Finite Group Databasebeta

← #501638

Subgroups — #501638

Order 512.

765 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C211yesmaximal, characteristic, complemented
2C2×#2215C211yesmaximal, characteristic
3C2×#2210C211yesmaximal, characteristic, complemented
4C22×C4⋊C16C211yesmaximal, characteristic, complemented
5#33173C211yesmaximal, characteristic, complemented
6C4×#466C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13475C211yesmaximal, characteristic, complemented
9#13449C211yesmaximal, characteristic, complemented
10#13448C211yesmaximal, characteristic, complemented
11#13254C211yesmaximal, characteristic, complemented
12#13178C211yesmaximal, characteristic, complemented
13#13060C211yesmaximal, characteristic, complemented
14#13059C211yesmaximal, characteristic, complemented
15#12813C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#495C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#466C2211yescharacteristic
23C2×#466C2211yescharacteristic
24C2×#465—21no—
25C2×C4×D8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2850—21no—
28C4×#121—21no—
29#2217—24no—
30#2215—24no—
31#2210—24no—
32C2×#367C2211yescharacteristic
33C2×#366C2211yescharacteristic
34C2×C4⋊C16—24no—
35C2×C4⋊C16C2211yescharacteristic
36C2×C4⋊C16C2211yescharacteristic
37C2×C4⋊C16C2211yescharacteristic
38C2×C4⋊C16C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×#357C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1969C2211yescharacteristic
50#1968—21no—
51#1958—21no—
52#1942C2211yescharacteristic
53#1934—21no—
54#1893—21no—
55#1846C2211yescharacteristic
56#1808C2211yescharacteristic
57C4×#107C2211yescharacteristic
58C4×#102—21no—
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62#1380C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×SD16—21no—
65C22×D8C2311yescharacteristic
66C22×D8C2311yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#124—21no—
70C2×C22⋊Q8—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×C4×D4—21no—
74C2×C4×D4—41no—
75C22×C42D411yescharacteristic
76C2×SD32—22no—
77C2×SD32—22no—
78C2×SD32—41no—
79C2×D16—22no—
80C2×D16—22no—
81C2×D16—41no—
82C22×C16—22no—
83C22×C16—22no—
84C22×C16—22no—
85C22×C16—22no—
86C22×C16—41no—
87C22×C16—41no—
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92#495—22no—
93#495—22no—
94#467—22no—
95#467—22no—
96#466—22no—
97#466—22no—
98#466—22no—
99#466D414yes—
100#465—44no—
101C4×D8—22no—
102C4×D8—22no—
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#106—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#390—21no—
124#390—21no—
125#389—21no—
126#388—41no—
127#388—41no—
128#388—41no—
129#386—41no—
130#386—41no—
131#386—41no—
132#382—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C22⋊C4—41no—
136#367—22no—
137#366—22no—
138C4⋊C16—22no—
139C4⋊C16—22no—
140C4⋊C16—22no—
141C4⋊C16D414yescomplemented
142#358—22no—
143#358—22no—
144#358—41no—
145#357—22no—
146#357—22no—
147#357—22no—
148#357—22no—
149#357—41no—
150#340—21no—
151#340—21no—
152#340—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—41no—
157C22×D4—41no—
158C23×C4—41no—
159C23×C4D811yescharacteristic
160C23×C4D811yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×SD16—42no—
169C2×SD16—42no—
170C2×SD16—42no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8C2×D412yes—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—41no—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D412yes—
194C22×C8#4112yes—
195C22×C8C2411yescharacteristic
196#124—42no—
197#124—42no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200#121—24no—
201#121—42no—
202#121—42no—
203#121—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—84no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—41no—
228C2×C42D811yescharacteristic
229C2×C42D811yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232SD32—44no—
233D16—44no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—44no—
245C2×C16—44no—
246C2×C16C2×D412yes—
247C2×C16C2×D412yes—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107C2×D412yes—
258#106—42no—
259#103—42no—
260#103—42no—
261#102—24no—
262#102—42no—
263#102—42no—
264#102—42no—
265#102—42no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—22no—
270C4×C8C2×D412yes—
271C4×C8#4112yes—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95—41no—
279C24—41no—
280C24—41no—
281C24C2×D811yescharacteristic
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×D4—22no—
289C2×D4—44no—
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—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4#12112yes—
326C22×C4C2×D811yescharacteristic
327C22×C4C2×D811yescharacteristic
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4C22×D411yescharacteristic
333Q16—41no—
334Q16—41no—
335Q16—41no—
336Q16—41no—
337SD16—84no—
338D8—22no—
339D8—22no—
340D8—41no—
341D8—41no—
342D8—41no—
343D8—41no—
344D8—41no—
345D8—41no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—44no—
358C2×C8#12112yes—
359C2×C8#12112yes—
360C2×C8C22×D411yescharacteristic
361C2×C8C22×D411yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C2×C82−1+411yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—44no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C42—21no—
390C42—21no—
391C42—22no—
392C42—22no—
393C42—82no—
394C42#12112yes—
395C42#13512yes—
396C16—22no—
397C16—22no—
398C16—41no—
399C16—41no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—42no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23#45912yes—
419C23#50812yes—
420C23#50912yes—
421C23C22×D811yescharacteristic
422Q8—81no—
423Q8—81no—
424Q8—81no—
425Q8—81no—
426D4—42no—
427D4—42no—
428D4—88no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—164no—
436C2×C4—28no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—164no—
472C2×C4#45912yes—
473C2×C4#46612yes—
474C2×C4#46612yes—
475C2×C4#50812yes—
476C2×C4C2×#12111yescharacteristic
477C2×C4C2×#12111yescharacteristic
478C2×C4#53711yescharacteristic
479C2×C4C2×#13511yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
481C2×C4#57511yescharacteristic
482C8—22no—
483C8—22no—
484C8—41no—
485C8D4211yescharacteristic
486C8#54711yescharacteristic
487C8#54811yescharacteristic
488C8#55311yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22#221014yescomplemented
510C22C2×#45911yescharacteristic
511C22C2×#46611yescharacteristic
512C22#324511yescharacteristic
513C22C2×#50811yescharacteristic
514C22C2×#50811yescharacteristic
515C22C2×#50911yescharacteristic
516C22#341411yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—82no—
523C4—161no—
524C4—161no—
525C4—161no—
526C4—161no—
527C4—161no—
528C4#241611yescharacteristic
529C4#303511yescharacteristic
530C4#305511yescharacteristic
531C4#324711yescharacteristic
532C2—24no—
533C2—162no—
534C2—321no—
535C2#2475011yescharacteristic
536C2C2×#221011yescharacteristic
537C2#3357911yescharacteristic
538C2#3360011yescharacteristic
539C2#3360511yescharacteristic
540C2#3381311yescharacteristic
541C2#3381611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#2215C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2217C2semidirect
3. C2×#2210C2semidirect
4. C22×C4⋊C16C2semidirect
5. #33173C2semidirect
6. C4×#466C2semidirect
7. #13476C2semidirect
8. #13475C2semidirect
9. #13449C2semidirect
10. #13448C2semidirect
11. #13254C2semidirect
12. #13178C2semidirect
13. #13060C2semidirect
14. #13059C2semidirect
15. #12813C2semidirect
141. C4⋊C16D4semidirect
509. C22#2210semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
C2×#22152
C2×#22102
C22×C4⋊C162
#331732
C4×#4662
#134762
#134752
#134492
#134482
#132542
#131782
#130602
#130592
#128132