Finite Group Databasebeta

← #205504

Subgroups — #205504

Order 512.

997 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60693C211yesmaximal, characteristic
2C4⋊C4×M16C211yesmaximal, characteristic, complemented
3#17795C211yesmaximal, characteristic, complemented
4#17546C212yesmaximal, complemented
5#17498C212yesmaximal, complemented
6#9754C212yesmaximal, complemented
7#9753C211yesmaximal, characteristic, complemented
8#9752C211yesmaximal, characteristic, complemented
9#9751C212yesmaximal, complemented
10#9750C211yesmaximal, characteristic, complemented
11#9749C211yesmaximal, characteristic, complemented
12#3444C2212yes—
13#3185C412yescomplemented
14#3184C412yescomplemented
15#3183—22no—
16#3183—22no—
17#3182C412yescomplemented
18#3182C412yescomplemented
19#3160C2211yescharacteristic
20#3147C2211yescharacteristic
21#3131C2212yes—
22#3054—21no—
23#3054—21no—
24#2943—21no—
25#2943—21no—
26#2920C2212yes—
27C2×#423C2211yescharacteristic
28C2×C4×M16—21no—
29C2×C4×M16C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1771—21no—
32#1771—21no—
33#1770C2211yescharacteristic
34#1770C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37#1728—21no—
38#1569—22no—
39#1569—22no—
40#1564C2211yescharacteristic
41#1564C2211yescharacteristic
42#1564C2212yes—
43#1509—24no—
44#1504—24no—
45#1503C2212yes—
46#1503C2212yes—
47#1502C2212yes—
48#1502C2212yes—
49#1495—24no—
50#1489—24no—
51#1481C2212yes—
52#1481C2212yes—
53#1480C2212yes—
54#1480C2212yes—
55C2×#135—21no—
56C2×#135—21no—
57C22×M16—21no—
58C22×M16D411yescharacteristic
59C22×M16D411yescharacteristic
60#533C2×C412yes—
61#533C2×C412yes—
62#533C2312yes—
63#532C2×C412yes—
64#532C2×C412yes—
65#532C2312yes—
66#529—22no—
67#529—22no—
68#529—22no—
69#529—22no—
70#529—22no—
71#529—22no—
72#525—22no—
73#525—22no—
74#525—22no—
75C4×#41—22no—
76C4×#41—22no—
77C4×#41—22no—
78C22×C4⋊C4—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#500C2×C411yescharacteristic, complemented
82#500C2×C411yescharacteristic, complemented
83#499—21no—
84#499—21no—
85#498C2×C412yescomplemented
86#495—21no—
87#495—21no—
88#494C2×C411yescharacteristic, complemented
89#493C2×C411yescharacteristic, complemented
90#492C2×C412yescomplemented
91#488—22no—
92#488—22no—
93#487C2×C412yescomplemented
94#487C2×C412yescomplemented
95#486C2×C412yescomplemented
96#486C2×C412yescomplemented
97#470—22no—
98#470—22no—
99#469—22no—
100#469—22no—
101#468—22no—
102#468—22no—
103#468—22no—
104#468—22no—
105#442—22no—
106#442—22no—
107#441—22no—
108#441—22no—
109#440—22no—
110#440—22no—
111#440—22no—
112#440—22no—
113#432—22no—
114#432—22no—
115#431—22no—
116#431C2312yes—
117#431C2312yes—
118#427—21no—
119#427—21no—
120#423—22no—
121#423—22no—
122#423D414yes—
123C2×C4⋊C8—21no—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126#416—22no—
127#416—22no—
128#407—22no—
129C4×M16—22no—
130C4×M16—44no—
131C4×M16C2×C412yes—
132C4×M16C2×C412yes—
133C4×M16D412yes—
134C4×M16D412yes—
135C2×#97—21no—
136C2×#97—21no—
137C2×#97D411yescharacteristic
138C2×#97D411yescharacteristic
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4D412yes—
146C4×C4⋊C4C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C2×#95D412yes—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#335—22no—
154#335—21no—
155#335—21no—
156#334—22no—
157#334—21no—
158#334—21no—
159#328—24no—
160#328—24no—
161#327—24no—
162#327—24no—
163Q8⋊C8—24no—
164Q8⋊C8—24no—
165D4⋊C8—24no—
166D4⋊C8—24no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C23×C4—22no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174#135—42no—
175#135—42no—
176#135—42no—
177#135—42no—
178C2×Q16—21no—
179C2×Q16—21no—
180C2×SD16—22no—
181C2×SD16—22no—
182C2×D8—21no—
183C2×D8—21no—
184C2×M16—24no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—41no—
188C2×M16—41no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16C22⋊C412yes—
193C2×M16C22⋊C412yes—
194C2×M16C22⋊C412yes—
195C2×M16C22⋊C412yes—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C4⋊Q8C22×C412yes—
203C4⋊Q8C22×C412yes—
204C4⋊Q8C22×C412yes—
205#127C22×C412yes—
206#126—42no—
207#126—42no—
208#125—22no—
209#125—22no—
210#125—22no—
211#124—22no—
212#124—22no—
213#124—22no—
214#123—42no—
215#123—42no—
216#123—42no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223#121—22no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—22no—
228#121—22no—
229#120—42no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—42no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117—42no—
252#117—42no—
253#117—42no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—44no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42C22×C412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42#4111yescharacteristic
270C2×C42#4112yes—
271C2×C42C2411yescharacteristic
272#107—21no—
273#107—21no—
274#107—21no—
275#107—21no—
276#106—22no—
277#106—22no—
278C4⋊C8—24no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—42no—
284C4⋊C8C2×D412yes—
285C4⋊C8C2×D412yes—
286#103—22no—
287#103—22no—
288#103—22no—
289#103—22no—
290#102—22no—
291#102—22no—
292#102—22no—
293#102—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—42no—
297#97—42no—
298#97—42no—
299#97—42no—
300#97C22⋊C412yes—
301#97C22⋊C412yes—
302#97C22⋊C412yes—
303#97C22⋊C412yes—
304C4×C8—22no—
305C4×C8—22no—
306C4×C8—42no—
307C4×C8—42no—
308C4×C8C22×C412yes—
309C4×C8C22×C411yescharacteristic
310C4×C8C22×C411yescharacteristic
311C4×C8C2×D412yes—
312C4×C8C2×D411yescharacteristic
313C4×C8C2×D411yescharacteristic
314#95—22no—
315#95C2×D412yes—
316#95C2×D412yes—
317C24#12111yescharacteristic
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—84no—
323#41—84no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4—82no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4#12112yes—
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12412yes—
355C22×C4#12712yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359Q16—42no—
360Q16—42no—
361SD16—44no—
362SD16—44no—
363D8—42no—
364D8—42no—
365M16—24no—
366M16—24no—
367M16—42no—
368M16—42no—
369M16—48no—
370M16—84no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8C2×C22⋊C412yes—
386C2×C8C2×C22⋊C412yes—
387C2×C8C2×C22⋊C411yescharacteristic
388C2×C8C2×C22⋊C411yescharacteristic
389C2×C8C2×C22⋊C411yescharacteristic
390C2×C8C2×C22⋊C411yescharacteristic
391C2×C8AGL(1,ℤ/8)12yes—
392C2×C8#13712yes—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C42—44no—
418C42—42no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—42no—
424C42—42no—
425C42C2×C22⋊C412yes—
426C42C4×D412yes—
427C42C4×D412yes—
428C42#12112yes—
429C42#12412yes—
430C42#12712yes—
431C42C23×C411yescharacteristic
432C42C22×D411yescharacteristic
433C42C22×D411yescharacteristic
434C42C2×#4111yescharacteristic
435C23—21no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23#39012yes—
441C23C2×#12111yescharacteristic
442Q8—42no—
443Q8—42no—
444Q8—44no—
445Q8—82no—
446D4—42no—
447D4—42no—
448D4—44no—
449D4—82no—
450D4—84no—
451D4—84no—
452D4—82no—
453D4—82no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—88no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#39012yes—
483C2×C4#39012yes—
484C2×C4#39012yes—
485C2×C4C22×C22⋊C411yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×#12411yescharacteristic
490C2×C4C2×#12711yescharacteristic
491C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
492C2×C4C2×#13711yescharacteristic
493C2×C4#57511yescharacteristic
494C2×C4#57511yescharacteristic
495C8—22no—
496C8—22no—
497C8—21no—
498C8—21no—
499C8—42no—
500C8—42no—
501C8—82no—
502C8—82no—
503C8#41712yes—
504C8#41712yes—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—42no—
511C22—42no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22C2×#39011yescharacteristic
516C22#303611yescharacteristic
517C22#303711yescharacteristic
518C4—22no—
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—21no—
524C4—44no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4C2×#41711yescharacteristic
530C4#289611yescharacteristic
531C4#295612yes—
532C4#303611yescharacteristic
533C4#303711yescharacteristic
534C2—21no—
535C2—21no—
536C2—42no—
537C2—162no—
538C2#2124111yescharacteristic
539C2#2124411yescharacteristic
540C2#2125811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60693C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×M16C2semidirect
3. #17795C2semidirect
4. #17546C2semidirect
5. #17498C2semidirect
6. #9754C2semidirect
7. #9753C2semidirect
8. #9752C2semidirect
9. #9751C2semidirect
10. #9750C2semidirect
11. #9749C2semidirect
13. #3185C4semidirect
14. #3184C4semidirect
17. #3182C4semidirect
18. #3182C4semidirect
81. #500C2×C4semidirect
82. #500C2×C4semidirect
85. #498C2×C4semidirect
88. #494C2×C4semidirect
89. #493C2×C4semidirect
90. #492C2×C4semidirect
93. #487C2×C4semidirect
94. #487C2×C4semidirect
95. #486C2×C4semidirect
96. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#606932
C4⋊C4×M162
#177952
#175462
#174982
#97542
#97532
#97522
#97512
#97502
#97492