Finite Group Databasebeta

← #278052

Subgroups — #278052

Order 512.

773 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic, complemented
2#24835C212yesmaximal
3#21487C211yesmaximal, characteristic
4C2×#1689C211yesmaximal, characteristic, complemented
5C22×#334C211yesmaximal, characteristic, complemented
6#10588C212yesmaximal, complemented
7#10484C212yesmaximal, complemented
8#9742C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9129C212yesmaximal, complemented
11#8652C211yesmaximal, characteristic, complemented
12C2×#494C2211yescharacteristic
13C2×#474C2212yes—
14C2×#467—21no—
15C2×#467—21no—
16C2×#436C2211yescharacteristic
17C2×#435—21no—
18C22×C4⋊C8C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2857C2212yes—
21C4×#121C2211yescharacteristic
22#2086—22no—
23#2067—22no—
24#1988C2212yes—
25#1969—21no—
26#1969—21no—
27#1956C2212yes—
28#1954C2211yescharacteristic
29#1947C2212yes—
30#1939C2211yescharacteristic
31#1923—21no—
32#1893—22no—
33#1869—21no—
34#1845C2211yescharacteristic
35#1808—22no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1770C2211yescharacteristic
39#1753—21no—
40#1689—24no—
41C2×#334—24no—
42C2×#334C2211yescharacteristic
43C2×#334C2212yes—
44C2×#334C2211yescharacteristic
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×Q16—21no—
50C22×Q16—21no—
51C22×M16—21no—
52C23×C8—22no—
53C23×C8D411yescharacteristic
54C23×C8D411yescharacteristic
55C2×C4⋊Q8C2312yes—
56C2×C22⋊Q8—22no—
57C2×#121D411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#494—22no—
62#494—22no—
63#474—24no—
64#474—42no—
65#467—44no—
66#467—44no—
67#436—22no—
68#436—22no—
69#435—44no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×#107—22no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—22no—
98#390D411yescharacteristic
99#389—22no—
100#389—22no—
101#389D411yescharacteristic
102#388—21no—
103#386—21no—
104#384C2312yes—
105#380—22no—
106#380—22no—
107#380D411yescharacteristic
108C4×C4⋊C4C2311yescharacteristic
109C4×C22⋊C4—21no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#336—21no—
115#336—21no—
116#336—21no—
117#334—24no—
118#334—22no—
119#334D414yescomplemented
120#332—24no—
121#332—42no—
122C22×Q8—22no—
123C22×D4—21no—
124C22×D4C2×D411yescharacteristic
125C23×C4—21no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×Q16—42no—
130C2×Q16—42no—
131C2×Q16—42no—
132C2×Q16—42no—
133C2×Q16—42no—
134C2×Q16—42no—
135C2×M16—42no—
136C2×M16—42no—
137C2×M16—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—41no—
147C22×C8—44no—
148C22×C8—44no—
149C22×C8—82no—
150C22×C8D814yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C22⋊Q8—44no—
160C22⋊Q8—44no—
161#121—24no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—42no—
167C4×D4—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—42no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42C2411yescharacteristic
193#107—44no—
194#107—42no—
195#107—42no—
196#107—82no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—22no—
201C4⋊C8—21no—
202C4⋊C8—22no—
203C4⋊C8—21no—
204C4⋊C8—44no—
205C4⋊C8—42no—
206C4⋊C8C2×D414yes—
207#103—44no—
208#103—44no—
209#102—44no—
210#102—82no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215#97—42no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8C2×D412yes—
221C4×C8#4112yes—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—42no—
226#95—42no—
227C24—21no—
228C24#12111yescharacteristic
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4C2×D812yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4#13714yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273Q16—84no—
274Q16—84no—
275M16—84no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—28no—
281C2×C8—21no—
282C2×C8—24no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—44no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—88no—
298C2×C8#12112yes—
299C2×C8#12112yes—
300C2×C8C2×D812yes—
301C2×C8C2×D812yes—
302C2×C8#13512yes—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—84no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—84no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—41no—
333C42C22×D411yescharacteristic
334C42C2×#4111yescharacteristic
335C42H(5,𝔽2)11yescharacteristic
336C422−1+411yescharacteristic
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23#45912yes—
349C23#47012yes—
350C23#47312yes—
351C23C2×#12111yescharacteristic
352Q8—82no—
353Q8—82no—
354Q8—82no—
355Q8—82no—
356D4—42no—
357D4—42no—
358D4—84no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
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—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4#45912yes—
396C2×C4#47012yes—
397C2×C4C2×#12111yescharacteristic
398C2×C4#53711yescharacteristic
399C2×C4D4211yescharacteristic
400C2×C4#54711yescharacteristic
401C2×C4#54811yescharacteristic
402C2×C4#55311yescharacteristic
403C2×C4C22×D811yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C4C2×#13712yes—
407C2×C4#57511yescharacteristic
408C2×C4#57712yes—
409C8—22no—
410C8—22no—
411C8—44no—
412C8—41no—
413C8—81no—
414C8—82no—
415C8—81no—
416C8#46614yes—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22#168914yescomplemented
431C22#241611yescharacteristic
432C22C2×#45911yescharacteristic
433C22#303511yescharacteristic
434C22C2×#47011yescharacteristic
435C22#306011yescharacteristic
436C22C2×#47311yescharacteristic
437C22#306711yescharacteristic
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—162no—
447C4—162no—
448C4C2×#46611yescharacteristic
449C4#305511yescharacteristic
450C4#324511yescharacteristic
451C4#324711yescharacteristic
452C4#326612yes—
453C4#334412yes—
454C2—24no—
455C2—161no—
456C2C2×#168911yescharacteristic
457C2#1915811yescharacteristic
458C2#1918911yescharacteristic
459C2#1922811yescharacteristic
460C2#2475011yescharacteristic
461C2#2478311yescharacteristic
462C2#2480411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33172C2semidirect
4. C2×#1689C2semidirect
5. C22×#334C2semidirect
6. #10588C2semidirect
7. #10484C2semidirect
8. #9742C2semidirect
9. #9620C2semidirect
10. #9129C2semidirect
11. #8652C2semidirect
119. #334D4semidirect
430. C22#1689semidirect

Maximal subgroups

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

subgroupindex
#331722
#248352
#214872
C2×#16892
C22×#3342
#105882
#104842
#97422
#96202
#91292
#86522