Finite Group Databasebeta

← #352111

Subgroups — #352111

Order 512.

868 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#33209C211yesmaximal, characteristic
2#33209C211yesmaximal, characteristic
3#33207C211yesmaximal, characteristic
4#33172C211yesmaximal, characteristic
5#33043C211yesmaximal, characteristic
6#33043C211yesmaximal, characteristic
7C2×#1723C211yesmaximal, characteristic, complemented
8C22×#322C211yesmaximal, characteristic
9#9841C211yesmaximal, characteristic, complemented
10#9836C211yesmaximal, characteristic, complemented
11#9835C211yesmaximal, characteristic, complemented
12#9834C211yesmaximal, characteristic, complemented
13#9833C211yesmaximal, characteristic, complemented
14#9833C211yesmaximal, characteristic, complemented
15#8820C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#487C2211yescharacteristic
22C2×#467—22no—
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1969—22no—
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1956C2211yescharacteristic
43#1942—22no—
44#1942—22no—
45#1941—22no—
46#1933—22no—
47#1933—22no—
48#1892—22no—
49#1892—22no—
50#1845C2211yescharacteristic
51#1845C2211yescharacteristic
52#1845C2211yescharacteristic
53#1844C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1808—22no—
57#1808—22no—
58#1807—22no—
59#1723—24no—
60C2×#322—24no—
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C22×Q16—22no—
66C23×C8D412yes—
67C23×C8D412yes—
68C23×C8D412yes—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C22×C42C2311yescharacteristic
78#500—22no—
79#500—22no—
80#500—22no—
81#500—22no—
82#498—22no—
83#498—22no—
84#494—22no—
85#494—22no—
86#487—22no—
87#487—22no—
88#487—22no—
89#487—22no—
90#467—48no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8D412yes—
111C2×C4×C8D412yes—
112C2×C4×C8D412yes—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391—21no—
120#391—21no—
121#391—21no—
122#391—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380—21no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#322—22no—
154#322—22no—
155#322—22no—
156#322D414yes—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×Q16—44no—
165C2×Q16—44no—
166C2×Q16—44no—
167C22×C8—24no—
168C22×C8—24no—
169C22×C8—24no—
170C22×C8—24no—
171C22×C8—24no—
172C22×C8—24no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8Q1618yescomplemented
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2411yescharacteristic
238#107—44no—
239#107—44no—
240#107—44no—
241#107—44no—
242#107—44no—
243#106—44no—
244#106—44no—
245#103—44no—
246#103—44no—
247#103—44no—
248#103—44no—
249#103—44no—
250#103—44no—
251C4×C8—24no—
252C4×C8—24no—
253C4×C8—24no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8C2×D412yes—
261C4×C8C2×D412yes—
262C4×C8C2×D412yes—
263C4×C8#4112yes—
264C4×C8#4112yes—
265C4×C8#4112yes—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24#12711yescharacteristic
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4#12711yescharacteristic
309C22×C4#12711yescharacteristic
310C22×C4#12711yescharacteristic
311C22×C4C2×Q1614yes—
312C22×C4AGL(1,ℤ/8)14yes—
313C22×C4#13714yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317Q16—88no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—216no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—48no—
332C2×C8—48no—
333C2×C8#12114yes—
334C2×C8#12114yes—
335C2×C8#12114yes—
336C2×C8C2×Q1614yes—
337C2×C8C2×Q1614yes—
338C2×C8#13514yes—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42C22×D411yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23#49412yes—
390C23#49612yes—
391C23#49712yes—
392C23C2×#12711yescharacteristic
393Q8—82no—
394Q8—82no—
395Q8—82no—
396Q8—82no—
397Q8—82no—
398Q8—82no—
399Q8—82no—
400Q8—82no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4#49412yes—
438C2×C4#49612yes—
439C2×C4#49712yes—
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4#53211yescharacteristic
444C2×C4#53211yescharacteristic
445C2×C4#53211yescharacteristic
446C2×C4C22×Q1612yes—
447C2×C4C2×#13512yes—
448C2×C4C2×AGL(1,ℤ/8)12yes—
449C2×C4C2×#13712yes—
450C2×C4#57512yes—
451C2×C4#57512yes—
452C8—24no—
453C8—24no—
454C8—24no—
455C8—42no—
456C8—42no—
457C8#46718yescomplemented
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22#172314yescomplemented
465C22#285611yescharacteristic
466C22C2×#49411yescharacteristic
467C22#314711yescharacteristic
468C22C2×#49611yescharacteristic
469C22C2×#49711yescharacteristic
470C22#315011yescharacteristic
471C22#315011yescharacteristic
472C4—22no—
473C4—22no—
474C4—22no—
475C4—161no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4C2×#46712yes—
484C4#305512yes—
485C4#305612yes—
486C4#317012yes—
487C4#318012yes—
488C4#318112yes—
489C2—24no—
490C2C2×#172311yescharacteristic
491C2#1961811yescharacteristic
492C2#1961911yescharacteristic
493C2#1962011yescharacteristic
494C2#3317211yescharacteristic
495C2#3318311yescharacteristic
496C2#3318411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. C2×#1723C2semidirect
9. #9841C2semidirect
10. #9836C2semidirect
11. #9835C2semidirect
12. #9834C2semidirect
13. #9833C2semidirect
14. #9833C2semidirect
15. #8820C2semidirect
175. C22×C8Q16semidirect
457. C8#467semidirect
464. C22#1723semidirect

Maximal subgroups

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

subgroupindex
#332092
#332072
#331722
#330432
C2×#17232
C22×#3222
#98412
#98362
#98352
#98342
#98332
#88202