Finite Group Databasebeta

← #501639

Subgroups — #501639

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C211yesmaximal, characteristic
2C2×#2216C211yesmaximal, characteristic, complemented
3C2×#2211C211yesmaximal, characteristic, complemented
4C22×C4⋊C16C211yesmaximal, characteristic, complemented
5#33173C211yesmaximal, characteristic
6C4×#467C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13450C211yesmaximal, characteristic, complemented
10#13449C211yesmaximal, characteristic, complemented
11#13254C211yesmaximal, characteristic, complemented
12#13178C211yesmaximal, characteristic, complemented
13#13060C211yesmaximal, characteristic, complemented
14#13059C211yesmaximal, characteristic
15#12814C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#495C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#467C2211yescharacteristic
23C2×#466C2211yescharacteristic
24C2×#465—21no—
25C2×C4×Q16C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2850—21no—
28C4×C22⋊Q8—21no—
29#2217—24no—
30#2216—24no—
31#2211—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×#358C2211yescharacteristic
43C2×#358C2211yescharacteristic
44C2×#357C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47#1970C2211yescharacteristic
48#1969C2211yescharacteristic
49#1969C2211yescharacteristic
50#1968—21no—
51#1958—21no—
52#1942C2211yescharacteristic
53#1933—21no—
54#1892—21no—
55#1846C2211yescharacteristic
56#1808C2211yescharacteristic
57C4×#107C2211yescharacteristic
58C4×#103—21no—
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62#1380C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×Q16C2311yescharacteristic
65C22×SD16—21no—
66C22×D8C2311yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#124—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×#121—21no—
73C2×C4×Q8—21no—
74C22×C42D411yescharacteristic
75C2×Q32—22no—
76C2×Q32—22no—
77C2×Q32—41no—
78C2×SD32—22no—
79C2×SD32—22no—
80C2×SD32—41no—
81C22×C16—22no—
82C22×C16—22no—
83C22×C16—22no—
84C22×C16—22no—
85C22×C16—41no—
86C22×C16—41no—
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91#495—22no—
92#495—22no—
93#467—22no—
94#467—22no—
95#467—22no—
96#467D414yes—
97#466—22no—
98#466—22no—
99#465—44no—
100C4×Q16—22no—
101C4×Q16—22no—
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#106—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#391—21no—
123#390—21no—
124#389—21no—
125#388—41no—
126#388—41no—
127#387—41no—
128#386—41no—
129#386—41no—
130#385—41no—
131#382—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—41no—
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—22no—
145#358—22no—
146#358—41no—
147#357—22no—
148#357—22no—
149#357—41no—
150#340—21no—
151#340—21no—
152#340—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×D4—21no—
156C22×D4—41no—
157C23×C4D811yescharacteristic
158C23×C4D811yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16C2×D412yes—
172C2×SD16—42no—
173C2×SD16—42no—
174C2×SD16—42no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—41no—
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D412yes—
192C22×C8#4112yes—
193C22×C8C2411yescharacteristic
194#124—42no—
195#124—42no—
196C22⋊Q8—24no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200#121—42no—
201#121—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C22⋊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×C42—22no—
222C2×C42—22no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42D811yescharacteristic
226C2×C42D811yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229Q32—44no—
230SD32—44no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—44no—
242C2×C16—44no—
243C2×C16C2×D412yes—
244C2×C16C2×D412yes—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107C2×D412yes—
255#106—42no—
256#103—24no—
257#103—42no—
258#103—42no—
259#103—42no—
260#103—42no—
261#102—42no—
262#102—42no—
263C4×C8—22no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—22no—
267C4×C8C2×D412yes—
268C4×C8#4112yes—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278C24—41no—
279C24C2×D811yescharacteristic
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4#12112yes—
319C22×C4C2×D811yescharacteristic
320C22×C4C2×D811yescharacteristic
321C22×C4C2×D811yescharacteristic
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4AGL(1,ℤ/8)12yes—
325C22×C4C22×D411yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330Q16—41no—
331Q16—41no—
332Q16—41no—
333Q16—41no—
334SD16—84no—
335D8—41no—
336D8—41no—
337D8—41no—
338D8—41no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—44no—
351C2×C8#12112yes—
352C2×C8#12112yes—
353C2×C8C22×D411yescharacteristic
354C2×C8C22×D411yescharacteristic
355C2×C8C2×#4111yescharacteristic
356C2×C8C2×#4111yescharacteristic
357C2×C8H(5,𝔽2)11yescharacteristic
358C2×C82−1+411yescharacteristic
359C4⋊C4—22no—
360C4⋊C4—41no—
361C4⋊C4—44no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C22⋊C4—44no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C42—21no—
382C42—21no—
383C42—22no—
384C42—22no—
385C42—82no—
386C42—82no—
387C42#12112yes—
388C42#13512yes—
389C16—22no—
390C16—22no—
391C16—41no—
392C16—41no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23#45912yes—
405C23#50812yes—
406C23#51012yes—
407C23C22×D811yescharacteristic
408Q8—42no—
409Q8—42no—
410Q8—81no—
411Q8—81no—
412Q8—81no—
413Q8—81no—
414Q8—81no—
415Q8—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420D4—164no—
421C2×C4—28no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4#45912yes—
460C2×C4#46612yes—
461C2×C4#46612yes—
462C2×C4#50812yes—
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4#53711yescharacteristic
466C2×C4C2×#13511yescharacteristic
467C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
468C2×C4#57511yescharacteristic
469C8—22no—
470C8—22no—
471C8—41no—
472C8D4211yescharacteristic
473C8#54711yescharacteristic
474C8#54811yescharacteristic
475C8#55311yescharacteristic
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22—161no—
486C22—161no—
487C22—161no—
488C22—161no—
489C22#221114yescomplemented
490C22C2×#45911yescharacteristic
491C22C2×#46611yescharacteristic
492C22#324511yescharacteristic
493C22C2×#50811yescharacteristic
494C22C2×#50811yescharacteristic
495C22C2×#51011yescharacteristic
496C22#341411yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—41no—
502C4—82no—
503C4—82no—
504C4—161no—
505C4—161no—
506C4—161no—
507C4—161no—
508C4—161no—
509C4#241611yescharacteristic
510C4#303511yescharacteristic
511C4#305511yescharacteristic
512C4#324711yescharacteristic
513C2—24no—
514C2—321no—
515C2#2475011yescharacteristic
516C2C2×#221111yescharacteristic
517C2#3357911yescharacteristic
518C2#3360011yescharacteristic
519C2#3360611yescharacteristic
520C2#3381411yescharacteristic
521C2#3381611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
2. C2×#2216C2semidirect
3. C2×#2211C2semidirect
4. C22×C4⋊C16C2semidirect
6. C4×#467C2semidirect
7. #13477C2semidirect
8. #13474C2semidirect
9. #13450C2semidirect
10. #13449C2semidirect
11. #13254C2semidirect
12. #13178C2semidirect
13. #13060C2semidirect
15. #12814C2semidirect
141. C4⋊C16D4semidirect
489. C22#2211semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
C2×#22162
C2×#22112
C22×C4⋊C162
#331732
C4×#4672
#134772
#134742
#134502
#134492
#132542
#131782
#130602
#130592
#128142