Finite Group Databasebeta

← #226738

Subgroups — #226738

Order 512.

605 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#22829C211yesmaximal, characteristic, complemented
2#22651C211yesmaximal, characteristic
3#21279C211yesmaximal, characteristic
4#9555C211yesmaximal, characteristic, complemented
5#9553C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8949C211yesmaximal, characteristic
9#8945C211yesmaximal, characteristic
10#8786C211yesmaximal, characteristic, complemented
11#8569C211yesmaximal, characteristic
12#8568C211yesmaximal, characteristic, complemented
13#8262C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic
15C8×#102C211yesmaximal, characteristic
16C2×C4×SD16C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#424—21no—
20#2295C2211yescharacteristic
21#2293C2211yescharacteristic
22#1984C2211yescharacteristic
23#1984C2211yescharacteristic
24#1973C411yescharacteristic
25#1973C411yescharacteristic
26#1944C411yescharacteristic
27#1944C411yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1936C411yescharacteristic
31#1936C411yescharacteristic
32#1935C411yescharacteristic
33#1935C411yescharacteristic
34#1917—21no—
35#1916—21no—
36#1851—21no—
37#1850—21no—
38#1842C2211yescharacteristic
39#1840—21no—
40#1831C2211yescharacteristic
41#1798—21no—
42C4×#107C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1746—21no—
50C4×#97C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C4×SD16—22no—
74C4×SD16—22no—
75#436—22no—
76#436—41no—
77C8×D4—22no—
78C8×D4—41no—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82#424—44no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97C2×C411yescharacteristic
104C2×#97C2×C411yescharacteristic
105C2×#97C2×C411yescharacteristic
106C2×#97C2×C411yescharacteristic
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#388—21no—
123#388—21no—
124#387—21no—
125#387—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130#376—21no—
131C43C2311yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
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—
149#336D411yescharacteristic
150#336D411yescharacteristic
151Q8⋊C8—41no—
152Q8⋊C8—41no—
153D4⋊C8—22no—
154D4⋊C8—22no—
155#322—41no—
156C82—41no—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C2×SD16—22no—
161C2×SD16—22no—
162C2×SD16—41no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—81no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4#4112yes—
190C4×D4#4112yes—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#107—22no—
220#107—41no—
221#106—22no—
222C4⋊C8—22no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—41no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—81no—
237#103—22no—
238#103—22no—
239#103—22no—
240#102—22no—
241#102—22no—
242#102—22no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245#97—22no—
246#97—22no—
247#97—22no—
248#97—41no—
249#97—42no—
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—21no—
262C4×C8—42no—
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—
272#95—21no—
273C24—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4#11712yes—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
306SD16—44no—
307M16—84no—
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—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—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×C8—84no—
347C2×C8#13112yes—
348C2×C8#13112yes—
349C2×C8#13112yes—
350C2×C8AGL(1,ℤ/8)12yes—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—81no—
368C4⋊C4#11712yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—81no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C42H(5,𝔽2)11yescharacteristic
394C422−1+411yescharacteristic
395C23—22no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23C2×C4×D411yescharacteristic
403Q8—41no—
404Q8—41no—
405Q8—41no—
406Q8—41no—
407D4—22no—
408D4—22no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
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—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
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—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4C2×#11711yescharacteristic
451C2×C4#52011yescharacteristic
452C2×C4#54611yescharacteristic
453C2×C4#54811yescharacteristic
454C2×C4#55211yescharacteristic
455C2×C4#55311yescharacteristic
456C2×C4C2×#13111yescharacteristic
457C2×C4C2×#13111yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57811yescharacteristic
464C8—22no—
465C8—22no—
466C8—22no—
467C8—22no—
468C8—42no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22#235211yescharacteristic
486C22#293111yescharacteristic
487C22#293511yescharacteristic
488C22C4×AGL(1,ℤ/8)11yescharacteristic
489C22#294711yescharacteristic
490C22#295511yescharacteristic
491C22#295611yescharacteristic
492C4—21no—
493C4—21no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—42no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4#292211yescharacteristic
506C4#292411yescharacteristic
507C4#299211yescharacteristic
508C4#299411yescharacteristic
509C4#331011yescharacteristic
510C4#331511yescharacteristic
511C4#336711yescharacteristic
512C4#337611yescharacteristic
513C2—82no—
514C2#1796411yescharacteristic
515C2#1796911yescharacteristic
516C2#1807611yescharacteristic
517C2#1808111yescharacteristic
518C2#2275611yescharacteristic
519C2#2277811yescharacteristic
520C2#2278411yescharacteristic
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
1. #22829C2semidirect
4. #9555C2semidirect
5. #9553C2semidirect
6. #9540C2semidirect
7. #9533C2semidirect
10. #8786C2semidirect
12. #8568C2semidirect
13. #8262C2semidirect

Maximal subgroups

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

subgroupindex
#228292
#226512
#212792
#95552
#95532
#95402
#95332
#89492
#89452
#87862
#85692
#85682
#82622
#82082
C8×#1022