Finite Group Databasebeta

← #278014

Subgroups — #278014

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic, complemented
2#24700C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4C2×#1669C211yesmaximal, characteristic, complemented
5C2×#1564C211yesmaximal, characteristic, complemented
6#10588C212yesmaximal, complemented
7#10484C212yesmaximal, complemented
8#9733C211yesmaximal, characteristic, complemented
9#9614C211yesmaximal, characteristic, complemented
10#9130C212yesmaximal, complemented
11#8644C211yesmaximal, characteristic, complemented
12C2×#494C2211yescharacteristic
13C2×#474C2212yes—
14C2×#467—21no—
15C2×#467—21no—
16C2×C8×D4—21no—
17C2×C8×D4C2211yescharacteristic
18C2×#431C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2857C2212yes—
21C4×#121C2211yescharacteristic
22#2075—22no—
23#2033—22no—
24#1988C2212yes—
25#1969—21no—
26#1969—21no—
27#1956C2212yes—
28#1954C2211yescharacteristic
29#1947C2212yes—
30#1942—22no—
31#1939C2211yescharacteristic
32#1934—22no—
33#1923—21no—
34#1869—21no—
35#1845C2211yescharacteristic
36#1802C2212yes—
37#1797C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40#1669—24no—
41#1564—24no—
42C2×#335C2211yescharacteristic
43C2×#335C2211yescharacteristic
44C2×#334C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×Q16—21no—
50C22×Q16—21no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8C2312yes—
55C2×C22⋊Q8—22no—
56C2×#121D411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2311yescharacteristic
59C22×C42C2311yescharacteristic
60#494—22no—
61#494—22no—
62#474—24no—
63#474—42no—
64#467—44no—
65#467—44no—
66C8×D4—22no—
67C8×D4—22no—
68C8×D4—44no—
69#431—24no—
70#431—42no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—22no—
74C2×#107—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
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#335—22no—
118#335D414yescomplemented
119#334—24no—
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—21no—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8D814yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158#121—22no—
159#121—22no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—42no—
165C4×D4—42no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—42no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42C2411yescharacteristic
191#107—44no—
192#107—42no—
193#107—42no—
194#107—44no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—22no—
201C4⋊C8—21no—
202C4⋊C8—42no—
203C4⋊C8C2×D414yes—
204#103—44no—
205#103—44no—
206#102—44no—
207#102—44no—
208C22⋊C8—44no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—44no—
218C4×C8—42no—
219C4×C8C2×D412yes—
220C4×C8#4112yes—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—42no—
225#95—42no—
226C24—21no—
227C24#12111yescharacteristic
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4#13512yes—
243C22×C4—21no—
244C22×C4—21no—
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—21no—
255C22×C4—21no—
256C22×C4—22no—
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×C4AGL(1,ℤ/8)12yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276Q16—84no—
277Q16—84no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—28no—
283C2×C8—21no—
284C2×C8—24no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—84no—
300C2×C8#12112yes—
301C2×C8#12112yes—
302C2×C8C2×D812yes—
303C2×C8C2×D812yes—
304C2×C8#13512yes—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4#13512yes—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—84no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—41no—
336C42—41no—
337C42C22×D411yescharacteristic
338C42C2×#4111yescharacteristic
339C42H(5,𝔽2)11yescharacteristic
340C422−1+411yescharacteristic
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23#45912yes—
355C23C2×#12111yescharacteristic
356Q8—82no—
357Q8—82no—
358Q8—82no—
359Q8—82no—
360D4—22no—
361D4—22no—
362D4—48no—
363D4—41no—
364D4—41no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—42no—
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—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4#45912yes—
404C2×C4#47312yes—
405C2×C4C2×#12111yescharacteristic
406C2×C4#53711yescharacteristic
407C2×C4D4211yescharacteristic
408C2×C4#54711yescharacteristic
409C2×C4#54811yescharacteristic
410C2×C4#55311yescharacteristic
411C2×C4C22×D811yescharacteristic
412C2×C4C2×#13511yescharacteristic
413C2×C4C2×#13511yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4#57511yescharacteristic
417C2×C4#57811yescharacteristic
418C2×C4#57811yescharacteristic
419C8—22no—
420C8—22no—
421C8—44no—
422C8—41no—
423C8—81no—
424C8—82no—
425C8—81no—
426C8#46614yes—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22#241611yescharacteristic
442C22C2×#45911yescharacteristic
443C22#303511yescharacteristic
444C22#305011yescharacteristic
445C22#305911yescharacteristic
446C22C2×#47311yescharacteristic
447C22#306711yescharacteristic
448C4—22no—
449C4—21no—
450C4—21no—
451C4—22no—
452C4—42no—
453C4—41no—
454C4—81no—
455C4—81no—
456C4—162no—
457C4—162no—
458C4C2×#46611yescharacteristic
459C4#305511yescharacteristic
460C4#324511yescharacteristic
461C4#324711yescharacteristic
462C4#328311yescharacteristic
463C4#328511yescharacteristic
464C4#333611yescharacteristic
465C4#334211yescharacteristic
466C2—42no—
467C2—82no—
468C2#1910111yescharacteristic
469C2#1912911yescharacteristic
470C2#1918811yescharacteristic
471C2#1922011yescharacteristic
472C2#2475011yescharacteristic
473C2#2477911yescharacteristic
474C2#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. C2×#1564C2semidirect
6. #10588C2semidirect
7. #10484C2semidirect
8. #9733C2semidirect
9. #9614C2semidirect
10. #9130C2semidirect
11. #8644C2semidirect
118. #335D4semidirect

Maximal subgroups

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

subgroupindex
#331722
#247002
C8×#1212
C2×#16692
C2×#15642
#105882
#104842
#97332
#96142
#91302
#86442