—
36#1955C2212yes—
37#1954C2214yes—
38#1940C2212yes—
39#1940C2212yes—
40#1939C2214yes—
41#1924C2214yes—
42#1924C2214yes—
43#1924C2214yes—
44#1924C2214yes—
45#1923C2214yes—
46#1917C2214yes—
47#1917C2214yes—
48#1916C2214yes—
49#1870C2214yes—
50#1869C2214yes—
51#1869C2214yes—
52#1869C2214yes—
53#1850C2214yes—
54#1843C2211yescharacteristic
55#1832C2211yescharacteristic
56#1818C2214yes—
57#1798C2214yes—
58#1798C2212yes—
59#1771C2214yes—
60#1771C2212yes—
61#1770C2212yes—
62C8×C4⋊C4C2212yes—
63C8×C4⋊C4C2212yes—
64#1767C2212yes—
65#1753C2214yes—
66#1753C2212yes—
67#1753C2214yes—
68C8×C22⋊C4C2214yes—
69C8×C22⋊C4C2212yes—
70#1750C2214yes—
71#1746C2211yescharacteristic
72#1745C2211yescharacteristic
73#1726C2211yescharacteristic
74C42×C8C2211yescharacteristic
75C22×M16C2314yes—
76C22×M16C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2314yes—
79C2×C4×D4C2312yes—
80C2×C4×D4C2314yes—
81C2×C4×D4C2314yes—
82C2×#117C2311yescharacteristic
83C2×#117C2314yes—
84#436—28no—
85#436—48no—
86#435—48no—
87C8×D4—28no—
88#433—24no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2314yes—
93C2×C4⋊C8C2314yes—
94C2×C4⋊C8C2314yes—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2314yes—
97C2×C22⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2314yes—
103#405—24no—
104#405—48no—
105C2×#97C2314yes—
106C2×#97C2311yescharacteristic
107C2×#97C2311yescharacteristic
108C2×#97C2311yescharacteristic
109C2×#97C2314yes—
110C2×#97C2312yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2312yes—
114C2×C4×C8C2314yes—
115C2×C4×C8C2312yes—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#400C2×C414yes—
119#400C2×C414yes—
120#399C2×C414yes—
121#399C2×C414yes—
122#398C2×C414yes—
123#398C2×C414yes—
124#398C2×C412yes—
125#398C2×C414yes—
126#397C2×C414yes—
127#397C2×C414yes—
128#397C2×C414yes—
129#397C2×C414yes—
130#397C2×C412yes—
131#396C2×C414yes—
132#396C2×C414yes—
133#396C2×C414yes—
134#394C2×C414yes—
135#394C2×C414yes—
136#394C2×C414yes—
137#394C2×C414yes—
138#394C2×C414yes—
139#394C2×C414yes—
140#394C2×C412yes—
141#392C2×C412yes—
142#392C2×C414yes—
143#392C2×C414yes—
144#390C2×C414yes—
145#389C2×C414yes—
146#388C2×C414yes—
147#388C2×C414yes—
148#388C2×C414yes—
149#387C2×C414yes—
150C4×C4⋊C4C2314yes—
151C4×C4⋊C4C2312yes—
152C4×C4⋊C4C2312yes—
153C4×C22⋊C4C2314yes—
154C4×C22⋊C4C2312yes—
155C4×C22⋊C4C2314yes—
156#376C2311yescharacteristic
157C43C2311yescharacteristic
158#336C2312yes—
159#336C2314yes—
160#336C2314yes—
161#336C2312yes—
162#336C2314yes—
163#336C2314yes—
164#336C2314yes—
165#336C2314yes—
166#336C2312yes—
167#336C2312yes—
168#336C2314yes—
169C22×D4C22×C412yes—
170C22×D4C22×C414yes—
171C22×D4C22×C414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2414yes—
174C2×M16—22no—
175C2×M16—28no—
176C2×M16—28no—
177C2×M16—28no—
178C2×M16—42no—
179C22×C8—22no—
180C22×C8—28no—
181C22×C8—24no—
182C22×C8—44no—
183C22×C8—44no—
184C22×C8C2414yes—
185C22×C8C2411yescharacteristic
186C22×C8C2411yescharacteristic
187C22×C8C2414yes—
188C22×C8C2412yes—
189C22×C8C2412yes—
190C22×C8C2411yescharacteristic
191C22×C8C2411yescharacteristic
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—44no—
198#117—22no—
199#117—22no—
200#117—44no—
201#117—44no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C22×C414yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C22×C414yes—
211C2×C4⋊C4C22×C414yes—
212C2×C22⋊C4C22×C412yes—
213C2×C22⋊C4C22×C414yes—
214C2×C22⋊C4C22×C414yes—
215C2×C22⋊C4C22×C414yes—
216C2×C22⋊C4C22×C414yes—
217C2×C22⋊C4C22×C414yes—
218C2×C22⋊C4C22×C414yes—
219C2×C22⋊C4C22×C414yes—
220C2×C22⋊C4C22×C414yes—
221C2×C22⋊C4C22×C414yes—
222C2×C22⋊C4C22×C414yes—
223C2×C22⋊C4C22×C414yes—
224C2×C22⋊C4C22×C414yes—
225C2×C42C22×C414yes—
226C2×C42C2412yes—
227C2×C42C2414yes—
228C2×C42C2411yescharacteristic
229C2×C42C2412yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—44no—
235C4⋊C8—44no—
236C4⋊C8—44no—
237C22⋊C8—24no—
238C22⋊C8—28no—
239C22⋊C8—28no—
240C22⋊C8—44no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—44no—
244C22⋊C8—44no—
245#97—22no—
246#97—28no—
247#97—22no—
248#97—22no—
249#97—42no—
250#97—44no—
251C4×C8—22no—
252C4×C8—24no—
253C4×C8—24no—
254C4×C8—42no—
255C4×C8—44no—
256C4×C8#4114yes—
257C4×C8#4114yes—
258#95C22×C414yes—
259#95C22×C414yes—
260#95C22×C414yes—
261#95C22×C414yes—
262#95C22×C414yes—
263#95C22×C414yes—
264#95C22×C414yes—
265#95C22×C412yes—
266#95C22×C412yes—
267#95C22×C414yes—
268#95C22×C414yes—
269C24C23×C411yescharacteristic
270C24C23×C414yes—
271C2×D4—24no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4C23×C412yes—
290C22×C4C23×C414yes—
291C22×C4C23×C414yes—
292C22×C4C23×C414yes—
293C22×C4C23×C414yes—
294C22×C4C23×C414yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C23×C412yes—
297C22×C4C2511yescharacteristic
298M16—44no—
299M16—416no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—416no—
317C2×C8—44no—
318C2×C8C2×#4112yes—
319C2×C8C2×#4114yes—
320C2×C8C2×#4112yes—
321C2×C8H(5,𝔽2)14yes—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C22⋊C4—24no—
333C22⋊C4—28no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—22no—
351C42—22no—
352C42—24no—
353C42—44no—
354C42C2×#4112yes—
355C42C2×#4112yes—
356C23—21no—
357C23—24no—
358C23—24no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23C24×C411yescharacteristic
367D4—48no—
368D4—88no—
369D4—88no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4C4×#4114yes—
398C2×C4C22×#4111yescharacteristic
399C2×C4C22×#4111yescharacteristic
400C2×C4C2×H(5,𝔽2)12yes—
401C2×C4#58512yes—
402C8—22no—
403C8—24no—
404C8—41no—
405C8—44no—
406C8—41no—
407C8—42no—
408C8—44no—
409C8#54014yes—
410C22—41no—
411C22—44no—
412C22—44no—
413C22—42no—
414C22—44no—
415C22—44no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22C2×C4×#4111yescharacteristic
421C22C4×H(5,𝔽2)12yes—
422C22#357314yes—
423C4—24no—
424C4—44no—
425C4—41no—
426C4—42no—
427C4—44no—
428C4—44no—
429C4—44no—
430C4—44no—
431C4C2×#54011yescharacteristic
432C4#345511yescharacteristic
433C4#346012yes—
434C2—84no—
435C2—81no—
436C2C4×#54011yescharacteristic
437C2#6020612yes—
438C2#6032714yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radical#36692C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22362C2semidirect
3. #22334C2semidirect
4. #22300C2semidirect
5. #22261C2semidirect
6. #22134C2semidirect
7. #22015C2semidirect
8. #21927C2semidirect
9. #21684C2semidirect
10. #21677C2semidirect
11. #21571C2semidirect
12. #21496C2semidirect
13. #21493C2semidirect
14. #20485C2semidirect
15. #20474C2semidirect

Maximal subgroups

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

subgroupindex
#366922
#223622
#223342
#223002
#222612
#221342
#220152
#219272
#216842
#216772
#215712
#214962
#214932
#204852
#204742