Finite Group Databasebeta

← #287213

Subgroups — #287213

Order 512.

759 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29139C211yesmaximal, characteristic, complemented
2#28716C211yesmaximal, characteristic, complemented
3#28251C211yesmaximal, characteristic
4#23173C211yesmaximal, characteristic, complemented
5#10621C212yesmaximal, complemented
6#10472C212yesmaximal, complemented
7#9896C211yesmaximal, characteristic, complemented
8#9601C211yesmaximal, characteristic, complemented
9#9221C212yesmaximal, complemented
10#8648C212yesmaximal, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#486—21no—
13C2×#459—21no—
14C2×#459C2212yes—
15C2×C8×D4—21no—
16C2×C8×D4C2212yes—
17#2868C2211yescharacteristic
18C4×#127C2211yescharacteristic
19C4×#127C2211yescharacteristic
20#2071—22no—
21#2054—22no—
22#2023—22no—
23#1990—21no—
24#1957C2212yes—
25#1954—21no—
26#1954C2212yes—
27#1869—22no—
28#1842C2211yescharacteristic
29#1834C2212yes—
30#1802—21no—
31#1802—21no—
32#1802—21no—
33#1802C2212yes—
34#1802C2212yes—
35#1802C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#331C2211yescharacteristic
41C2×#331C2211yescharacteristic
42#1296C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1277C2212yes—
47C22×D8—22no—
48C23×C8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×#127D411yescharacteristic
53C2×#127C2311yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2312yes—
60#486—44no—
61#459—24no—
62#459—42no—
63#459—44no—
64C8×D4—24no—
65C8×D4—42no—
66C8×D4—44no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D411yescharacteristic
70C2×C4⋊C8D411yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2312yes—
88#390—21no—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390D412yes—
94#384—21no—
95#384—21no—
96#384—21no—
97#384D411yescharacteristic
98#384C2311yescharacteristic
99#384C2312yes—
100C43C2311yescharacteristic
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—21no—
107#336—21no—
108#336D412yes—
109#335—42no—
110#331—22no—
111#331—22no—
112C22×Q8—41no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4C2×D412yes—
119C23×C4—22no—
120C23×C4—41no—
121C2×D8—44no—
122C2×D8—44no—
123C2×D8—82no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—44no—
130C22×C8—44no—
131C22×C8—82no—
132C22×C8C2×D412yes—
133C22×C8C2×D412yes—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—81no—
137C4⋊Q8—81no—
138C4⋊Q8—82no—
139#127—22no—
140#127—21no—
141#127—21no—
142#127—21no—
143#127D814yes—
144#127#4112yes—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—81no—
156C4×D4—81no—
157C4×D4—82no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C2×D412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42C2411yescharacteristic
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—42no—
195C4⋊C8—81no—
196C4⋊C8#4112yes—
197#102—44no—
198#102—44no—
199#102—44no—
200#102—44no—
201#102—82no—
202#102—82no—
203C22⋊C8—44no—
204C22⋊C8—44no—
205C22⋊C8—82no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—42no—
209C4×C8—42no—
210C24—22no—
211C24—41no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8—82no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4—81no—
241C2×D4—81no—
242C2×D4C2×D814yes—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C2×#4111yescharacteristic
262D8—88no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—88no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—41no—
315C42C2×D812yes—
316C42#13512yes—
317C42#13512yes—
318C42AGL(1,ℤ/8)12yes—
319C42C2×#4111yescharacteristic
320C42C2×#4111yescharacteristic
321C42H(5,𝔽2)11yescharacteristic
322C422−1+411yescharacteristic
323C23—24no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23C2×#12111yescharacteristic
338Q8—164no—
339D4—24no—
340D4—24no—
341D4—42no—
342D4—42no—
343D4—42no—
344D4—48no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—84no—
349D4—82no—
350D4—82no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—164no—
378C2×C4#46312yes—
379C2×C4#46312yes—
380C2×C4#46612yes—
381C2×C4#46612yes—
382C2×C4#47112yes—
383C2×C4#52911yescharacteristic
384C2×C4#54211yescharacteristic
385C2×C4#54611yescharacteristic
386C2×C4#54611yescharacteristic
387C2×C4#54711yescharacteristic
388C2×C4#54811yescharacteristic
389C2×C4C22×D811yescharacteristic
390C2×C4C2×#13511yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
393C2×C4#57611yescharacteristic
394C2×C4#57611yescharacteristic
395C2×C4#57711yescharacteristic
396C2×C4#57811yescharacteristic
397C8—42no—
398C8—42no—
399C8—42no—
400C8—81no—
401C8—81no—
402C8—81no—
403C8—82no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22#242711yescharacteristic
419C22C2×#46311yescharacteristic
420C22#304111yescharacteristic
421C22C2×#46611yescharacteristic
422C22#306311yescharacteristic
423C22C2×#47111yescharacteristic
424C22#306811yescharacteristic
425C4—21no—
426C4—21no—
427C4—22no—
428C4—22no—
429C4—41no—
430C4—82no—
431C4—161no—
432C4—162no—
433C4#168114yes—
434C4#329411yescharacteristic
435C4#330211yescharacteristic
436C4#330511yescharacteristic
437C4#332511yescharacteristic
438C4#333411yescharacteristic
439C4#334011yescharacteristic
440C4#336111yescharacteristic
441C4#336611yescharacteristic
442C2—84no—
443C2—162no—
444C2C2×#168111yescharacteristic
445C2#1923611yescharacteristic
446C2#1938611yescharacteristic
447C2#1955311yescharacteristic
448C2#2532111yescharacteristic
449C2#2533811yescharacteristic
450C2#2535911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#28251C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29139C2semidirect
2. #28716C2semidirect
4. #23173C2semidirect
5. #10621C2semidirect
6. #10472C2semidirect
7. #9896C2semidirect
8. #9601C2semidirect
9. #9221C2semidirect
10. #8648C2semidirect
11. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#291392
#287162
#282512
#231732
#106212
#104722
#98962
#96012
#92212
#86482
C4×#3312