Finite Group Databasebeta

← #228417

Subgroups — #228417

Order 512.

590 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23759C211yesmaximal, characteristic
2#23051C211yesmaximal, characteristic
3#22640C211yesmaximal, characteristic
4#9594C211yesmaximal, characteristic, complemented
5#9553C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9534C211yesmaximal, characteristic, complemented
9#9071C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8834C211yesmaximal, characteristic
12#8283C211yesmaximal, characteristic, complemented
13C8×#103C211yesmaximal, characteristic, complemented
14C8×C22⋊C8C211yesmaximal, characteristic
15#7898C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20#2283C2211yescharacteristic
21#2101C411yescharacteristic
22#2097C411yescharacteristic
23#2085C411yescharacteristic
24#2077C411yescharacteristic
25#2075C411yescharacteristic
26#2074C411yescharacteristic
27#2036C411yescharacteristic
28#2035C411yescharacteristic
29#1984C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1924—21no—
34#1924—21no—
35#1923—21no—
36#1923—21no—
37#1917—21no—
38#1840—21no—
39#1797C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#106C2211yescharacteristic
42#1771—21no—
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46#1768—21no—
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×C82C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C8×Q8—22no—
72C8×Q8—41no—
73#432—22no—
74#432—41no—
75C2×#107C2×C411yescharacteristic
76C2×#107C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8C2×C411yescharacteristic
94C2×C22⋊C8C2×C411yescharacteristic
95C2×C22⋊C8D411yescharacteristic
96C2×C22⋊C8D411yescharacteristic
97C2×#97—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—41no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#387—21no—
114#385—21no—
115#383—21no—
116#382C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118#381—21no—
119#380C2×C411yescharacteristic
120#380C2×C411yescharacteristic
121#380C2×C411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—21no—
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
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#336—21no—
145#336—21no—
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—22no—
152Q8⋊C8—22no—
153Q8⋊C8—41no—
154C82—22no—
155C82—41no—
156C22×Q8C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C22×C8—21no—
161C22×C8—24no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22⋊Q8—22no—
178C22⋊Q8—41no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42C2411yescharacteristic
209#107—22no—
210#107—41no—
211#106—22no—
212#106—41no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8#4112yes—
225#103—22no—
226#103—22no—
227#103—22no—
228#103—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—42no—
234C22⋊C8—81no—
235#97—42no—
236#97—42no—
237#97—81no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—42no—
250C4×C8—42no—
251C4×C8—81no—
252C4×C8#4112yes—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266C24C4×D411yescharacteristic
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8#13112yes—
273C22×C4—21no—
274C22×C4—22no—
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×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
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—48no—
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#13112yes—
343C2×C8#13112yes—
344C2×C8#13512yes—
345C2×C8#13512yes—
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—21no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—81no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—81no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C42H(5,𝔽2)11yescharacteristic
385C422−1+411yescharacteristic
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#44312yes—
393C23C2×C4×D411yescharacteristic
394Q8—22no—
395Q8—22no—
396Q8—41no—
397Q8—41no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
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×C4C4×#4111yescharacteristic
444C2×C4#52011yescharacteristic
445C2×C4#54611yescharacteristic
446C2×C4#54711yescharacteristic
447C2×C4#55111yescharacteristic
448C2×C4#55211yescharacteristic
449C2×C4C2×#13111yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4C2×#13111yescharacteristic
452C2×C4#56811yescharacteristic
453C2×C4C2×#13511yescharacteristic
454C2×C4C2×#13511yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57811yescharacteristic
457C8—24no—
458C8—22no—
459C8—22no—
460C8—22no—
461C8—22no—
462C8—42no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—42no—
468C8—81no—
469C8—81no—
470C8—81no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22#235911yescharacteristic
479C22#293111yescharacteristic
480C22#293311yescharacteristic
481C22C4×#13511yescharacteristic
482C22#294711yescharacteristic
483C22C2×#44311yescharacteristic
484C22#295611yescharacteristic
485C4—21no—
486C4—21no—
487C4—41no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4#296611yescharacteristic
499C4#296811yescharacteristic
500C4#299211yescharacteristic
501C4#299411yescharacteristic
502C4#330111yescharacteristic
503C4#330211yescharacteristic
504C4#333711yescharacteristic
505C4#334211yescharacteristic
506C2—42no—
507C2#1795911yescharacteristic
508C2#1796911yescharacteristic
509C2#1799911yescharacteristic
510C2#1801711yescharacteristic
511C2#2290511yescharacteristic
512C2#2291911yescharacteristic
513C2#2294811yescharacteristic
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. #9594C2semidirect
5. #9553C2semidirect
6. #9548C2semidirect
7. #9537C2semidirect
8. #9534C2semidirect
10. #8846C2semidirect
12. #8283C2semidirect
13. C8×#103C2semidirect

Maximal subgroups

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

subgroupindex
#237592
#230512
#226402
#95942
#95532
#95482
#95372
#95342
#90712
#88462
#88342
#82832
C8×#1032
C8×C22⋊C82
#78982