Finite Group Databasebeta

← #274666

Subgroups — #274666

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32903C211yesmaximal, characteristic
2#27906C211yesmaximal, characteristic
3Q8×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×#1803C211yesmaximal, characteristic, complemented
5#10620C212yesmaximal, complemented
6#10475C212yesmaximal, complemented
7#9222C212yesmaximal, complemented
8#9133C211yesmaximal, characteristic
9#8830C211yesmaximal, characteristic, complemented
10#8724C211yesmaximal, characteristic, complemented
11#8643C212yesmaximal, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×#481—21no—
14C2×#481—21no—
15C2×#460C2211yescharacteristic
16C2×#460C2211yescharacteristic
17C2×#448—21no—
18C2×C8×Q8C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103—21no—
21#2856C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2094—22no—
24#1984—22no—
25#1984C2212yes—
26#1957C2212yes—
27#1880—21no—
28#1880—21no—
29#1874—21no—
30#1844C2212yes—
31#1803—21no—
32#1803—21no—
33#1803C2212yes—
34#1803C2211yescharacteristic
35#1803C2212yes—
36#1803C2211yescharacteristic
37#1797—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C4×C22⋊C8—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C22×SD16—21no—
51C22×SD16—21no—
52C23×C8—21no—
53C2×#127C2311yescharacteristic
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—24no—
60C2×C4×Q8—42no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#481—44no—
73#481—44no—
74#460—24no—
75#460—42no—
76#448—44no—
77C8×Q8—24no—
78C8×Q8—42no—
79C2×#106—22no—
80C2×#106—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—44no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8—22no—
102C2×C4×C8C2312yes—
103#390—21no—
104#390—21no—
105#390—21no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384C2311yescharacteristic
110#384C2312yes—
111#340—22no—
112#340—22no—
113#340—21no—
114#340—21no—
115#336—22no—
116#336—22no—
117#336—21no—
118#336—21no—
119#336D412yes—
120#334—42no—
121#331—42no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8SD1614yes—
129C22×Q8C2×D411yescharacteristic
130C22×Q8C2×D411yescharacteristic
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4#4111yescharacteristic
141C2×SD16—42no—
142C2×SD16—42no—
143C2×SD16—42no—
144C2×SD16—42no—
145C2×SD16—42no—
146C2×SD16—42no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156#127—22no—
157#127—22no—
158#121—42no—
159#121—42no—
160#121—42no—
161#121—42no—
162#121—82no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—44no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—44no—
172C4×Q8#4114yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—44no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#106—44no—
212#106—44no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—44no—
220C4⋊C8—42no—
221#103—44no—
222#103—42no—
223#103—42no—
224#103—44no—
225#102—42no—
226#102—44no—
227#102—44no—
228#102—42no—
229C22⋊C8—22no—
230C22⋊C8—42no—
231C22⋊C8—42no—
232C22⋊C8SD1614yescomplemented
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—44no—
236C24—41no—
237C24#12111yescharacteristic
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—24no—
243C2×Q8—28no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8#12312yes—
254C2×Q8C2×SD1612yes—
255C2×Q8C2×SD1612yes—
256C2×Q8AGL(1,ℤ/8)12yes—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C2×SD1612yes—
301C22×C4C2×SD1612yes—
302C22×C4#13512yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307SD16—84no—
308SD16—84no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—84no—
327C2×C8C2×SD1614yes—
328C4⋊C4—21no—
329C4⋊C4—24no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4#12312yes—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C42—21no—
354C42—21no—
355C42—24no—
356C42—22no—
357C42—22no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42C2×#4112yes—
362C42H(5,𝔽2)12yes—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23#46012yes—
375C23#46512yes—
376C23#47112yes—
377C23C2×#12111yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8—22no—
382Q8—22no—
383Q8—48no—
384Q8—41no—
385Q8—41no—
386Q8#48114yescomplemented
387D4—81no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—82no—
392D4—82no—
393D4—162no—
394D4—164no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—44no—
416C2×C4—48no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—84no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—164no—
432C2×C4#46012yes—
433C2×C4#46512yes—
434C2×C4C2×#12311yescharacteristic
435C2×C4#53411yescharacteristic
436C2×C4#54612yes—
437C2×C4#55112yes—
438C2×C4C22×SD1611yescharacteristic
439C2×C4C22×SD1611yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57812yes—
445C8—24no—
446C8—42no—
447C8—81no—
448C8—81no—
449C8—82no—
450C8—82no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22#166614yes—
465C22#241211yescharacteristic
466C22C2×#46011yescharacteristic
467C22#304411yescharacteristic
468C22C2×#46511yescharacteristic
469C22#305511yescharacteristic
470C22C2×#47111yescharacteristic
471C22#306811yescharacteristic
472C4—22no—
473C4—22no—
474C4—24no—
475C4—22no—
476C4—42no—
477C4—41no—
478C4—42no—
479C4—81no—
480C4—161no—
481C4—162no—
482C4C2×#48111yescharacteristic
483C4#309111yescharacteristic
484C4#318711yescharacteristic
485C4#318811yescharacteristic
486C4#329812yes—
487C4#330812yes—
488C2—24no—
489C2—321no—
490C2C2×#166611yescharacteristic
491C2#1917011yescharacteristic
492C2#1940311yescharacteristic
493C2#1947411yescharacteristic
494C2#2452811yescharacteristic
495C2#2455311yescharacteristic
496C2#2457311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
3. Q8×C22⋊C8C2semidirect
4. C2×#1803C2semidirect
5. #10620C2semidirect
6. #10475C2semidirect
7. #9222C2semidirect
9. #8830C2semidirect
10. #8724C2semidirect
11. #8643C2semidirect
232. C22⋊C8SD16semidirect
386. Q8#481semidirect

Maximal subgroups

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

subgroupindex
#329032
#279062
Q8×C22⋊C82
C2×#18032
#106202
#104752
#92222
#91332
#88302
#87242
#86432