Finite Group Databasebeta

← #323855

Subgroups — #323855

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic, complemented
2#24750C212yesmaximal, complemented
3C8×#121C211yesmaximal, characteristic, complemented
4#10649C211yesmaximal, characteristic, complemented
5#10615C211yesmaximal, characteristic, complemented
6#10536C211yesmaximal, characteristic, complemented
7#10260C211yesmaximal, characteristic, complemented
8#9563C212yesmaximal, complemented
9#9221C212yesmaximal, complemented
10#9133C212yesmaximal, complemented
11C2×#1277C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466—21no—
15C2×#466—22no—
16C2×#459C2212yes—
17C2×C8×D4—21no—
18C2×C8×D4C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2856C2212yes—
22C4×#121C2211yescharacteristic
23#2067—22no—
24#2054—22no—
25#1982C2212yes—
26#1970—21no—
27#1970—21no—
28#1957C2212yes—
29#1954C2211yescharacteristic
30#1947C2212yes—
31#1939C2211yescharacteristic
32#1923—21no—
33#1893—22no—
34#1869—21no—
35#1845C2211yescharacteristic
36#1802C2212yes—
37#1797C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40C2×#331C2212yes—
41C2×#331C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×D8—22no—
48C22×D8—21no—
49C22×D8—21no—
50C23×C8—22no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×#127C2312yes—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121D411yescharacteristic
59C2×C4×D4—21no—
60C2×C4×D4C2311yescharacteristic
61C22×C42C2311yescharacteristic
62#493—42no—
63#466—44no—
64#466—44no—
65#466—48no—
66#459—24no—
67#459—42no—
68C8×D4—42no—
69C8×D4—44no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—42no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#390—22no—
98#390—22no—
99#390—22no—
100#390D411yescharacteristic
101#389D411yescharacteristic
102#388—21no—
103#386—21no—
104#384C2312yes—
105#380D411yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C4×C22⋊C4—21no—
108#340—22no—
109#340—22no—
110#336—21no—
111#336—21no—
112#336—21no—
113#331—24no—
114#331—42no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—42no—
120C22×D4C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4#4111yescharacteristic
125C2×D8—44no—
126C2×D8—42no—
127C2×D8—42no—
128C2×D8—42no—
129C2×D8—42no—
130C2×D8—44no—
131C2×D8—42no—
132C2×D8—42no—
133C2×D8—82no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—44no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—44no—
147C22×C8—42no—
148C22×C8—82no—
149C22×C8D814yescomplemented
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154#127—22no—
155#127—22no—
156#127—22no—
157#127—22no—
158#121—22no—
159#121—44no—
160#121—44no—
161#121—44no—
162#121—44no—
163#121—84no—
164#121D814yescomplemented
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—42no—
168C4×D4—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
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×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42C2411yescharacteristic
191#107—42no—
192#107—42no—
193#107—44no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—44no—
200C4⋊C8—42no—
201C4⋊C8#4114yes—
202#102—44no—
203#102—44no—
204#102—44no—
205#102—44no—
206#102—44no—
207#102—82no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—42no—
217#95—21no—
218#95—21no—
219#95—21no—
220C24—21no—
221C24—42no—
222C24#12111yescharacteristic
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—82no—
244C2×D4—82no—
245C2×D4—84no—
246C2×D4—84no—
247C2×D4—84no—
248C2×D4C2×D812yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4C2×D812yes—
272C22×C4C2×D812yes—
273C22×C4#13512yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278D8—84no—
279D8—84no—
280D8—88no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—24no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—84no—
305C2×C8—88no—
306C2×C8C2×D812yes—
307C2×C8C2×D812yes—
308C2×C8#13512yes—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4C2×D812yes—
319C22⋊C4—24no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C22⋊C4—84no—
327C42—21no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42—41no—
332C42—41no—
333C42C2×#4112yes—
334C42H(5,𝔽2)12yes—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23#45912yes—
354C23#46612yes—
355C23#47212yes—
356C23C2×#12111yescharacteristic
357D4—24no—
358D4—48no—
359D4—42no—
360D4—82no—
361D4—82no—
362D4—82no—
363D4—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367D4—82no—
368D4—162no—
369D4—162no—
370D4—168no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—44no—
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#45912yes—
404C2×C4#46612yes—
405C2×C4#53211yescharacteristic
406C2×C4#53411yescharacteristic
407C2×C4#54612yes—
408C2×C4#55312yes—
409C2×C4C22×D811yescharacteristic
410C2×C4C22×D811yescharacteristic
411C2×C4C2×#13511yescharacteristic
412C2×C4C2×#13511yescharacteristic
413C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
414C2×C4#57511yescharacteristic
415C2×C4#57712yes—
416C8—22no—
417C8—22no—
418C8—22no—
419C8—41no—
420C8—44no—
421C8—81no—
422C8—81no—
423C8—82no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—162no—
438C22—162no—
439C22—162no—
440C22—162no—
441C22—162no—
442C22—162no—
443C22—162no—
444C22#166714yes—
445C22#254311yescharacteristic
446C22C2×#45911yescharacteristic
447C22#303511yescharacteristic
448C22C2×#46611yescharacteristic
449C22#306011yescharacteristic
450C22C2×#47211yescharacteristic
451C22#306711yescharacteristic
452C4—22no—
453C4—22no—
454C4—22no—
455C4—42no—
456C4—41no—
457C4—81no—
458C4—81no—
459C4—162no—
460C4#316911yescharacteristic
461C4#318011yescharacteristic
462C4#318611yescharacteristic
463C4#318811yescharacteristic
464C4#329412yes—
465C4#330112yes—
466C2—24no—
467C2—82no—
468C2—322no—
469C2C2×#166711yescharacteristic
470C2#1912511yescharacteristic
471C2#1918911yescharacteristic
472C2#1922711yescharacteristic
473C2#2855111yescharacteristic
474C2#2857211yescharacteristic
475C2#2859211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
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. #33171C2semidirect
2. #24750C2semidirect
3. C8×#121C2semidirect
4. #10649C2semidirect
5. #10615C2semidirect
6. #10536C2semidirect
7. #10260C2semidirect
8. #9563C2semidirect
9. #9221C2semidirect
10. #9133C2semidirect
11. C2×#1277C2semidirect
149. C22×C8D8semidirect
164. #121D8semidirect

Maximal subgroups

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

subgroupindex
#331712
#247502
C8×#1212
#106492
#106152
#105362
#102602
#95632
#92212
#91332
C2×#12772