Finite Group Databasebeta

← #278016

Subgroups — #278016

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic, complemented
2#24835C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4C2×#1669C211yesmaximal, characteristic, complemented
5C22×#335C211yesmaximal, characteristic, complemented
6#10592C212yesmaximal, complemented
7#10485C212yesmaximal, complemented
8#9733C211yesmaximal, characteristic, complemented
9#9614C211yesmaximal, characteristic, complemented
10#9129C212yesmaximal, complemented
11#8644C211yesmaximal, characteristic, complemented
12C2×#494C2211yescharacteristic
13C2×#474C2212yes—
14C2×#467—21no—
15C2×#467—21no—
16C2×C8×D4—21no—
17C2×C8×D4C2211yescharacteristic
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
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40#1669—24no—
41C2×#335—24no—
42C2×#335C2211yescharacteristic
43C2×#335C2212yes—
44C2×#335C2211yescharacteristic
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×Q16—21no—
50C22×Q16—21no—
51C23×C8—22no—
52C23×C8—21no—
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—
67C8×D4—22no—
68C8×D4—22no—
69C8×D4—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×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—21no—
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#335—24no—
118#335—22no—
119#335D414yescomplemented
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—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—41no—
144C22×C8—44no—
145C22×C8—44no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
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—22no—
162#121Q1614yescomplemented
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—42no—
168C4×D4—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—42no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42C2411yescharacteristic
194#107—44no—
195#107—42no—
196#107—42no—
197#107—82no—
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—22no—
202C4⋊C8—21no—
203C4⋊C8—22no—
204C4⋊C8—21no—
205C4⋊C8—44no—
206C4⋊C8—42no—
207C4⋊C8C2×D414yes—
208#103—44no—
209#103—44no—
210#102—44no—
211#102—82no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215C22⋊C8—42no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—42no—
221C4×C8C2×D412yes—
222C4×C8#4112yes—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—42no—
227#95—42no—
228C24—21no—
229C24#12111yescharacteristic
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×D4—24no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4C2×Q1612yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4C2×D812yes—
272C22×C4C2×Q1612yes—
273C22×C4#13512yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278Q16—84no—
279Q16—84no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—28no—
285C2×C8—21no—
286C2×C8—24no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—88no—
302C2×C8—84no—
303C2×C8#12112yes—
304C2×C8#12112yes—
305C2×C8C2×D812yes—
306C2×C8C2×D812yes—
307C2×C8#13512yes—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C4⋊C4—84no—
324C4⋊C4C2×Q1612yes—
325C22⋊C4—24no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—84no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—41no—
339C42—41no—
340C42C22×D411yescharacteristic
341C42C2×#4111yescharacteristic
342C42H(5,𝔽2)11yescharacteristic
343C422−1+411yescharacteristic
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23#45912yes—
356C23#46712yes—
357C23#47312yes—
358C23C2×#12111yescharacteristic
359Q8—82no—
360Q8—82no—
361Q8—82no—
362Q8—82no—
363D4—22no—
364D4—22no—
365D4—48no—
366D4—41no—
367D4—41no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#45912yes—
407C2×C4#46712yes—
408C2×C4C2×#12111yescharacteristic
409C2×C4#53711yescharacteristic
410C2×C4D4211yescharacteristic
411C2×C4#54711yescharacteristic
412C2×C4#54811yescharacteristic
413C2×C4#55311yescharacteristic
414C2×C4C22×D811yescharacteristic
415C2×C4C22×Q1611yescharacteristic
416C2×C4C2×#13511yescharacteristic
417C2×C4C2×#13511yescharacteristic
418C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
419C2×C4#57511yescharacteristic
420C2×C4#57611yescharacteristic
421C2×C4#57811yescharacteristic
422C8—22no—
423C8—22no—
424C8—44no—
425C8—41no—
426C8—81no—
427C8—82no—
428C8—81no—
429C8#46614yes—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22#166914yescomplemented
444C22#241611yescharacteristic
445C22C2×#45911yescharacteristic
446C22#303511yescharacteristic
447C22C2×#46711yescharacteristic
448C22#306111yescharacteristic
449C22C2×#47311yescharacteristic
450C22#306711yescharacteristic
451C4—22no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—41no—
457C4—81no—
458C4—81no—
459C4—162no—
460C4—162no—
461C4C2×#46611yescharacteristic
462C4#305511yescharacteristic
463C4#324511yescharacteristic
464C4#324711yescharacteristic
465C4#328011yescharacteristic
466C4D4×Q1611yescharacteristic
467C4#333611yescharacteristic
468C4#333811yescharacteristic
469C2—24no—
470C2—82no—
471C2C2×#166911yescharacteristic
472C2#1912611yescharacteristic
473C2#1919011yescharacteristic
474C2#1922311yescharacteristic
475C2#2475011yescharacteristic
476C2#2478111yescharacteristic
477C2#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×#1669C2semidirect
5. C22×#335C2semidirect
6. #10592C2semidirect
7. #10485C2semidirect
8. #9733C2semidirect
9. #9614C2semidirect
10. #9129C2semidirect
11. #8644C2semidirect
119. #335D4semidirect
162. #121Q16semidirect
443. C22#1669semidirect

Maximal subgroups

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

subgroupindex
#331722
#248352
C8×#1212
C2×#16692
C22×#3352
#105922
#104852
#97332
#96142
#91292
#86442