Finite Group Databasebeta

← #498567

Subgroups — #498567

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2250C211yesmaximal, characteristic
2C2×#2177C211yesmaximal, characteristic, complemented
3C2×C4×SD32C211yesmaximal, characteristic, complemented
4#23186C211yesmaximal, characteristic, complemented
5#23185C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic, complemented
7#13252C211yesmaximal, characteristic, complemented
8#13238C211yesmaximal, characteristic, complemented
9#13235C211yesmaximal, characteristic, complemented
10#13014C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#13012C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic, complemented
14#12674C211yesmaximal, characteristic, complemented
15#12663C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×D16C2211yescharacteristic
19C22×D16C2211yescharacteristic
20C2×#495C2211yescharacteristic
21C2×#493C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×C4×D8C2211yescharacteristic
25C2×C4×D8C2211yescharacteristic
26#2367—21no—
27#2366—21no—
28#2250C418yescomplemented
29#2177—24no—
30C4×SD32—24no—
31C2×#367C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#346C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×C4×C16C2211yescharacteristic
45#1983—21no—
46#1982—21no—
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1970C2211yescharacteristic
50#1969C2211yescharacteristic
51#1880—21no—
52#1879—21no—
53#1879—21no—
54#1877—21no—
55C4×#107C2211yescharacteristic
56C4×#107C2211yescharacteristic
57C8×C4⋊C4C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×D8—21no—
65C22×D8C2311yescharacteristic
66C22×D8C2311yescharacteristic
67C22×D8C2311yescharacteristic
68C2×#127—21no—
69C2×#124—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—41no—
75C2×SD32—22no—
76C2×SD32—22no—
77C2×SD32—41no—
78C2×SD32—41no—
79C2×SD32C2×C414yescomplemented
80C2×SD32C2×C414yescomplemented
81C2×D16—22no—
82C2×D16—22no—
83C2×D16—41no—
84C2×D16—41no—
85C2×D16C2×C414yescomplemented
86C2×D16C2×C414yescomplemented
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91#495—22no—
92#495C2×C414yescomplemented
93#493—22no—
94#493C2×C414yescomplemented
95C4×Q16—22no—
96C4×Q16—41no—
97C4×D8—22no—
98C4×D8—22no—
99C4×D8—22no—
100C4×D8—41no—
101C4×D8—41no—
102C4×D8—41no—
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390—41no—
124#388—41no—
125#388—41no—
126#386—41no—
127#386—41no—
128#385—41no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D411yescharacteristic
132C4×C22⋊C4—41no—
133#367D414yescomplemented
134#366—22no—
135C4⋊C16—22no—
136#358—22no—
137#358—41no—
138#357—22no—
139#357—22no—
140#357—22no—
141#357—41no—
142#357—41no—
143#357—41no—
144#346—22no—
145#346C2×C414yescomplemented
146C4×C16—22no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—41no—
156C22×D4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16C22×C412yes—
166C2×SD16—24no—
167C2×SD16—81no—
168C2×SD16—81no—
169C2×D8—24no—
170C2×D8—21no—
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×D8—21no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8—81no—
186C2×D8—81no—
187C2×D8C22×C412yes—
188C2×D8C22×C412yes—
189C2×D8C22×C412yes—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2411yescharacteristic
194#127—24no—
195#127—42no—
196#124—24no—
197#124—42no—
198C4×Q8—42no—
199C4×Q8—81no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—81no—
204C4×D4—81no—
205C4×D4—81no—
206C4×D4—84no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4D811yescharacteristic
213C2×C4⋊C4D811yescharacteristic
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—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231SD32—28no—
232SD32—44no—
233D16—28no—
234D16—44no—
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×C16C22×C412yes—
244C2×C16C22×C412yes—
245C2×C16C2×D412yes—
246C2×C16#4112yes—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107C2×D412yes—
254#107C2×D412yes—
255C4⋊C8—22no—
256#103—42no—
257#103—81no—
258#102—42no—
259#102—42no—
260#102—42no—
261#102—81no—
262#102—81no—
263#102—81no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8C22×C412yes—
269C4×C8#4112yes—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95D811yescharacteristic
275#95D811yescharacteristic
276C24—41no—
277C24—41no—
278C24—41no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—41no—
289C2×D4—41no—
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—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—161no—
308C2×D4—161no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—161no—
321C22×C4—161no—
322C22×C4—161no—
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C2×D811yescharacteristic
326C22×C4C2×D811yescharacteristic
327C22×C4C2×D811yescharacteristic
328C22×C4C2×D811yescharacteristic
329C22×C4C22×D411yescharacteristic
330Q16—22no—
331Q16—22no—
332Q16—41no—
333Q16—41no—
334SD16—84no—
335D8—22no—
336D8—22no—
337D8—22no—
338D8—22no—
339D8—22no—
340D8—22no—
341D8—41no—
342D8—41no—
343D8—41no—
344D8—41no—
345D8—41no—
346D8—41no—
347D8—84no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8C4×D412yes—
357C2×C8C23×C411yescharacteristic
358C2×C8C22×D411yescharacteristic
359C2×C8C2×#4111yescharacteristic
360C2×C8C2×#4111yescharacteristic
361C2×C8H(5,𝔽2)11yescharacteristic
362C2×C8H(5,𝔽2)11yescharacteristic
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—82no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C22⋊C4—161no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—82no—
390C42—82no—
391C42—161no—
392C42C4×D412yes—
393C42#13512yes—
394C16—22no—
395C16—41no—
396C16—41no—
397C16C4×D414yescomplemented
398C23—42no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
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—81no—
419C23C22×D811yescharacteristic
420Q8—42no—
421Q8—42no—
422Q8—81no—
423Q8—81no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—88no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—81no—
436D4—81no—
437D4—164no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
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—164no—
469C2×C4—164no—
470C2×C4—164no—
471C2×C4C4×D812yes—
472C2×C4C4×D812yes—
473C2×C4#50912yes—
474C2×C4C2×C4×D411yescharacteristic
475C2×C4D4211yescharacteristic
476C2×C4#55011yescharacteristic
477C2×C4C2×#13511yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57611yescharacteristic
480C8—22no—
481C8—41no—
482C8—41no—
483C8C2×C4×D411yescharacteristic
484C8#51811yescharacteristic
485C8#55011yescharacteristic
486C8#55911yescharacteristic
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22C2×C4×D811yescharacteristic
509C22D4×D811yescharacteristic
510C22#339111yescharacteristic
511C22C2×#50911yescharacteristic
512C22#341411yescharacteristic
513C22#341511yescharacteristic
514C22#341611yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4—161no—
525C4—161no—
526C4#236611yescharacteristic
527C4#294811yescharacteristic
528C4#328711yescharacteristic
529C4#339611yescharacteristic
530C2—162no—
531C2—162no—
532C2—162no—
533C2#2318511yescharacteristic
534C2C4×#50911yescharacteristic
535C2#3345411yescharacteristic
536C2#3383111yescharacteristic
537C2#3383411yescharacteristic
538C2#3396611yescharacteristic
539C2#3396811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2250C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2177C2semidirect
3. C2×C4×SD32C2semidirect
4. #23186C2semidirect
5. #23185C2semidirect
6. #13254C2semidirect
7. #13252C2semidirect
8. #13238C2semidirect
9. #13235C2semidirect
10. #13014C2semidirect
11. #13013C2semidirect
12. #13012C2semidirect
13. #13012C2semidirect
14. #12674C2semidirect
15. #12663C2semidirect
28. #2250C4semidirect
79. C2×SD32C2×C4semidirect
80. C2×SD32C2×C4semidirect
85. C2×D16C2×C4semidirect
86. C2×D16C2×C4semidirect
92. #495C2×C4semidirect
94. #493C2×C4semidirect
133. #367D4semidirect
145. #346C2×C4semidirect
397. C16C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22502
C2×#21772
C2×C4×SD322
#231862
#231852
#132542
#132522
#132382
#132352
#130142
#130132
#130122
#126742
#126632