Finite Group Databasebeta

← #318927

Subgroups — #318927

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29258C211yesmaximal, characteristic
2#28284C211yesmaximal, characteristic, complemented
3#21713C211yesmaximal, characteristic, complemented
4#21646C211yesmaximal, characteristic
5#10602C211yesmaximal, characteristic, complemented
6#10598C211yesmaximal, characteristic, complemented
7#10591C211yesmaximal, characteristic
8#10587C211yesmaximal, characteristic
9#9746C211yesmaximal, characteristic
10#9738C211yesmaximal, characteristic
11#9619C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#8808C211yesmaximal, characteristic, complemented
14#8591C211yesmaximal, characteristic, complemented
15#8585C211yesmaximal, characteristic
16C2×#486C2211yescharacteristic
17C2×#477—21no—
18C2×#446—21no—
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2094—21no—
23#2068—21no—
24#2068—21no—
25#2023—21no—
26#1990C2211yescharacteristic
27#1985—21no—
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1981C2211yescharacteristic
31#1981C2211yescharacteristic
32#1957C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1940—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1936—21no—
41#1936—21no—
42#1864—21no—
43#1864—21no—
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1834C2211yescharacteristic
47#1831C2211yescharacteristic
48#1803—21no—
49#1771—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53#1762—21no—
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#486—22no—
76#486Q814yescomplemented
77#477—44no—
78#446—44no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#97—21no—
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×#97D411yescharacteristic
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#391—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389D411yescharacteristic
124#389D411yescharacteristic
125#384D411yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C43C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#335—41no—
150#334—22no—
151#334—41no—
152#332—41no—
153#332—41no—
154#331—41no—
155#331—41no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C23×C4—41no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—22no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—41no—
177C4⋊Q8C2×Q812yes—
178#127—21no—
179#127—21no—
180#127—21no—
181#127C2×Q812yes—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—81no—
189C4×D4—81no—
190C4×D4—82no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—41no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—42no—
223#107—42no—
224#106—81no—
225#106—81no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—81no—
244#103—42no—
245#103—42no—
246#103—81no—
247#103—81no—
248#102—24no—
249#102—24no—
250#102—42no—
251#102—42no—
252#102—42no—
253#102—42no—
254#102—81no—
255#102—81no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—41no—
260#97—42no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—42no—
264C4×C8C2×Q812yes—
265C4×C8#4112yes—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—41no—
271#95—41no—
272C24—41no—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—81no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—81no—
297C2×D4—81no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
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—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8C22⋊Q812yes—
343C2×C8C22⋊Q812yes—
344C2×C8AGL(1,ℤ/8)12yes—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4AGL(1,ℤ/8)12yes—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—82no—
389C42#13512yes—
390C42#13712yes—
391C42C22×Q811yescharacteristic
392C42C2×#4111yescharacteristic
393C42H(5,𝔽2)11yescharacteristic
394C422−1+411yescharacteristic
395C23—42no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23C2×#12111yescharacteristic
403Q8—42no—
404Q8—42no—
405Q8—84no—
406Q8—81no—
407Q8—81no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—82no—
413D4—82no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—164no—
452C2×C4#46412yes—
453C2×C4#46412yes—
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4#53711yescharacteristic
456C2×C4D4×Q811yescharacteristic
457C2×C4#55011yescharacteristic
458C2×C4#55411yescharacteristic
459C2×C4#55611yescharacteristic
460C2×C4C2×#13511yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C2×C4#57511yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57611yescharacteristic
467C2×C4#57711yescharacteristic
468C8—22no—
469C8—22no—
470C8—22no—
471C8—41no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22#253811yescharacteristic
486C22C2×#46411yescharacteristic
487C22#303611yescharacteristic
488C22#305511yescharacteristic
489C22#306211yescharacteristic
490C22#306811yescharacteristic
491C22#306911yescharacteristic
492C4—22no—
493C4—21no—
494C4—21no—
495C4—22no—
496C4—41no—
497C4—42no—
498C4—41no—
499C4—81no—
500C4—82no—
501C4—81no—
502C4—81no—
503C4—161no—
504C4—161no—
505C4#307611yescharacteristic
506C4#307911yescharacteristic
507C4#324711yescharacteristic
508C4#325211yescharacteristic
509C4#338411yescharacteristic
510C4#338911yescharacteristic
511C4#339911yescharacteristic
512C4#340111yescharacteristic
513C2—162no—
514C2#1917711yescharacteristic
515C2#1924211yescharacteristic
516C2#1930611yescharacteristic
517C2#1936811yescharacteristic
518C2#2827611yescharacteristic
519C2#2828211yescharacteristic
520C2#2829811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×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
2. #28284C2semidirect
3. #21713C2semidirect
5. #10602C2semidirect
6. #10598C2semidirect
11. #9619C2semidirect
12. #9613C2semidirect
13. #8808C2semidirect
14. #8591C2semidirect
76. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#292582
#282842
#217132
#216462
#106022
#105982
#105912
#105872
#97462
#97382
#96192
#96132
#88082
#85912
#85852