Finite Group Databasebeta

← #188526

Subgroups — #188526

Order 512.

1338 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
1#34206C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3C2×#1517C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic, complemented
6#8577C214yesmaximal, complemented
7#8571C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2367—22no—
15#2366C412yescomplemented
16#2365C412yescomplemented
17#2354C412yescomplemented
18#2352—22no—
19D4×C4⋊C4C412yescomplemented
20#2325—22no—
21#2311—21no—
22C4×#121C2214yes—
23C2×#389—21no—
24C2×#389—21no—
25#2285C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—22no—
30#1939—22no—
31#1797C2214yes—
32#1517—24no—
33#1502—24no—
34C2×#327C2212yes—
35C2×#327C2214yes—
36C2×#327C2212yes—
37#1444—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2214yes—
43C22×#41—21no—
44#532—22no—
45#532—22no—
46#532D414yes—
47#532Q814yes—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51C2×#121—21no—
52C2×#121C2312yes—
53#520—44no—
54#520—44no—
55C2×C4×Q8—22no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C414yes—
61C2×#117—21no—
62C2×#117—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75#431—22no—
76#431—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—24no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#391—22no—
85#390—22no—
86#390C2×C414yes—
87#389—44no—
88#389—44no—
89#389C2×C414yes—
90#388—24no—
91#388—24no—
92#387—24no—
93#386—24no—
94#386—24no—
95#385—24no—
96#384C2×C411yescharacteristic
97#384C2×C411yescharacteristic
98#382—21no—
99#382—22no—
100#382—22no—
101#380—21no—
102#380—22no—
103#380—22no—
104#380C2312yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C414yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C22⋊C4—24no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95D412yes—
119C2×#95C2311yescharacteristic
120C2×#95C2311yescharacteristic
121C2×#95C2312yes—
122#336—24no—
123#327—24no—
124#327—24no—
125#327—28no—
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22×C412yes—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—22no—
150C4⋊Q8C4⋊C414yescomplemented
151#127—22no—
152#127C4⋊C414yescomplemented
153#124—24no—
154#124—24no—
155#121—24no—
156#121—24no—
157#121—22no—
158#121—22no—
159#121C2×D414yes—
160#121C2×Q814yes—
161C4×Q8—48no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—48no—
166C4×D4—48no—
167C4×D4—48no—
168C4×D4—84no—
169C4×D4C4⋊C418yescomplemented
170#117—42no—
171#117—42no—
172#117—48no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4D814yes—
199C2×C4⋊C4SD1614yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—82no—
215C2×C42C22×C414yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D414yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×Q812yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—84no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C4×C8—42no—
233C4×C8—42no—
234#95—24no—
235#95—22no—
236#95—24no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—44no—
245#95—42no—
246#95—44no—
247#95C22⋊C414yes—
248#95C22⋊C414yes—
249C24—22no—
250C24C22×D411yescharacteristic
251#41—48no—
252#41—48no—
253#41—48no—
254#41—48no—
255C2×Q8—24no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4C2×C4⋊C414yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C414yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4C2×D814yes—
325C22×C4C2×D812yes—
326C22×C4C2×SD1614yes—
327C22×C4C2×SD1612yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4#13712yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—48no—
349C4⋊C4—42no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C4⋊C4—84no—
357C4⋊C4#10218yes—
358C4⋊C4#10218yes—
359C4⋊C4C2×C4⋊C414yes—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—28no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—48no—
366C22⋊C4—84no—
367C22⋊C4—84no—
368C42—44no—
369C42—44no—
370C42—48no—
371C42—44no—
372C42—84no—
373C42C2×C4⋊C412yes—
374C42C2×C4⋊C412yes—
375C42#11712yes—
376C42#11712yes—
377C42#12112yes—
378C42#12112yes—
379C42C22⋊Q812yes—
380C42C22⋊Q812yes—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—24no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23C22×C22⋊C411yescharacteristic
393C23C2×#12111yescharacteristic
394C23C2×C22⋊Q811yescharacteristic
395C23C22×D811yescharacteristic
396C23C22×SD1611yescharacteristic
397C23C2×AGL(1,ℤ/8)11yescharacteristic
398C23C2×#13711yescharacteristic
399Q8—48no—
400D4—44no—
401D4—44no—
402D4—44no—
403D4—44no—
404D4—44no—
405D4—44no—
406D4—48no—
407D4—84no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—416no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
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—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—168no—
463C2×C4#38012yes—
464C2×C4#38012yes—
465C2×C4#38012yes—
466C2×C4#38012yes—
467C2×C4C2×#10218yes—
468C2×C4C2×#10212yes—
469C2×C4C2×#10212yes—
470C2×C4#41712yes—
471C2×C4#41712yes—
472C2×C4#45914yes—
473C2×C4#46114yes—
474C2×C4#47414yes—
475C2×C4#47614yes—
476C2×C4C22×C4⋊C411yescharacteristic
477C2×C4C2×#11711yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×C22⋊Q811yescharacteristic
480C8—82no—
481C8—82no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22C2×#38011yescharacteristic
502C22#288311yescharacteristic
503C22#288411yescharacteristic
504C22C22×#10211yescharacteristic
505C22C2×#41711yescharacteristic
506C22#290111yescharacteristic
507C22#290111yescharacteristic
508C22C2×#45911yescharacteristic
509C22C2×#46111yescharacteristic
510C22C2×#47411yescharacteristic
511C22C2×#47611yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—22no—
518C4—28no—
519C4—44no—
520C4—44no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—162no—
526C4#180214yes—
527C4#180214yes—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—84no—
533C2#1761311yescharacteristic
534C2#1761811yescharacteristic
535C2#1762811yescharacteristic
536C2#1763011yescharacteristic
537C2#2063411yescharacteristic
538C2C2×#180211yescharacteristic
539C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
5. #13954C2semidirect
6. #8577C2semidirect
7. #8571C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
15. #2366C4semidirect
16. #2365C4semidirect
17. #2354C4semidirect
19. D4×C4⋊C4C4semidirect
150. C4⋊Q8C4⋊C4semidirect
152. #127C4⋊C4semidirect
169. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219512
C2×#15172
C2×#15022
#139542
#85772
#85712
#80262
#80142