Finite Group Databasebeta

← #279579

Subgroups — #279579

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29983C211yesmaximal, characteristic, complemented
2#28508C211yesmaximal, characteristic
3#22721C211yesmaximal, characteristic, complemented
4#22147C211yesmaximal, characteristic
5#10652C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10599C211yesmaximal, characteristic
8#10595C211yesmaximal, characteristic
9#9896C211yesmaximal, characteristic, complemented
10#9603C211yesmaximal, characteristic, complemented
11#9230C211yesmaximal, characteristic
12#9222C211yesmaximal, characteristic
13#8653C211yesmaximal, characteristic, complemented
14#8645C211yesmaximal, characteristic, complemented
15#8588C211yesmaximal, characteristic
16C2×#474—21no—
17C2×#474—21no—
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21#2868C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2295C2211yescharacteristic
24#2099—21no—
25#2096—21no—
26#2093—21no—
27#2093—21no—
28#2058—21no—
29#2024—21no—
30#1982C2211yescharacteristic
31#1968—21no—
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1940—21no—
38#1939—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1934—21no—
42#1877—21no—
43#1864—21no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1843—21no—
47#1842C2211yescharacteristic
48#1831C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1768—21no—
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×#127C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#474—44no—
77#474—44no—
78#460—22no—
79#460Q814yescomplemented
80#446—22no—
81#446—41no—
82C8×Q8—22no—
83C8×Q8—41no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391D411yescharacteristic
121#391D411yescharacteristic
122#390—21no—
123#390—21no—
124#390—21no—
125#389D411yescharacteristic
126#389D411yescharacteristic
127#387—21no—
128#387—21no—
129#384—21no—
130#384—21no—
131#384—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137#376—21no—
138C43C2311yescharacteristic
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—41no—
153#334—41no—
154#331—41no—
155#331—41no—
156C22×Q8—41no—
157C22×Q8C2×D411yescharacteristic
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C23×C4—41no—
162C2×SD16—24no—
163C2×SD16—81no—
164C2×SD16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—81no—
178C4⋊Q8—81no—
179C4⋊Q8—82no—
180#127—21no—
181#127—21no—
182#127—21no—
183#127C2×Q812yes—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8C2×Q812yes—
187C4×Q8#4112yes—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—81no—
191C4×D4—81no—
192C4×D4—82no—
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—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—42no—
224#107—42no—
225#107—42no—
226#107—42no—
227#106—81no—
228#106—81no—
229#106—81no—
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—81no—
246C4⋊C8—81no—
247C4⋊C8—81no—
248C4⋊C8SD1614yescomplemented
249C4⋊C8C2×Q812yes—
250#103—81no—
251#103—81no—
252#103—81no—
253#102—24no—
254#102—42no—
255#102—42no—
256#102—42no—
257#102—42no—
258#102—81no—
259#102—81no—
260#97—42no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8—81no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24—41no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8C22⋊Q812yes—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
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—41no—
297C2×D4—41no—
298C2×D4—81no—
299C2×D4—81no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317SD16—84no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
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×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8C2×SD1612yes—
347C2×C8C2×SD1612yes—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
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—81no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4C22⋊Q812yes—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—22no—
380C42—22no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—22no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42C2×SD1612yes—
392C42AGL(1,ℤ/8)12yes—
393C42C22×Q811yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C422−1+411yescharacteristic
397C23—42no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23C2×#12111yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—41no—
408Q8—41no—
409Q8—164no—
410D4—42no—
411D4—42no—
412D4—42no—
413D4—42no—
414D4—82no—
415D4—82no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
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—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—81no—
452C2×C4—164no—
453C2×C4#46012yes—
454C2×C4#46012yes—
455C2×C4#47212yes—
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4#53711yescharacteristic
458C2×C4#54611yescharacteristic
459C2×C4D4×Q811yescharacteristic
460C2×C4#55211yescharacteristic
461C2×C4#55411yescharacteristic
462C2×C4C22×SD1611yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4#57611yescharacteristic
465C2×C4#57611yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57711yescharacteristic
468C2×C4#57811yescharacteristic
469C2×C4#57811yescharacteristic
470C8—22no—
471C8—22no—
472C8—41no—
473C8—41no—
474C8—42no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8—81no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—161no—
484C22—161no—
485C22—161no—
486C22—161no—
487C22#241711yescharacteristic
488C22C2×#46011yescharacteristic
489C22#304611yescharacteristic
490C22#305811yescharacteristic
491C22#306311yescharacteristic
492C22C2×#47211yescharacteristic
493C22#306911yescharacteristic
494C4—22no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—41no—
502C4—81no—
503C4—81no—
504C4—161no—
505C4—161no—
506C4—161no—
507C4#308111yescharacteristic
508C4#308311yescharacteristic
509C4#325711yescharacteristic
510C4#326211yescharacteristic
511C4#329811yescharacteristic
512C4#333211yescharacteristic
513C4Q8×SD1611yescharacteristic
514C4#338611yescharacteristic
515C2—162no—
516C2#1920311yescharacteristic
517C2#1923911yescharacteristic
518C2#1954611yescharacteristic
519C2#1959111yescharacteristic
520C2#2481311yescharacteristic
521C2#2484711yescharacteristic
522C2#2485511yescharacteristic
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
1. #29983C2semidirect
3. #22721C2semidirect
5. #10652C2semidirect
6. #10651C2semidirect
9. #9896C2semidirect
10. #9603C2semidirect
13. #8653C2semidirect
14. #8645C2semidirect
79. #460Q8semidirect
248. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#299832
#285082
#227212
#221472
#106522
#106512
#105992
#105952
#98962
#96032
#92302
#92222
#86532
#86452
#85882