Finite Group Databasebeta

← #498141

Subgroups — #498141

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2186C211yesmaximal, characteristic
2C2×C4×SD32C211yesmaximal, characteristic, complemented
3C2×#2139C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic, complemented
5#22451C211yesmaximal, characteristic, complemented
6#13239C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic, complemented
10#13013C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12938C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×D16C2211yescharacteristic
19C22×M32C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C22×#107C2211yescharacteristic
25#2361—21no—
26#2341—21no—
27#2186C418yescomplemented
28C4×SD32—24no—
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31#2139—24no—
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1942C2211yescharacteristic
47#1925—21no—
48#1880—21no—
49#1880—21no—
50#1874—21no—
51#1862—21no—
52#1808C2211yescharacteristic
53#1788—21no—
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×#41—21no—
63C2×#135—22no—
64C2×#135—22no—
65C2×#135—22no—
66C2×#135—41no—
67C2×#135—41no—
68C2×#135C2×C414yescomplemented
69C22×Q16C2311yescharacteristic
70C22×SD16—21no—
71C22×D8C2311yescharacteristic
72C22×D8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×#121—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×D4—21no—
78C2×C4×D4—41no—
79C22×C4⋊C4—21no—
80C2×SD32—22no—
81C2×SD32—22no—
82C2×SD32C2×C414yescomplemented
83C2×D16—41no—
84C2×D16—41no—
85C2×D16C2×C414yescomplemented
86C2×M32—22no—
87C2×M32—22no—
88C2×M32—41no—
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92C22×C16C2311yescharacteristic
93#466—22no—
94#466C2×C414yes—
95C4×Q16—22no—
96C4×Q16—22no—
97C4×Q16—41no—
98C4×Q16—41no—
99C4×D8—22no—
100C4×D8—41no—
101C2×#107—22no—
102C2×#107—21no—
103C2×#107—41no—
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#103—21no—
110C2×#103—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#390—21no—
120#390—21no—
121#390—21no—
122#390—41no—
123#390—41no—
124#389—21no—
125#388—41no—
126#386—41no—
127#385—41no—
128#380—21no—
129#380—41no—
130C4×C4⋊C4—21no—
131C4×C22⋊C4—41no—
132C4×C22⋊C4D411yescharacteristic
133#367—22no—
134#367—41no—
135#358—22no—
136#358C2×C414yescomplemented
137#358D414yescomplemented
138#357—22no—
139#357—22no—
140#357—22no—
141#357—41no—
142#357C2×C414yescomplemented
143C22⋊C16C2×C414yescomplemented
144#346—41no—
145C4×C16—22no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C2×#41—24no—
152C2×#41—42no—
153C2×#41—42no—
154C2×#41—42no—
155C2×#41—81no—
156C2×#41—81no—
157C22×Q8—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—41no—
161C22×D4—41no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166#135—28no—
167#135—44no—
168#135—44no—
169#135—44no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16C22×C412yes—
174C2×Q16C2×D412yes—
175C2×Q16#4112yes—
176C2×SD16—24no—
177C2×SD16—42no—
178C2×SD16—42no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8C22×C412yes—
190C2×D8C22×C412yes—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8C22×C412yes—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8#4112yes—
200C22×C8C2411yescharacteristic
201#121—24no—
202#121—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—81no—
206C4×Q8—81no—
207C4×D4—42no—
208C4×D4—81no—
209C4×D4—84no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—81no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4D811yescharacteristic
231C2×C22⋊C4D811yescharacteristic
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237SD32—28no—
238D16—44no—
239M32—44no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16C22×C412yes—
250C2×C16C22×C412yes—
251C2×C16C2×D412yes—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107—42no—
261#107C22×C412yes—
262#107C2×D412yes—
263#103—42no—
264#103—42no—
265#102—24no—
266#102—42no—
267#102—81no—
268#102—81no—
269#102—81no—
270C22⋊C8—22no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95D811yescharacteristic
279#95D811yescharacteristic
280C24—41no—
281C24—41no—
282C24C2×D811yescharacteristic
283#41—48no—
284#41—84no—
285#41—84no—
286#41—84no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—44no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—161no—
329C22×C4—161no—
330C22×C4C4×D412yes—
331C22×C4C2×D811yescharacteristic
332C22×C4C2×D811yescharacteristic
333C22×C4C2×D811yescharacteristic
334C22×C4C2×D811yescharacteristic
335C22×C4C2×D811yescharacteristic
336C22×C4#13512yes—
337C22×C4C22×D411yescharacteristic
338Q16—22no—
339Q16—22no—
340Q16—22no—
341Q16C4×D414yescomplemented
342SD16—48no—
343D8—22no—
344D8—22no—
345D8—22no—
346D8—22no—
347D8—41no—
348D8—41no—
349D8—41no—
350D8—41no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8C4×D412yes—
364C2×C8C23×C411yescharacteristic
365C2×C8C22×D411yescharacteristic
366C2×C8C2×#4111yescharacteristic
367C2×C8C2×#4111yescharacteristic
368C2×C8H(5,𝔽2)11yescharacteristic
369C2×C8H(5,𝔽2)11yescharacteristic
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C22⋊C4—22no—
385C22⋊C4—41no—
386C22⋊C4—44no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—161no—
392C22⋊C4—161no—
393C22⋊C4—161no—
394C22⋊C4—161no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—82no—
400C42—82no—
401C42—82no—
402C16—22no—
403C16—22no—
404C16—22no—
405C16—41no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23C4×D812yes—
427C23C22×D811yescharacteristic
428Q8—24no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—88no—
437D4—88no—
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—88no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—164no—
479C2×C4—164no—
480C2×C4C4×D812yes—
481C2×C4#50912yes—
482C2×C4C2×C4×D411yescharacteristic
483C2×C4D4211yescharacteristic
484C2×C4#54611yescharacteristic
485C2×C4C2×#13511yescharacteristic
486C2×C4#57611yescharacteristic
487C2×C4#57711yescharacteristic
488C8—22no—
489C8—41no—
490C8—41no—
491C8C2×C4×D411yescharacteristic
492C8#51811yescharacteristic
493C8#54611yescharacteristic
494C8#55111yescharacteristic
495C22—22no—
496C22—22no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—161no—
509C22—161no—
510C22—161no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22C2×C4×D811yescharacteristic
517C22D4×D811yescharacteristic
518C22#329411yescharacteristic
519C22C2×#50911yescharacteristic
520C22#341411yescharacteristic
521C22#341611yescharacteristic
522C22#341611yescharacteristic
523C4—22no—
524C4—22no—
525C4—41no—
526C4—44no—
527C4—41no—
528C4—41no—
529C4—82no—
530C4—82no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4#235811yescharacteristic
535C4#294611yescharacteristic
536C4#328211yescharacteristic
537C4#329711yescharacteristic
538C2—42no—
539C2—162no—
540C2—162no—
541C2#2287011yescharacteristic
542C2C4×#50911yescharacteristic
543C2#3346011yescharacteristic
544C2#3383111yescharacteristic
545C2#3383711yescharacteristic
546C2#3389311yescharacteristic
547C2#3389911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2186C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×SD32C2semidirect
3. C2×#2139C2semidirect
4. #23025C2semidirect
5. #22451C2semidirect
6. #13239C2semidirect
7. #13147C2semidirect
8. #13131C2semidirect
9. #13061C2semidirect
10. #13013C2semidirect
11. #13012C2semidirect
12. #12938C2semidirect
13. #12920C2semidirect
14. #12638C2semidirect
15. #12612C2semidirect
27. #2186C4semidirect
68. C2×#135C2×C4semidirect
82. C2×SD32C2×C4semidirect
85. C2×D16C2×C4semidirect
136. #358C2×C4semidirect
137. #358D4semidirect
142. #357C2×C4semidirect
143. C22⋊C16C2×C4semidirect
341. Q16C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21862
C2×C4×SD322
C2×#21392
#230252
#224512
#132392
#131472
#131312
#130612
#130132
#130122
#129382
#129202
#126382
#126122