Finite Group Databasebeta

← #212693

Subgroups — #212693

Order 512.

599 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21675C211yesmaximal, characteristic, complemented
2C4×#488C211yesmaximal, characteristic
3#21634C211yesmaximal, characteristic
4#9875C211yesmaximal, characteristic, complemented
5#9645C211yesmaximal, characteristic
6#9558C211yesmaximal, characteristic
7#9557C211yesmaximal, characteristic, complemented
8#9554C211yesmaximal, characteristic
9#9553C211yesmaximal, characteristic, complemented
10#8785C211yesmaximal, characteristic, complemented
11#8210C211yesmaximal, characteristic
12#8206C211yesmaximal, characteristic
13#8203C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#488C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C4×#125C2211yescharacteristic
20C4×#124C2211yescharacteristic
21#1991C411yescharacteristic
22#1984C2211yescharacteristic
23#1958C411yescharacteristic
24#1954C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1934C411yescharacteristic
28#1934C411yescharacteristic
29#1933C411yescharacteristic
30#1933C411yescharacteristic
31#1924—21no—
32#1924—21no—
33#1917—21no—
34#1917—21no—
35#1835C411yescharacteristic
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38#1771—21no—
39#1770—21no—
40#1770C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46#1753—21no—
47#1753—21no—
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×C82C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×#125C2×C411yescharacteristic
69C2×#124C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72#488—24no—
73C8×Q8—22no—
74C8×Q8—22no—
75#436—22no—
76#436—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#103C2×C411yescharacteristic
85C2×#103C2×C411yescharacteristic
86C2×#103C2×C411yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#102C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×#97D411yescharacteristic
97C2×#97D411yescharacteristic
98C2×#97D411yescharacteristic
99C2×#97D411yescharacteristic
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2×C411yescharacteristic
103C2×C4×C8C2×C411yescharacteristic
104C2×C4×C8C2×C411yescharacteristic
105C2×C4×C8C2×C411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#388—21no—
117#388—21no—
118#387—21no—
119#387—21no—
120#382C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149Q8⋊C8—22no—
150Q8⋊C8—22no—
151D4⋊C8—22no—
152D4⋊C8—22no—
153#322—22no—
154#322—22no—
155C82—22no—
156C82—22no—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177#125—22no—
178#125—41no—
179#124—22no—
180#124—41no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42C2411yescharacteristic
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226#103—22no—
227#103—22no—
228#103—41no—
229#103M1614yescomplemented
230#102—22no—
231#102—22no—
232#102—22no—
233#102—41no—
234C22⋊C8—42no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237#97—22no—
238#97—22no—
239#97—22no—
240#97—22no—
241#97—42no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—41no—
253C4×C8—41no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8#4112yes—
257C4×C8#4112yes—
258C4×C8#4112yes—
259C4×C8#4112yes—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8C2×M1612yes—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303M16—84no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8#12412yes—
339C2×C8#12412yes—
340C2×C8C2×M1612yes—
341C2×C8#13112yes—
342C2×C8#13512yes—
343C2×C8#13512yes—
344C2×C8#13512yes—
345C2×C8#13512yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4C2×M1612yes—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—81no—
369C22⋊C4—81no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C42C2×#4111yescharacteristic
389C42C2×#4111yescharacteristic
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23C2×C4×D411yescharacteristic
398Q8—22no—
399Q8—22no—
400Q8—41no—
401Q8—41no—
402D4—41no—
403D4—41no—
404D4—41no—
405D4—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
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—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4C4×#4111yescharacteristic
446C2×C4C4×#4111yescharacteristic
447C2×C4C2×#12411yescharacteristic
448C2×C4#52911yescharacteristic
449C2×C4#52911yescharacteristic
450C2×C4#53211yescharacteristic
451C2×C4C22×M1611yescharacteristic
452C2×C4C2×#13111yescharacteristic
453C2×C4#56811yescharacteristic
454C2×C4#56811yescharacteristic
455C2×C4C2×#13511yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C8—22no—
460C8—22no—
461C8—22no—
462C8—22no—
463C8—22no—
464C8—42no—
465C8—42no—
466C8—41no—
467C8—41no—
468C8—81no—
469C8—81no—
470C8—81no—
471C8—81no—
472C8#48814yescomplemented
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22C4×#12411yescharacteristic
481C22D4×M1611yescharacteristic
482C22#293511yescharacteristic
483C22C4×#13511yescharacteristic
484C22C4×#13511yescharacteristic
485C22#295711yescharacteristic
486C22#295711yescharacteristic
487C4—21no—
488C4—21no—
489C4—41no—
490C4—42no—
491C4—41no—
492C4—42no—
493C4—41no—
494C4—41no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4#296511yescharacteristic
501C4#296711yescharacteristic
502C4#296811yescharacteristic
503C4#297011yescharacteristic
504C4C2×#48811yescharacteristic
505C4#313111yescharacteristic
506C4#316811yescharacteristic
507C4#316811yescharacteristic
508C2—161no—
509C2#1803711yescharacteristic
510C2#1803911yescharacteristic
511C2#1806911yescharacteristic
512C2#1807011yescharacteristic
513C2#2163211yescharacteristic
514C2C4×#48811yescharacteristic
515C2#2165911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21675C2semidirect
4. #9875C2semidirect
7. #9557C2semidirect
9. #9553C2semidirect
10. #8785C2semidirect
13. #8203C2semidirect
14. C8×#103C2semidirect
15. C8×#97C2semidirect
229. #103M16semidirect
472. C8#488semidirect

Maximal subgroups

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

subgroupindex
#216752
C4×#4882
#216342
#98752
#96452
#95582
#95572
#95542
#95532
#87852
#82102
#82062
#82032
C8×#1032
C8×#972