Finite Group Databasebeta

← #226845

Subgroups — #226845

Order 512.

684 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#22839C211yesmaximal, characteristic, complemented
2#22835C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4#9665C212yesmaximal
5#9545C212yesmaximal, complemented
6#9541C212yesmaximal, complemented
7#8807C211yesmaximal, characteristic, complemented
8#8565C211yesmaximal, characteristic, complemented
9#8565C211yesmaximal, characteristic, complemented
10#8288C212yesmaximal, complemented
11#8261C211yesmaximal, characteristic, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C42×D4C2211yescharacteristic
16C42×D4C2211yescharacteristic
17#1954C2212yes—
18#1954C2212yes—
19#1924—22no—
20#1923—22no—
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1817—21no—
24#1817—21no—
25C4×#107C411yescharacteristic
26C4×#107C411yescharacteristic
27C4×#107C2211yescharacteristic
28#1770C2212yes—
29C8×C4⋊C4C2212yes—
30C4×C4⋊C8C2211yescharacteristic
31C4×C4⋊C8C2211yescharacteristic
32C4×#102C411yescharacteristic
33C4×#102C411yescharacteristic
34C4×#102C411yescharacteristic
35C4×#102C411yescharacteristic
36C4×#102C2211yescharacteristic
37C4×#102C2211yescharacteristic
38#1749—21no—
39C4×C22⋊C8—21no—
40C42×C8C2211yescharacteristic
41C2×#335C2212yes—
42C2×#334C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C2×#322C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1273C2212yes—
49C22×D8C2×C411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C4×D8—24no—
59C4×D8—24no—
60C4×D8—22no—
61C4×D8—22no—
62C2×#107C2×C411yescharacteristic
63C2×#107C2×C411yescharacteristic
64C2×#107C2×C411yescharacteristic
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×#102C2×C411yescharacteristic
70C2×#102C2×C411yescharacteristic
71C2×#102C2×C411yescharacteristic
72C2×#102C2×C411yescharacteristic
73C2×#102C2×C411yescharacteristic
74C2×#102C2×C411yescharacteristic
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×C4×C8—22no—
81C2×C4×C8C2×C411yescharacteristic
82C2×C4×C8C2×C411yescharacteristic
83C2×C4×C8C2×C411yescharacteristic
84C2×C4×C8C2×C411yescharacteristic
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2312yes—
88C2×C4×C8C2311yescharacteristic
89C4×C4⋊C4C2×C411yescharacteristic
90C4×C4⋊C4C2×C411yescharacteristic
91C4×C4⋊C4C2×C411yescharacteristic
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4C2311yescharacteristic
94C4×C4⋊C4C2311yescharacteristic
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C43C2311yescharacteristic
102#340C2×C411yescharacteristic
103#340C2×C411yescharacteristic
104#340C2×C411yescharacteristic
105#340C2×C411yescharacteristic
106#340C2×C411yescharacteristic
107#340C2×C411yescharacteristic
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336D411yescharacteristic
116#336D411yescharacteristic
117#335—42no—
118#334—24no—
119D4⋊C8—42no—
120D4⋊C8—42no—
121#322—24no—
122C22×D4C22×C411yescharacteristic
123C22×D4C22×C411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8C22×C411yescharacteristic
137C22×C8C22×C411yescharacteristic
138C22×C8C22×C411yescharacteristic
139C22×C8C22×C411yescharacteristic
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4—41no—
156C4×D4—41no—
157C4×D4M1614yes—
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—82no—
177C2×C42—82no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185#107—22no—
186#107—22no—
187#107—22no—
188C4⋊C8—24no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—42no—
194#102—22no—
195#102—22no—
196#102—22no—
197#102—22no—
198#102—22no—
199#102M1614yes—
200C22⋊C8—44no—
201C22⋊C8—44no—
202C22⋊C8—82no—
203C22⋊C8—82no—
204#97—82no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—21no—
211C4×C8—22no—
212C4×C8—21no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—82no—
216C4×C8#4112yes—
217C4×C8#4112yes—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224C24—21no—
225C24—21no—
226C2×D4—21no—
227C2×D4—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×D4C2×M1612yes—
237C2×D4C2×M1612yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4111yescharacteristic
271D8—28no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—24no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8#11712yes—
297C2×C8#11712yes—
298C2×C8C2×M1612yes—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4C2×M1612yes—
310C4⋊C4C2×M1612yes—
311C22⋊C4—24no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C42—21no—
320C42—21no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—41no—
325C42—42no—
326C42—42no—
327C42—81no—
328C42—81no—
329C42—81no—
330C42—81no—
331C42C2×M1612yes—
332C42#13112yes—
333C42C2×#4111yescharacteristic
334C42C2×#4111yescharacteristic
335C42H(5,𝔽2)11yescharacteristic
336C422−1+411yescharacteristic
337C23—22no—
338C23—22no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C2×C4×D411yescharacteristic
352D4—22no—
353D4—22no—
354D4—22no—
355D4—22no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
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—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#43512yes—
398C2×C4#43512yes—
399C2×C4#44312yes—
400C2×C4C2×#11711yescharacteristic
401C2×C4#52011yescharacteristic
402C2×C4#54711yescharacteristic
403C2×C4#55011yescharacteristic
404C2×C4#55511yescharacteristic
405C2×C4#55611yescharacteristic
406C2×C4C22×M1611yescharacteristic
407C2×C4C22×M1611yescharacteristic
408C2×C4C2×#13111yescharacteristic
409C2×C4#56811yescharacteristic
410C2×C4#57512yes—
411C2×C4#57612yes—
412C8—22no—
413C8—22no—
414C8—22no—
415C8—44no—
416C8—42no—
417C8—42no—
418C8—41no—
419C8—82no—
420C8—82no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#235411yescharacteristic
436C22C2×#43511yescharacteristic
437C22D4×M1611yescharacteristic
438C22#294311yescharacteristic
439C22#294811yescharacteristic
440C22C2×#44311yescharacteristic
441C22#295711yescharacteristic
442C4—21no—
443C4—21no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—41no—
448C4—42no—
449C4—42no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4#292011yescharacteristic
455C4#292511yescharacteristic
456C4#298911yescharacteristic
457C4#299011yescharacteristic
458C4#335512yes—
459C4#340312yes—
460C2—82no—
461C2—82no—
462C2#1786811yescharacteristic
463C2#1787711yescharacteristic
464C2#1804111yescharacteristic
465C2#1805411yescharacteristic
466C2#2279711yescharacteristic
467C2#2280211yescharacteristic
468C2#2281611yescharacteristic
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×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22839C2semidirect
2. #22835C2semidirect
5. #9545C2semidirect
6. #9541C2semidirect
7. #8807C2semidirect
8. #8565C2semidirect
9. #8565C2semidirect
10. #8288C2semidirect
11. #8261C2semidirect

Maximal subgroups

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

subgroupindex
#228392
#228352
C42×D82
#96652
#95452
#95412
#88072
#85652
#82882
#82612