Finite Group Databasebeta

← #213110

Subgroups — #213110

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22268C211yesmaximal, characteristic, complemented
2#22219C211yesmaximal, characteristic, complemented
3#21727C211yesmaximal, characteristic, complemented
4#21690C211yesmaximal, characteristic
5C4×#492C211yesmaximal, characteristic, complemented
6C4×#486C211yesmaximal, characteristic, complemented
7C2×#1715C211yesmaximal, characteristic
8#9829C211yesmaximal, characteristic, complemented
9#9827C211yesmaximal, characteristic, complemented
10#9824C211yesmaximal, characteristic
11#9822C211yesmaximal, characteristic, complemented
12#9820C211yesmaximal, characteristic, complemented
13#9819C211yesmaximal, characteristic, complemented
14#8819C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×#492C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#442—22no—
23C2×C4×SD16—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1972—22no—
33#1968—22no—
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1944—22no—
43#1934—22no—
44#1926—22no—
45#1880—22no—
46#1877—22no—
47#1864—22no—
48#1845C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1776—22no—
55C4×#102—22no—
56#1744C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1715C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C22×SD16—22no—
65C22×SD16—22no—
66C22×D8—22no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79#498—22no—
80#498C2×C414yescomplemented
81#493—22no—
82#493C2×C414yes—
83#492—22no—
84#492—22no—
85#492C2×C414yescomplemented
86#492C2×C414yescomplemented
87#486—22no—
88#486—22no—
89#486C2×C414yescomplemented
90#486C2×C414yescomplemented
91#442—48no—
92C4×SD16—48no—
93C2×#107—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#97D412yes—
109C2×#97D412yes—
110C2×C4×C8D412yes—
111C2×C4×C8D412yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#391—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C43C2311yescharacteristic
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#336D412yes—
146#336D412yes—
147#322—22no—
148#322C2×C414yescomplemented
149C82—22no—
150C82—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C23×C4—41no—
160C23×C4—41no—
161C2×SD16—28no—
162C2×SD16—28no—
163C2×SD16—44no—
164C2×SD16—44no—
165C2×SD16—82no—
166C2×SD16—82no—
167C2×D8—28no—
168C2×D8—44no—
169C2×D8—44no—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C412yes—
184#127—21no—
185#127—21no—
186#127—21no—
187#127—21no—
188#127—21no—
189#127—21no—
190#127C22×C412yes—
191#127C22×C412yes—
192C4×Q8—44no—
193C4×Q8—82no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—82no—
199C4×D4—82no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
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—41no—
217C2×C4⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233#107—44no—
234#106—28no—
235#106—44no—
236#106—44no—
237#106—82no—
238#103—44no—
239#103—82no—
240#102—28no—
241#102—28no—
242#102—44no—
243#102—44no—
244#102—44no—
245#102—44no—
246#102—82no—
247#102—82no—
248#97—24no—
249#97SD1618yescomplemented
250C4×C8—24no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—42no—
258C4×C8C22×C412yes—
259C4×C8C22×C412yes—
260C4×C8C22×C412yes—
261C4×C8#4112yes—
262C4×C8#4112yes—
263C4×C8#4112yes—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268C24—41no—
269C24—41no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4#12711yescharacteristic
315C22×C4#12711yescharacteristic
316C22×C4#12711yescharacteristic
317C22×C4#12711yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321SD16—416no—
322SD16—88no—
323D8—416no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8C4×D414yes—
336C2×C8C4×D414yes—
337C2×C8C4×D414yes—
338C2×C8C2×SD1614yes—
339C2×C8C2×SD1614yes—
340C2×C8#13514yes—
341C2×C8AGL(1,ℤ/8)14yes—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—82no—
376C42—82no—
377C42C2×SD1614yes—
378C42C23×C411yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—42no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23C2×#12711yescharacteristic
397Q8—44no—
398Q8—44no—
399Q8—82no—
400Q8—82no—
401D4—44no—
402D4—44no—
403D4—44no—
404D4—44no—
405D4—44no—
406D4—44no—
407D4—82no—
408D4—82no—
409D4—82no—
410D4—82no—
411D4—82no—
412D4—82no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—42no—
424C2×C4—42no—
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—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—164no—
442C2×C4—164no—
443C2×C4#49212yes—
444C2×C4#49212yes—
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4#53211yescharacteristic
449C2×C4#53211yescharacteristic
450C2×C4#53211yescharacteristic
451C2×C4C22×SD1612yes—
452C2×C4C2×#13512yes—
453C2×C4C2×AGL(1,ℤ/8)12yes—
454C2×C4#57512yes—
455C2×C4#57612yes—
456C2×C4#57712yes—
457C8—24no—
458C8—24no—
459C8—24no—
460C8—24no—
461C8—42no—
462C8C4×SD1618yescomplemented
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22C4×#12711yescharacteristic
478C22C2×#49211yescharacteristic
479C22#314711yescharacteristic
480C22#315111yescharacteristic
481C22#315311yescharacteristic
482C22#315411yescharacteristic
483C22#315511yescharacteristic
484C4—22no—
485C4—22no—
486C4—41no—
487C4—41no—
488C4—82no—
489C4—82no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4C2×C4×SD1612yes—
495C4C4×AGL(1,ℤ/8)12yes—
496C4#294612yes—
497C4#316712yes—
498C4#317912yes—
499C4#318012yes—
500C2—162no—
501C2—162no—
502C2#1962111yescharacteristic
503C2#1962311yescharacteristic
504C2#1964711yescharacteristic
505C2#1965011yescharacteristic
506C2C4×#49211yescharacteristic
507C2#2170111yescharacteristic
508C2#2170411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22268C2semidirect
2. #22219C2semidirect
3. #21727C2semidirect
5. C4×#492C2semidirect
6. C4×#486C2semidirect
8. #9829C2semidirect
9. #9827C2semidirect
11. #9822C2semidirect
12. #9820C2semidirect
13. #9819C2semidirect
14. #8819C2semidirect
15. C8×#97C2semidirect
59. #1715C4semidirect
80. #498C2×C4semidirect
85. #492C2×C4semidirect
86. #492C2×C4semidirect
89. #486C2×C4semidirect
90. #486C2×C4semidirect
148. #322C2×C4semidirect
249. #97SD16semidirect
462. C8C4×SD16semidirect

Maximal subgroups

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

subgroupindex
#222682
#222192
#217272
#216902
C4×#4922
C4×#4862
C2×#17152
#98292
#98272
#98242
#98222
#98202
#98192
#88192
C8×#972