Finite Group Databasebeta

← #506217

Subgroups — #506217

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2235C211yesmaximal, characteristic, complemented
2C2×#2187C211yesmaximal, characteristic
3C4×#467C211yesmaximal, characteristic, complemented
4C2×#1970C211yesmaximal, characteristic
5#13476C212yesmaximal, complemented
6#13449C212yesmaximal, complemented
7#13131C212yesmaximal, complemented
8#13013C212yesmaximal, complemented
9#12866C211yesmaximal, characteristic, complemented
10#12783C211yesmaximal, characteristic, complemented
11#12731C211yesmaximal, characteristic
12C23×D8C2211yescharacteristic
13C22×SD32C2212yes—
14C2×#467C2211yescharacteristic
15C2×C4×Q16C2211yescharacteristic, complemented
16C22×#107C2211yescharacteristic
17C22×#102—21no—
18C22×C4×C8C2211yescharacteristic
19C4×C22⋊Q8—21no—
20C2×#390—21no—
21#2235—24no—
22#2187—24no—
23C2×#367C2212yescomplemented
24C2×C4⋊C16C2212yescomplemented
25C2×#358C2212yescomplemented
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×#357C2212yes—
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31#1970—21no—
32#1970—21no—
33#1970C2212yes—
34#1970C2212yes—
35#1969C2211yescharacteristic, complemented
36#1942C2211yescharacteristic, complemented
37#1933—21no—
38#1892—21no—
39#1808C2211yescharacteristic
40C4×#107C2211yescharacteristic, complemented
41C4×#103—21no—
42#1380C2212yescomplemented
43#1380C2212yescomplemented
44#1374C2212yescomplemented
45C23×D4—21no—
46C22×Q16C2311yescharacteristic
47C22×D8—24no—
48C22×D8—21no—
49C22×D8—22no—
50C22×D8—22no—
51C22×D8—42no—
52C22×D8C2312yes—
53C23×C8D411yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C4×Q8—21no—
57C22×C4⋊C4—21no—
58C22×C22⋊C4—41no—
59C22×C22⋊C4—41no—
60C22×C42D411yescharacteristic
61C2×SD32—24no—
62C2×SD32—24no—
63C2×SD32—42no—
64C22×C16C2312yes—
65C22×C16C2312yes—
66#467—22no—
67#467D414yes—
68C4×Q16—42no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—21no—
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2312yes—
75C2×#107C2311yescharacteristic
76C2×#103—21no—
77C2×#103—21no—
78C2×#103—21no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—44no—
82C2×C4×C8—42no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#391—21no—
88#390—22no—
89#390—22no—
90#390—41no—
91#390—41no—
92#389—21no—
93#387—41no—
94#385—41no—
95#380—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—41no—
98C4×C22⋊C4—41no—
99#367—24no—
100C4⋊C16—42no—
101#358—24no—
102#357—24no—
103#357—24no—
104#357—42no—
105C22⋊C16—22no—
106C22⋊C16—22no—
107#340—21no—
108#340—21no—
109#340—21no—
110C25—41no—
111C22×Q8—21no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4—41no—
119C22×D4—42no—
120C23×C4—41no—
121C23×C4D811yescharacteristic
122C23×C4D811yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×Q16—22no—
125C2×Q16—21no—
126C2×Q16—22no—
127C2×Q16C2×D412yes—
128C2×D8—24no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—48no—
135C2×D8—48no—
136C2×D8—42no—
137C2×D8—42no—
138C2×D8#4114yes—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D412yes—
148C22×C8#4112yes—
149C22×C8C2411yescharacteristic
150C22⋊Q8—24no—
151C22⋊Q8—42no—
152C4×Q8—42no—
153C4×Q8—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—84no—
171C2×C22⋊C4—84no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42D811yescharacteristic
177C2×C42D811yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180SD32—48no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187#107—21no—
188#107—22no—
189#107—22no—
190#107—22no—
191#107—22no—
192#107—42no—
193#107C2×D412yes—
194#103—24no—
195#103—42no—
196#103—42no—
197#102—44no—
198#102—44no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208C24—44no—
209C24—42no—
210C24—82no—
211C24—82no—
212C24—82no—
213C24—82no—
214C24—82no—
215C24—82no—
216C24C2×D811yescharacteristic
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×D4—24no—
224C2×D4—48no—
225C2×D4—48no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—82no—
233C2×D4—84no—
234C2×D4—84no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4SD3214yescomplemented
254C22×C4#12112yes—
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4C22×D411yescharacteristic
262Q16—24no—
263Q16—42no—
264D8—24no—
265D8—24no—
266D8—48no—
267D8—42no—
268D8—42no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—44no—
280C2×C8#12112yes—
281C2×C8C22×D411yescharacteristic
282C2×C8C2×#4112yes—
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)12yes—
285C4⋊C4—22no—
286C4⋊C4—41no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C22⋊C4—44no—
298C22⋊C4—84no—
299C22⋊C4—84no—
300C22⋊C4—82no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—41no—
308C42—41no—
309C42—82no—
310C42—82no—
311C16—42no—
312C16—42no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—44no—
318C23—88no—
319C23—88no—
320C23—88no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23—164no—
328C23—164no—
329C23—164no—
330C23—164no—
331C23#45912yes—
332C23C2×SD3212yes—
333C23#50912yes—
334C23C22×D811yescharacteristic
335Q8—42no—
336Q8—42no—
337Q8—81no—
338Q8—81no—
339D4—44no—
340D4—44no—
341D4—88no—
342D4—82no—
343D4—82no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—164no—
378C2×C4#45912yes—
379C2×C4C2×SD3212yes—
380C2×C4C2×SD3212yes—
381C2×C4C2×#12111yescharacteristic
382C2×C4#52511yescharacteristic
383C2×C4#53411yescharacteristic
384C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
385C2×C4#57511yescharacteristic
386C2×C4#57511yescharacteristic
387C8—22no—
388C8—41no—
389C8—41no—
390C8#54612yes—
391C8#55112yes—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—84no—
399C22—84no—
400C22—84no—
401C22—1616no—
402C22—162no—
403C22—162no—
404C22—162no—
405C22—162no—
406C22#220714yes—
407C22C2×#45911yescharacteristic
408C22#310011yescharacteristic
409C22#318611yescharacteristic
410C22C22×SD3211yescharacteristic
411C22C2×#50911yescharacteristic
412C22#341411yescharacteristic
413C22#341411yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4—82no—
421C4—162no—
422C4—161no—
423C4—161no—
424C4#253411yescharacteristic
425C4#304011yescharacteristic
426C4#311411yescharacteristic
427C4#319411yescharacteristic
428C2—24no—
429C2—164no—
430C2#2811411yescharacteristic
431C2C2×#220711yescharacteristic
432C2#3358411yescharacteristic
433C2#3366111yescharacteristic
434C2#3367411yescharacteristic
435C2#3375311yescharacteristic
436C2#3376011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2235C2semidirect
3. C4×#467C2semidirect
5. #13476C2semidirect
6. #13449C2semidirect
7. #13131C2semidirect
8. #13013C2semidirect
9. #12866C2semidirect
10. #12783C2semidirect
15. C2×C4×Q16C22semidirect
23. C2×#367C22semidirect
24. C2×C4⋊C16C22semidirect
25. C2×#358C22semidirect
35. #1969C22semidirect
36. #1942C22semidirect
40. C4×#107C22semidirect
42. #1380C22semidirect
43. #1380C22semidirect
44. #1374C22semidirect
253. C22×C4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
C2×#21872
C4×#4672
C2×#19702
#134762
#134492
#131312
#130132
#128662
#127832
#127312