Finite Group Databasebeta

← #226639

Subgroups — #226639

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22839C211yesmaximal, characteristic
2#22721C211yesmaximal, characteristic, complemented
3#21114C211yesmaximal, characteristic
4#9558C211yesmaximal, characteristic
5#9556C211yesmaximal, characteristic
6#9540C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic
9#8948C211yesmaximal, characteristic
10#8789C211yesmaximal, characteristic, complemented
11#8570C211yesmaximal, characteristic, complemented
12#8563C211yesmaximal, characteristic
13#8265C211yesmaximal, characteristic, complemented
14#8209C211yesmaximal, characteristic, complemented
15#8203C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19#2295C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1940—21no—
24#1939—21no—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1928C411yescharacteristic
28#1928C411yescharacteristic
29#1924—21no—
30#1924—21no—
31#1919C411yescharacteristic
32#1919C411yescharacteristic
33#1877C411yescharacteristic
34#1877C411yescharacteristic
35#1862C411yescharacteristic
36#1862C411yescharacteristic
37#1843—21no—
38#1842C2211yescharacteristic
39#1831C2211yescharacteristic
40#1817—21no—
41C4×#107C2211yescharacteristic
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44#1768—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1749—21no—
50#1744C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C22×C42—21no—
73#446—22no—
74#446—41no—
75C8×Q8—22no—
76C8×Q8—41no—
77C4×SD16—22no—
78C4×SD16—22no—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×C4⋊C8—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⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#387—21no—
120#387—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129#376—21no—
130C43C2311yescharacteristic
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#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150Q8⋊C8—22no—
151Q8⋊C8—22no—
152D4⋊C8—41no—
153D4⋊C8—41no—
154#322—41no—
155#322—41no—
156C22×Q8C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C2×SD16—22no—
162C2×SD16—22no—
163C2×SD16—41no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8#4112yes—
181C4×Q8#4112yes—
182C4×D4—22no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—81no—
208C2×C42—81no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#107—41no—
217#107—41no—
218#106—22no—
219C4⋊C8—22no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—41no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236#103—22no—
237#103—41no—
238#103—41no—
239#102—22no—
240#102—22no—
241#102—41no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—81no—
245C22⋊C8—81no—
246#97—42no—
247#97—81no—
248#97—81no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8#11712yes—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4#13112yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4111yescharacteristic
311SD16—44no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
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—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
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×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8#13712yes—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—42no—
368C4⋊C4—81no—
369C4⋊C4#11712yes—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C42—22no—
375C42—22no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—22no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—81no—
391C42—81no—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42H(5,𝔽2)11yescharacteristic
395C422−1+411yescharacteristic
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C2×C4×D411yescharacteristic
404Q8—22no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
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—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—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#44412yes—
453C2×C4C2×#11711yescharacteristic
454C2×C4#52011yescharacteristic
455C2×C4#54611yescharacteristic
456C2×C4#54811yescharacteristic
457C2×C4#55211yescharacteristic
458C2×C4#55311yescharacteristic
459C2×C4C2×#13111yescharacteristic
460C2×C4#56811yescharacteristic
461C2×C4#56811yescharacteristic
462C2×C4#56811yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C2×C4#57511yescharacteristic
465C2×C4#57611yescharacteristic
466C2×C4#57711yescharacteristic
467C8—22no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8—81no—
480C8—81no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22#235211yescharacteristic
489C22#293011yescharacteristic
490C22#293311yescharacteristic
491C22C4×#13711yescharacteristic
492C22#294611yescharacteristic
493C22C2×#44411yescharacteristic
494C22#295711yescharacteristic
495C4—22no—
496C4—21no—
497C4—21no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—42no—
502C4—42no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4#292311yescharacteristic
509C4#292411yescharacteristic
510C4#299211yescharacteristic
511C4#299511yescharacteristic
512C4#330911yescharacteristic
513C4#332011yescharacteristic
514C4#337111yescharacteristic
515C4#337511yescharacteristic
516C2—82no—
517C2#1794111yescharacteristic
518C2#1794611yescharacteristic
519C2#1800911yescharacteristic
520C2#1802011yescharacteristic
521C2#2275311yescharacteristic
522C2#2277511yescharacteristic
523C2#2278211yescharacteristic
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×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22721C2semidirect
6. #9540C2semidirect
7. #9537C2semidirect
10. #8789C2semidirect
11. #8570C2semidirect
13. #8265C2semidirect
14. #8209C2semidirect
15. #8203C2semidirect

Maximal subgroups

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

subgroupindex
#228392
#227212
#211142
#95582
#95562
#95402
#95372
#89522
#89482
#87892
#85702
#85632
#82652
#82092
#82032