Finite Group Databasebeta

← #203885

Subgroups — #203885

Order 512.

664 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#21985C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic
3#20824C211yesmaximal, characteristic
4#20482C211yesmaximal, characteristic
5C2×#1565C211yesmaximal, characteristic
6#9622C211yesmaximal, characteristic, complemented
7#9621C211yesmaximal, characteristic, complemented
8#9619C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9614C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9612C211yesmaximal, characteristic, complemented
14#8039C211yesmaximal, characteristic
15#8038C211yesmaximal, characteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22#2283C2211yescharacteristic
23#1949—21no—
24#1949—21no—
25#1948C2211yescharacteristic
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1854—21no—
34#1854—21no—
35#1846—21no—
36#1845C2211yescharacteristic
37#1844C2211yescharacteristic
38#1835—21no—
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1809—21no—
42#1809—21no—
43#1808—21no—
44#1807—21no—
45#1776—21no—
46#1776—21no—
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1753—21no—
52#1753—21no—
53#1746—21no—
54#1726—21no—
55#1565C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68C22×M16D411yescharacteristic
69C22×M16D411yescharacteristic
70C23×C8—21no—
71C2×#117—21no—
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74#431—22no—
75#431—22no—
76#431C2×C414yes—
77#431Q814yes—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C4×M16C2×C414yes—
102C4×M16Q814yes—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97D411yescharacteristic
106C2×#97D411yescharacteristic
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#383—21no—
118#382—21no—
119#382—21no—
120#382D411yescharacteristic
121#382D411yescharacteristic
122#381—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380D411yescharacteristic
128#380D411yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136#376—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#335C2×C414yes—
154#335C2×C414yes—
155#334C2×C414yes—
156#334C2×C414yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—41no—
163C2×M16—41no—
164C2×M16C4⋊C414yes—
165C2×M16C4⋊C414yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—24no—
173C22×C8—81no—
174C22×C8—81no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179#117—81no—
180#117—81no—
181#117—82no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C22×C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q812yes—
210C2×C42C2×Q812yes—
211C2×C42#4112yes—
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214#107—42no—
215#107—42no—
216#107—81no—
217#107—81no—
218#107—81no—
219#107—81no—
220#106—42no—
221#106—42no—
222#106—81no—
223#106—81no—
224#106—81no—
225#106—81no—
226C4⋊C8—24no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8C22×C412yes—
234C4⋊C8C22×C412yes—
235C4⋊C8C22×C412yes—
236C4⋊C8C22×C412yes—
237C4⋊C8C2×Q812yes—
238C4⋊C8C2×Q812yes—
239C22⋊C8—24no—
240C22⋊C8—24no—
241#97—42no—
242#97—42no—
243#97C4⋊C414yes—
244#97C4⋊C414yes—
245C4×C8—24no—
246C4×C8—24no—
247C4×C8—42no—
248C4×C8—42no—
249C4×C8C22×C412yes—
250C4×C8C22×C412yes—
251C4×C8C2×Q812yes—
252C4×C8C2×Q812yes—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24#12111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4C2×C4⋊C412yes—
285C22×C4C4×D412yes—
286C22×C4C4×Q812yes—
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4C22⋊Q812yes—
291C22×C4#12512yes—
292C22×C4C4⋊Q812yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296M16—28no—
297M16—44no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—84no—
323C2×C8C2×C4⋊C412yes—
324C2×C8C2×C4⋊C412yes—
325C2×C8C2×C4⋊C412yes—
326C2×C8C2×C4⋊C412yes—
327C2×C8#13512yes—
328C2×C8#13512yes—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
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—81no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C22⋊C4—81no—
358C22⋊C4—81no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—82no—
366C42—82no—
367C42C2×C4⋊C412yes—
368C42C4×D412yes—
369C42C4×Q812yes—
370C42C22⋊Q812yes—
371C42#12512yes—
372C42C4⋊Q812yes—
373C42C23×C411yescharacteristic
374C42C22×Q811yescharacteristic
375C42C22×Q811yescharacteristic
376C42C2×#4111yescharacteristic
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23#38912yes—
384C23C2×#12111yescharacteristic
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
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#38912yes—
428C2×C4#38912yes—
429C2×C4#38912yes—
430C2×C4C22×C4⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12511yescharacteristic
435C2×C4C2×C4⋊Q811yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4#57511yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57811yescharacteristic
444C8—22no—
445C8—22no—
446C8—22no—
447C8—42no—
448C8—42no—
449C8—42no—
450C8—42no—
451C8—42no—
452C8—42no—
453C8#42714yes—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22C2×#38911yescharacteristic
462C22#303511yescharacteristic
463C22#303511yescharacteristic
464C22#306211yescharacteristic
465C22#306311yescharacteristic
466C22#306811yescharacteristic
467C22#306911yescharacteristic
468C4—22no—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4C2×#42711yescharacteristic
482C4#291111yescharacteristic
483C4#294611yescharacteristic
484C4#294711yescharacteristic
485C4#307611yescharacteristic
486C4#307611yescharacteristic
487C4#316311yescharacteristic
488C4#316411yescharacteristic
489C2—42no—
490C2#1924811yescharacteristic
491C2#1924911yescharacteristic
492C2#1925211yescharacteristic
493C2#1925311yescharacteristic
494C2#2116011yescharacteristic
495C2#2116911yescharacteristic
496C2#2118211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ 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
6. #9622C2semidirect
7. #9621C2semidirect
8. #9619C2semidirect
9. #9617C2semidirect
10. #9616C2semidirect
11. #9614C2semidirect
12. #9613C2semidirect
13. #9612C2semidirect
55. #1565C4semidirect

Maximal subgroups

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

subgroupindex
#219852
#219622
#208242
#204822
C2×#15652
#96222
#96212
#96192
#96172
#96162
#96142
#96132
#96122
#80392
#80382