Finite Group Databasebeta

← #498133

Subgroups — #498133

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2185C211yesmaximal, characteristic
2C2×C4×SD32C211yesmaximal, characteristic, complemented
3C22×#358C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic
5C22⋊C4×Q16C211yesmaximal, characteristic, complemented
6#13237C211yesmaximal, characteristic, complemented
7#13148C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic
10#13014C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#12938C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C23×Q16C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C22×#107C2211yescharacteristic
25#2361—21no—
26Q8×C22⋊C4—21no—
27#2185C418yescomplemented
28C4×SD32—24no—
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#358—24no—
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1925—21no—
48#1878—21no—
49#1878—21no—
50#1874—21no—
51#1862—21no—
52#1808C2211yescharacteristic
53#1787—21no—
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C23×Q8—21no—
63C22×Q16—22no—
64C22×Q16—22no—
65C22×Q16—21no—
66C22×Q16—22no—
67C22×Q16—41no—
68C22×Q16—41no—
69C22×Q16C2×C414yescomplemented
70C22×Q16C2311yescharacteristic
71C22×Q16C2311yescharacteristic
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×SD32—41no—
83C2×SD32—41no—
84C2×SD32C2×C414yescomplemented
85C2×SD32C2×C414yescomplemented
86C22×C16—22no—
87C22×C16—22no—
88C22×C16—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×#103—21no—
112C2×#103—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391—21no—
120#391—21no—
121#391—41no—
122#391—41no—
123#390—21no—
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#358—22no—
137#358—22no—
138#358C2×C414yescomplemented
139#358C2×C414yescomplemented
140#358D414yescomplemented
141#357—22no—
142#357—41no—
143C22⋊C16C2×C414yescomplemented
144C4×C16—22no—
145C4×C16—41no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C22×Q8—24no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—41no—
157C22×Q8—42no—
158C22×Q8—81no—
159C22×Q8—81no—
160C22×D4—21no—
161C22×D4—41no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C2×Q16—28no—
167C2×Q16—24no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—44no—
177C2×Q16—44no—
178C2×Q16—44no—
179C2×Q16—42no—
180C2×Q16—42no—
181C2×Q16C22×C412yes—
182C2×Q16C22×C412yes—
183C2×Q16C2×D412yes—
184C2×Q16#4112yes—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8—21no—
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⋊C4D811yescharacteristic
227C2×C22⋊C4D811yescharacteristic
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233SD32—28no—
234SD32—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×C16—21no—
244C2×C16—44no—
245C2×C16C22×C412yes—
246C2×C16C22×C412yes—
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—42no—
257#107C22×C412yes—
258#107C2×D412yes—
259#103—42no—
260#103—42no—
261#103—81no—
262#103—81no—
263#102—24no—
264#102—42no—
265#102—81no—
266C22⋊C8—22no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95D811yescharacteristic
279#95D811yescharacteristic
280C24—41no—
281C24C2×D811yescharacteristic
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—48no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—44no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—82no—
297C2×Q8—82no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×D4—22no—
302C2×D4—44no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—82no—
309C2×D4—82no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
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—
341Q16—22no—
342Q16—22no—
343Q16—48no—
344Q16—41no—
345Q16—41no—
346Q16C4×D414yescomplemented
347D8—22no—
348D8—22no—
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—44no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—161no—
390C22⋊C4SD3214yescomplemented
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—82no—
396C42—82no—
397C42—82no—
398C16—22no—
399C16—22no—
400C16—22no—
401C16—41no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—42no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23C4×D812yes—
414C23C2×SD3212yes—
415C23#50812yes—
416C23C22×D811yescharacteristic
417Q8—24no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—88no—
424Q8—81no—
425Q8—81no—
426D4—42no—
427D4—42no—
428D4—88no—
429D4—81no—
430D4—81no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—88no—
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—164no—
470C2×C4—164no—
471C2×C4C4×D812yes—
472C2×C4C2×SD3212yes—
473C2×C4C2×SD3212yes—
474C2×C4C2×C4×D411yescharacteristic
475C2×C4D4211yescharacteristic
476C2×C4#54611yescharacteristic
477C2×C4C2×#13511yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57711yescharacteristic
480C8—22no—
481C8—41no—
482C8—41no—
483C8C2×C4×D411yescharacteristic
484C8#51811yescharacteristic
485C8#54611yescharacteristic
486C8#55111yescharacteristic
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22C4×SD3214yescomplemented
501C22C2×C4×D811yescharacteristic
502C22D4×D811yescharacteristic
503C22#329411yescharacteristic
504C22C22×SD3211yescharacteristic
505C22C2×#50811yescharacteristic
506C22#341611yescharacteristic
507C22#341711yescharacteristic
508C4—22no—
509C4—22no—
510C4—22no—
511C4—41no—
512C4—41no—
513C4—44no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4—161no—
518C4—161no—
519C4—161no—
520C4#235811yescharacteristic
521C4#294611yescharacteristic
522C4#328211yescharacteristic
523C4#329711yescharacteristic
524C2—24no—
525C2—162no—
526C2#2287011yescharacteristic
527C2C2×C4×SD3211yescharacteristic
528C2#3345511yescharacteristic
529C2D4×SD3211yescharacteristic
530C2#3385111yescharacteristic
531C2#3386511yescharacteristic
532C2#3386811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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. C22×#358C2semidirect
5. C22⋊C4×Q16C2semidirect
6. #13237C2semidirect
7. #13148C2semidirect
8. #13132C2semidirect
10. #13014C2semidirect
11. #13013C2semidirect
12. #12938C2semidirect
13. #12920C2semidirect
14. C4×#358C2semidirect
15. C16×C22⋊C4C2semidirect
27. #2185C4semidirect
69. C22×Q16C2×C4semidirect
84. C2×SD32C2×C4semidirect
85. C2×SD32C2×C4semidirect
138. #358C2×C4semidirect
139. #358C2×C4semidirect
140. #358D4semidirect
143. C22⋊C16C2×C4semidirect
346. Q16C4×D4semidirect
390. C22⋊C4SD32semidirect
500. C22C4×SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#21852
C2×C4×SD322
C22×#3582
#230252
C22⋊C4×Q162
#132372
#131482
#131322
#130592
#130142
#130132
#129382
#129202
C4×#3582
C16×C22⋊C42