Finite Group Databasebeta

← #203846

Subgroups — #203846

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21987C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#20548C211yesmaximal, characteristic
5C2×#1572C211yesmaximal, characteristic
6#9619C211yesmaximal, characteristic, complemented
7#9619C211yesmaximal, characteristic, complemented
8#9618C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9613C211yesmaximal, characteristic, complemented
11#9613C211yesmaximal, characteristic, complemented
12#9612C211yesmaximal, characteristic, complemented
13#9611C211yesmaximal, characteristic, complemented
14#8043C211yesmaximal, characteristic
15#8042C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2283C2211yescharacteristic
23#1948—21no—
24#1948—21no—
25#1948C2211yescharacteristic
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1854—21no—
34#1854—21no—
35#1853—21no—
36#1853—21no—
37#1853—21no—
38#1852—21no—
39#1847—21no—
40#1846—21no—
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1836—21no—
44#1835—21no—
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1768—21no—
52#1768—21no—
53#1753—21no—
54#1753—21no—
55#1572C418yescomplemented
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×M16—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432C2×C414yes—
76#431—22no—
77#431C2×C414yes—
78#431C2×C414yes—
79#431Q814yes—
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×#106—21no—
91C2×#106—21no—
92#423—22no—
93#423Q814yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8D411yescharacteristic
106C2×C22⋊C8D411yescharacteristic
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#384—21no—
116#384—21no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#383—21no—
123#382—21no—
124#382D411yescharacteristic
125#382D411yescharacteristic
126#381—21no—
127#380—21no—
128#380—21no—
129#380D411yescharacteristic
130#380D411yescharacteristic
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×C414yescomplemented
154#335C2×C414yescomplemented
155#334C2×C414yescomplemented
156#334C2×C414yescomplemented
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×M16—24no—
162C2×M16—81no—
163C2×M16—81no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174#117—81no—
175#117—81no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
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—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—81no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C42—21no—
204C2×C42—21no—
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—81no—
215#107—81no—
216#107—81no—
217#107—81no—
218#106—81no—
219#106—81no—
220#106—81no—
221#106—81no—
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⋊C8C4⋊C414yes—
234C4⋊C8C4⋊C414yes—
235C4⋊C8C22×C412yes—
236C4⋊C8C22×C412yes—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C2×Q812yes—
240C4⋊C8C2×Q812yes—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8C4⋊C414yes—
244C22⋊C8C4⋊C414yes—
245#97—24no—
246#97—81no—
247#97—81no—
248C4×C8—24no—
249C4×C8—81no—
250C4×C8—81no—
251C4×C8C22×C412yes—
252C4×C8C22×C412yes—
253C4×C8C2×Q812yes—
254C4×C8C2×Q812yes—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
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×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C4⋊C412yes—
288C22×C4C4×D412yes—
289C22×C4C4×Q812yes—
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4C22⋊Q812yes—
294C22×C4#12512yes—
295C22×C4C4⋊Q812yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4111yescharacteristic
299M16—84no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
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—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8C2×C4⋊C412yes—
327C2×C8C2×C4⋊C412yes—
328C2×C8C2×C4⋊C412yes—
329C2×C8C2×C4⋊C412yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
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—41no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C22⋊C4—81no—
362C22⋊C4—81no—
363C22⋊C4—82no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—82no—
369C42C2×C4⋊C412yes—
370C42C4×D412yes—
371C42C4×Q812yes—
372C42C22⋊Q812yes—
373C42#12512yes—
374C42C4⋊Q812yes—
375C42C23×C411yescharacteristic
376C42C22×Q811yescharacteristic
377C42C22×Q811yescharacteristic
378C42C2×#4111yescharacteristic
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23#38912yes—
386C23C2×#12111yescharacteristic
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—42no—
404C2×C4—42no—
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—41no—
416C2×C4—41no—
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—164no—
430C2×C4#38912yes—
431C2×C4#38912yes—
432C2×C4#38912yes—
433C2×C4C22×C4⋊C411yescharacteristic
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×C4×Q811yescharacteristic
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4C2×#12511yescharacteristic
438C2×C4C2×C4⋊Q811yescharacteristic
439C2×C4#57611yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57811yescharacteristic
447C8—22no—
448C8—22no—
449C8—22no—
450C8—22no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—42no—
455C8—42no—
456C8—42no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22C2×#38911yescharacteristic
465C22#304511yescharacteristic
466C22#304611yescharacteristic
467C22#305711yescharacteristic
468C22#305711yescharacteristic
469C22#306811yescharacteristic
470C22#306911yescharacteristic
471C4—22no—
472C4—22no—
473C4—21no—
474C4—21no—
475C4—22no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4#291211yescharacteristic
485C4#291311yescharacteristic
486C4#295211yescharacteristic
487C4#295211yescharacteristic
488C4#308111yescharacteristic
489C4#308211yescharacteristic
490C4#316311yescharacteristic
491C4#316311yescharacteristic
492C2—42no—
493C2#1953611yescharacteristic
494C2#1953711yescharacteristic
495C2#1954411yescharacteristic
496C2#1954511yescharacteristic
497C2#2115511yescharacteristic
498C2#2117511yescharacteristic
499C2#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. #9619C2semidirect
7. #9619C2semidirect
8. #9618C2semidirect
9. #9617C2semidirect
10. #9613C2semidirect
11. #9613C2semidirect
12. #9612C2semidirect
13. #9611C2semidirect
55. #1572C4semidirect
153. #335C2×C4semidirect
154. #335C2×C4semidirect
155. #334C2×C4semidirect
156. #334C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219872
#219852
#219812
#205482
C2×#15722
#96192
#96182
#96172
#96132
#96122
#96112
#80432
#80422