Finite Group Databasebeta

← #188567

Subgroups — #188567

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34242C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic, complemented
3#20427C211yesmaximal, characteristic, complemented
4C2×#1529C211yesmaximal, characteristic
5C2×#1524C211yesmaximal, characteristic
6#8583C212yesmaximal, complemented
7#8582C212yesmaximal, complemented
8#8581C212yesmaximal, complemented
9#8580C212yesmaximal, complemented
10#8034C211yesmaximal, characteristic, complemented
11#8028C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#424—21no—
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×#97C2211yescharacteristic
18#2373—22no—
19#2363C412yes—
20#2362C412yes—
21#2355C411yescharacteristic
22#2354—21no—
23#2354C411yescharacteristic
24#2352—21no—
25Q8×C4⋊C4C411yescharacteristic
26D4×C4⋊C4C411yescharacteristic
27#2335—22no—
28#2314—21no—
29C4×#125C2212yes—
30C4×#124C2212yes—
31#2285C2211yescharacteristic
32#2274—21no—
33C4×#117C2211yescharacteristic
34#1939—21no—
35#1939—21no—
36#1851—22no—
37#1768—22no—
38#1767C2212yes—
39#1767C2212yes—
40#1767C2212yes—
41#1767C2212yes—
42#1729—21no—
43#1529—24no—
44#1524—24no—
45C2×#330C2211yescharacteristic
46C2×#330C2212yes—
47C2×#330C2211yescharacteristic
48C2×#329C2211yescharacteristic
49C2×#329C2212yes—
50C2×#329C2211yescharacteristic
51#1295C2212yes—
52#1295C2212yes—
53#1295C2212yes—
54#1295C2212yes—
55C22×M16—21no—
56C22×M16—21no—
57C23×C8—21no—
58#529—22no—
59#529—22no—
60#529D414yes—
61#529Q814yes—
62C2×#126—21no—
63C2×#125C2311yescharacteristic
64C2×#124C2311yescharacteristic
65C2×#123—21no—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C412yes—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117C2×C412yes—
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C42C2311yescharacteristic
80#432—22no—
81#432—22no—
82#424—28no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C22⋊C8—21no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16—22no—
92C4×M16—22no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—21no—
96C2×#97C2311yescharacteristic
97C2×#97C2311yescharacteristic
98C2×#97C2311yescharacteristic
99C2×#97C2312yes—
100C2×#97C2311yescharacteristic
101C2×#97C2312yes—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#391C2×C412yes—
105#390—22no—
106#390C2×C412yes—
107#389—22no—
108#389C2×C412yes—
109#389C2×C412yes—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#386—22no—
121#385—22no—
122#384—21no—
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#382—21no—
126#382C2312yes—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4C2×C412yes—
139C4×C4⋊C4C2×C412yes—
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D411yescharacteristic
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4C2×C412yes—
148C4×C22⋊C4D412yes—
149#376D412yes—
150C43C2312yes—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—22no—
155#330—22no—
156#330—22no—
157#330—24no—
158#329—22no—
159#329—22no—
160#329—24no—
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—22no—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16—42no—
175C22×C8—24no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—42no—
183C22×C8—42no—
184#126—24no—
185#126—24no—
186#125—21no—
187#125—21no—
188#125C2×D412yes—
189#125C2×Q812yes—
190#124—21no—
191#124—21no—
192#124C2×D412yes—
193#124C2×Q812yes—
194#123—24no—
195#123—24no—
196C22⋊Q8—22no—
197C22⋊Q8C4⋊C414yes—
198#121—22no—
199#121C4⋊C414yes—
200C4×Q8—22no—
201C4×Q8—82no—
202C4×Q8C4⋊C414yes—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—82no—
207C4×D4C4⋊C414yes—
208#117—24no—
209#117—24no—
210#117—24no—
211#117—42no—
212#117—44no—
213#117—82no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4C22⋊C412yes—
244C2×C22⋊C4C22⋊C412yes—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42C22×C412yes—
251C2×C42C22×C412yes—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264C4⋊C8—28no—
265C4⋊C8—28no—
266C4⋊C8—44no—
267C4⋊C8—44no—
268C4⋊C8—42no—
269C22⋊C8—42no—
270#97—21no—
271#97—21no—
272#97—21no—
273#97—21no—
274#97—22no—
275#97—22no—
276#97—21no—
277#97—21no—
278#97—22no—
279#97—21no—
280#97—21no—
281#97—22no—
282#97—42no—
283C4×C8—42no—
284C4×C8—42no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95C22⋊C412yes—
294#95C22⋊C412yes—
295#95C22⋊C412yes—
296#95C22⋊C412yes—
297C24—21no—
298C24C2×C22⋊C411yescharacteristic
299C2×Q8—21no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8C2×C4⋊C412yes—
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4C2×C4⋊C412yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C4⋊C412yes—
335C22×C4#11712yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C22⋊Q812yes—
339C22×C4C22⋊Q812yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343M16—48no—
344M16—48no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—48no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4#10414yes—
385C4⋊C4C2×C4⋊C412yes—
386C4⋊C4C2×C4⋊C412yes—
387C22⋊C4—24no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—44no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—24no—
401C42—24no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—44no—
408C42—82no—
409C42—82no—
410C42—82no—
411C42C2×C4⋊C412yes—
412C42#11712yes—
413C42#12112yes—
414C42#12112yes—
415C42C22⋊Q812yes—
416C42C22⋊Q812yes—
417C42C22×D411yescharacteristic
418C42C22×Q811yescharacteristic
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23#38012yes—
435C23C22×C22⋊C411yescharacteristic
436Q8—42no—
437Q8—42no—
438D4—42no—
439D4—42no—
440D4—84no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—164no—
485C2×C4#38012yes—
486C2×C4#38012yes—
487C2×C4#38012yes—
488C2×C4C2×#10414yes—
489C2×C4C2×#10412yes—
490C2×C4#42112yes—
491C2×C4C22×C4⋊C411yescharacteristic
492C2×C4C2×#11711yescharacteristic
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×C22⋊Q811yescharacteristic
496C2×C4C2×C22⋊Q811yescharacteristic
497C8—42no—
498C8—42no—
499C8—44no—
500C8—42no—
501C8—42no—
502C8—82no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22C2×#38011yescharacteristic
518C22#288811yescharacteristic
519C22#288911yescharacteristic
520C22C22×#10411yescharacteristic
521C22C2×#42111yescharacteristic
522C22#290111yescharacteristic
523C22#290111yescharacteristic
524C4—21no—
525C4—21no—
526C4—22no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—24no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—82no—
540C4—82no—
541C4#180614yes—
542C2—42no—
543C2—82no—
544C2#1765311yescharacteristic
545C2#1765511yescharacteristic
546C2#1768111yescharacteristic
547C2#1768311yescharacteristic
548C2#2064011yescharacteristic
549C2C2×#180611yescharacteristic
550C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21356C2semidirect
3. #20427C2semidirect
6. #8583C2semidirect
7. #8582C2semidirect
8. #8581C2semidirect
9. #8580C2semidirect
10. #8034C2semidirect
11. #8028C2semidirect

Maximal subgroups

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

subgroupindex
#342422
#213562
#204272
C2×#15292
C2×#15242
#85832
#85822
#85812
#85802
#80342
#80282