Finite Group Databasebeta

← #203349

Subgroups — #203349

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21270C211yesmaximal, characteristic, complemented
2C42×D8C211yesmaximal, characteristic
3#20494C211yesmaximal, characteristic, complemented
4#9542C212yesmaximal, complemented
5#9536C212yesmaximal, complemented
6#8951C212yesmaximal
7#8849C212yesmaximal, complemented
8#8273C211yesmaximal, characteristic, complemented
9#8174C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11#8132C211yesmaximal, characteristic, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C2×#436C2212yes—
16C2×#432—21no—
17C2×C4×M16—21no—
18C42×D4C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#1954C2212yes—
21#1939C2212yes—
22#1924—22no—
23C4×#107C411yescharacteristic
24C4×#107C411yescharacteristic
25C4×#107C2211yescharacteristic
26C8×C4⋊C4C2212yes—
27#1767C2211yescharacteristic
28#1767C2211yescharacteristic
29C4×#102C411yescharacteristic
30C4×#102C411yescharacteristic
31C4×#102C411yescharacteristic
32C4×#102C411yescharacteristic
33C4×#102C2211yescharacteristic
34C4×#102C2211yescharacteristic
35#1753—22no—
36#1749—21no—
37#1749—21no—
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×#334C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1277C2212yes—
47#1273C2212yes—
48#1273C2212yes—
49C22×D8C2×C411yescharacteristic
50C22×M16—22no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C4×D8—24no—
60C4×D8—24no—
61C4×D8—24no—
62#436—24no—
63#436—24no—
64#432—44no—
65C2×#107C2×C411yescharacteristic
66C2×#107C2×C411yescharacteristic
67C2×#107C2×C411yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×#102C2×C411yescharacteristic
72C2×#102C2×C411yescharacteristic
73C2×#102C2×C411yescharacteristic
74C2×#102C2×C411yescharacteristic
75C2×#102C2×C411yescharacteristic
76C2×#102C2×C411yescharacteristic
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C4×M16—44no—
81C2×#97—21no—
82C2×#97—21no—
83C2×#97D411yescharacteristic
84C2×#97D411yescharacteristic
85C2×#97D412yes—
86C2×C4×C8—22no—
87C2×C4×C8C2×C411yescharacteristic
88C2×C4×C8C2×C411yescharacteristic
89C2×C4×C8C2×C411yescharacteristic
90C2×C4×C8C2×C411yescharacteristic
91C2×C4×C8C2312yes—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2312yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C4⋊C4C2×C411yescharacteristic
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C4×C22⋊C4—21no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C43C2311yescharacteristic
108#340C2×C411yescharacteristic
109#340C2×C411yescharacteristic
110#340C2×C411yescharacteristic
111#340C2×C411yescharacteristic
112#340C2×C411yescharacteristic
113#340C2×C411yescharacteristic
114#336—22no—
115#336—22no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#334—24no—
121D4⋊C8—24no—
122D4⋊C8—24no—
123C22×D4C22×C411yescharacteristic
124C22×D4C22×C411yescharacteristic
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C2×D8—22no—
132C2×D8—22no—
133C2×D8—22no—
134C2×M16—28no—
135C2×M16—44no—
136C2×M16—44no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C22×C411yescharacteristic
141C22×C8C22×C411yescharacteristic
142C22×C8C22×C411yescharacteristic
143C22×C8C22×C411yescharacteristic
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C4×D4#4114yes—
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187#107—22no—
188#107—22no—
189#107M1614yescomplemented
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—44no—
194C4⋊C8#4114yes—
195#102—22no—
196#102—22no—
197#102—22no—
198#102—22no—
199#102—22no—
200#102M1614yescomplemented
201C22⋊C8—28no—
202C22⋊C8—44no—
203C22⋊C8—44no—
204#97—24no—
205#97—22no—
206#97—22no—
207#97—42no—
208#97—42no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218C4×C8—41no—
219C4×C8—44no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226C24—21no—
227C24—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4#11712yes—
239C2×D4C2×M1612yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273D8—28no—
274M16—416no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8C2×M1612yes—
300C2×C8C2×M1612yes—
301C2×C8#13514yes—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4#11712yes—
311C4⋊C4C2×M1612yes—
312C4⋊C4C2×M1612yes—
313C4⋊C4C2×M1612yes—
314C22⋊C4—24no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42AGL(1,ℤ/8)12yes—
336C42AGL(1,ℤ/8)12yes—
337C42C2×#4112yes—
338C42C2×#4112yes—
339C23—22no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23C2×C4×D411yescharacteristic
354D4—22no—
355D4—22no—
356D4—22no—
357D4—22no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—41no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#44212yes—
401C2×C4#44212yes—
402C2×C4C2×#11711yescharacteristic
403C2×C4C4×#4111yescharacteristic
404C2×C4#52512yes—
405C2×C4#52912yes—
406C2×C4C22×M1611yescharacteristic
407C2×C4C22×M1611yescharacteristic
408C2×C4#56811yescharacteristic
409C2×C4#56811yescharacteristic
410C2×C4C2×#13512yes—
411C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
413C8—24no—
414C8—22no—
415C8—22no—
416C8—44no—
417C8—42no—
418C8—41no—
419C8—41no—
420C8—82no—
421C8—82no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#229011yescharacteristic
437C22D4×M1611yescharacteristic
438C22D4×M1611yescharacteristic
439C22C4×#13511yescharacteristic
440C22C2×#44211yescharacteristic
441C22#295711yescharacteristic
442C22#295711yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—81no—
453C4—81no—
454C4#292011yescharacteristic
455C4#292311yescharacteristic
456C4#297011yescharacteristic
457C4#297111yescharacteristic
458C4#310512yes—
459C4#310612yes—
460C2—82no—
461C2—82no—
462C2#1803711yescharacteristic
463C2#1803911yescharacteristic
464C2#1804411yescharacteristic
465C2#1804411yescharacteristic
466C2#2110811yescharacteristic
467C2#2111811yescharacteristic
468C2#2114011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21270C2semidirect
3. #20494C2semidirect
4. #9542C2semidirect
5. #9536C2semidirect
7. #8849C2semidirect
8. #8273C2semidirect
9. #8174C2semidirect
10. #8174C2semidirect
11. #8132C2semidirect
189. #107M16semidirect
200. #102M16semidirect

Maximal subgroups

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

subgroupindex
#212702
C42×D82
#204942
#95422
#95362
#89512
#88492
#82732
#81742
#81322