Finite Group Databasebeta

← #228781

Subgroups — #228781

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757C211yesmaximal, characteristic
2#23055C211yesmaximal, characteristic
3#22600C211yesmaximal, characteristic
4#9595C211yesmaximal, characteristic, complemented
5#9553C211yesmaximal, characteristic, complemented
6#9543C211yesmaximal, characteristic, complemented
7#9540C211yesmaximal, characteristic, complemented
8#9533C211yesmaximal, characteristic, complemented
9#9073C211yesmaximal, characteristic
10#8848C211yesmaximal, characteristic, complemented
11#8834C211yesmaximal, characteristic
12C8×#107C211yesmaximal, characteristic, complemented
13#8209C211yesmaximal, characteristic, complemented
14#8148C211yesmaximal, characteristic
15#7900C211yesmaximal, characteristic
16C2×#446C2211yescharacteristic
17C2×#433—21no—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C4×C22⋊Q8C2211yescharacteristic
21#2285C2211yescharacteristic
22#2100C411yescharacteristic
23#2095C411yescharacteristic
24#2083C411yescharacteristic
25#2076C411yescharacteristic
26#2076C411yescharacteristic
27#2072C411yescharacteristic
28#2036C411yescharacteristic
29#2036C411yescharacteristic
30#1984C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1924—21no—
35#1923—21no—
36#1917—21no—
37#1917—21no—
38#1916—21no—
39#1818—21no—
40#1797C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43#1770—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×#117—21no—
70C22×C42C2311yescharacteristic
71#446—22no—
72#446—41no—
73#433—44no—
74#432—22no—
75#432—41no—
76#431—22no—
77#431—41no—
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#103C2×C411yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8C2×C411yescharacteristic
99C2×C22⋊C8C2×C411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101C2×C22⋊C8D411yescharacteristic
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#387—21no—
116#385—21no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#382—21no—
120#380—21no—
121#380C2×C411yescharacteristic
122#380C2×C411yescharacteristic
123#380C2×C411yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
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#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#335—22no—
152#334—41no—
153Q8⋊C8—22no—
154Q8⋊C8—41no—
155#322—41no—
156C82—22no—
157C22×Q8C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22⋊Q8—22no—
176C22⋊Q8—41no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213#107—22no—
214#107—22no—
215#107—22no—
216#107—22no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8#4112yes—
230#103—22no—
231#103—22no—
232#103—41no—
233#103—41no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238C22⋊C8—42no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241#97—42no—
242#97—42no—
243#97—42no—
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—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8#4112yes—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267C24C4×D411yescharacteristic
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
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—22no—
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—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×C8—41no—
343C2×C8#13112yes—
344C2×C8#13112yes—
345C2×C8AGL(1,ℤ/8)12yes—
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—21no—
358C4⋊C4—21no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4#13112yes—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
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
388C42H(5,𝔽2)11yescharacteristic
389C422−1+411yescharacteristic
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23C2×C4×D411yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
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—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4C4×#4111yescharacteristic
448C2×C4#52011yescharacteristic
449C2×C4#54611yescharacteristic
450C2×C4#54711yescharacteristic
451C2×C4#55111yescharacteristic
452C2×C4#55211yescharacteristic
453C2×C4C2×#13111yescharacteristic
454C2×C4C2×#13111yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4#56811yescharacteristic
457C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
458C2×C4#57511yescharacteristic
459C2×C4#57811yescharacteristic
460C2×C4#57811yescharacteristic
461C8—22no—
462C8—22no—
463C8—22no—
464C8—42no—
465C8—41no—
466C8—42no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—81no—
472C8—81no—
473C8—81no—
474C8—81no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22#235911yescharacteristic
483C22#293111yescharacteristic
484C22#293311yescharacteristic
485C22C4×AGL(1,ℤ/8)11yescharacteristic
486C22#295011yescharacteristic
487C22#295511yescharacteristic
488C22#295711yescharacteristic
489C4—21no—
490C4—21no—
491C4—42no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4#296611yescharacteristic
503C4#296911yescharacteristic
504C4#299311yescharacteristic
505C4#299411yescharacteristic
506C4#332711yescharacteristic
507C4#333011yescharacteristic
508C4#334311yescharacteristic
509C4#334911yescharacteristic
510C2—81no—
511C2#1796411yescharacteristic
512C2#1797211yescharacteristic
513C2#1800711yescharacteristic
514C2#1803211yescharacteristic
515C2#2291511yescharacteristic
516C2#2294711yescharacteristic
517C2#2295111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9595C2semidirect
5. #9553C2semidirect
6. #9543C2semidirect
7. #9540C2semidirect
8. #9533C2semidirect
10. #8848C2semidirect
12. C8×#107C2semidirect
13. #8209C2semidirect

Maximal subgroups

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

subgroupindex
#237572
#230552
#226002
#95952
#95532
#95432
#95402
#95332
#90732
#88482
#88342
C8×#1072
#82092
#81482
#79002