Finite Group Databasebeta

← #198946

Subgroups — #198946

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3#21323C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5C2×#1518C211yesmaximal, characteristic
6#9290C214yesmaximal, complemented
7#9288C214yesmaximal, complemented
8C2×#1286C211yesmaximal, characteristic
9#7860C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—22no—
13#3230C412yescomplemented
14#3230C2212yes—
15#3209C412yescomplemented
16#3209C2212yes—
17#3189—22no—
18#3170C414yescomplemented
19#3104—22no—
20#3104—22no—
21Q8×M16—22no—
22Q8×M16—22no—
23#2962—22no—
24#2962—22no—
25#2961C2214yes—
26#2960—22no—
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2905—21no—
30C2×#423C2211yescharacteristic
31#2873—21no—
32C2×C4×M16C2211yescharacteristic
33C2×#382C2211yescharacteristic
34#1993C2212yes—
35#1993C2212yes—
36#1991—22no—
37#1736—21no—
38#1736—21no—
39#1550—22no—
40#1549C2212yes—
41#1548C2212yes—
42#1542—22no—
43#1518—22no—
44#1518C2214yes—
45C2×#328—22no—
46#1286C2214yes—
47#1275C2214yes—
48C22×M16—21no—
49C22×M16D411yescharacteristic
50C22×M16D411yescharacteristic
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533C2×C412yes—
62#533C2×C412yes—
63#533C2×C412yes—
64#533D412yes—
65#533D412yes—
66#533C2312yes—
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D412yes—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8D412yes—
81C2×#117—21no—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C42C2311yescharacteristic
85#500C2×C414yescomplemented
86#494C2×C414yescomplemented
87#487C2×C414yescomplemented
88#487C2×C414yescomplemented
89#484—24no—
90#484—24no—
91#477—24no—
92#477—24no—
93#467—28no—
94#462—24no—
95#462—24no—
96#451—24no—
97#451—24no—
98#446—24no—
99#446—24no—
100#446—28no—
101#446—28no—
102C8×Q8—24no—
103C8×Q8—24no—
104C4×Q16—28no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2×C412yes—
111#431C2312yes—
112#431C2312yes—
113#424—24no—
114#424—24no—
115#423—24no—
116#423—21no—
117#423—21no—
118#423D412yes—
119#423C2312yes—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8D412yes—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126#405—24no—
127#405—24no—
128C4×M16—24no—
129C4×M16—21no—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16—21no—
133C2×#97C2312yes—
134C2×C4×C8—21no—
135C2×C4×C8C2×C412yes—
136C2×C4×C8D412yes—
137#382—21no—
138#382—21no—
139#382—21no—
140#382C2312yes—
141C2×#95—21no—
142C2×#95—21no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#333—24no—
148#332—24no—
149#328—28no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8C22⋊C412yes—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C2×Q16—28no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—42no—
169C2×M16—42no—
170C2×M16Q1614yescomplemented
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—41no—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22⋊C412yes—
187C4⋊Q8C22⋊C412yes—
188C4⋊Q8C22×C412yes—
189C4⋊Q8C22×C412yes—
190C4⋊Q8C22×C412yes—
191C4⋊Q8C22×C412yes—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—22no—
196#125—22no—
197#125—22no—
198#125—41no—
199#125—41no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8C22⋊C414yes—
215C22⋊Q8C2×D414yes—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8C22⋊C414yes—
231C4×Q8C2×D414yes—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—41no—
239#117—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4C22⋊C412yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—41no—
258C2×C42—22no—
259C2×C42—41no—
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42C2411yescharacteristic
272#107—28no—
273C4⋊C8—28no—
274C4⋊C8—28no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—41no—
282C4⋊C8C2×D414yes—
283#103—28no—
284#103—28no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—41no—
289#97—28no—
290#97—22no—
291#97—22no—
292C4×C8—28no—
293C4×C8—22no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—41no—
298C4×C8C22×C414yes—
299#95—22no—
300#95—22no—
301#95—41no—
302#95—41no—
303C24#12111yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8C2×C22⋊C414yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12311yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4C2×Q1612yes—
342C22×C4#13512yes—
343C22×C4#13712yes—
344C22×C4#13712yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348Q16—88no—
349M16—24no—
350M16—24no—
351M16—42no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16—44no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—22no—
360C2×C8—24no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—44no—
364C2×C8—42no—
365C2×C8—82no—
366C2×C8—82no—
367C2×C8C2×Q1614yes—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—82no—
394C4⋊C4C2×C22⋊C414yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C42—24no—
400C42—24no—
401C42—42no—
402C42—42no—
403C42C2×C22⋊C411yescharacteristic
404C42C2×C22⋊C411yescharacteristic
405C42C4×D411yescharacteristic
406C42C4×D411yescharacteristic
407C42C4×D411yescharacteristic
408C42C4×D411yescharacteristic
409C42#12011yescharacteristic
410C42#12011yescharacteristic
411C42#12111yescharacteristic
412C42#12111yescharacteristic
413C42#12311yescharacteristic
414C42#12311yescharacteristic
415C42C23×C411yescharacteristic
416C42C22×D411yescharacteristic
417C42C22×D411yescharacteristic
418C42C2×#4111yescharacteristic
419C23—41no—
420C23—41no—
421C23#38611yescharacteristic
422C23#38611yescharacteristic
423C23#46212yes—
424C23#46412yes—
425C23C2×#12111yescharacteristic
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—82no—
431Q8—82no—
432Q8—82no—
433Q8—82no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—164no—
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4#38611yescharacteristic
462C2×C4#38611yescharacteristic
463C2×C4#38611yescharacteristic
464C2×C4C4×Q1612yes—
465C2×C4C4×Q1612yes—
466C2×C4#44112yes—
467C2×C4#44112yes—
468C2×C4#45112yes—
469C2×C4#45112yes—
470C2×C4#45212yes—
471C2×C4#45212yes—
472C2×C4#46212yes—
473C2×C4#46412yes—
474C2×C4C22×C22⋊C411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×#12011yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4C22×Q1611yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×#13711yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C8—24no—
485C8—44no—
486C8—42no—
487C8—44no—
488C8—82no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—42no—
496C22—42no—
497C22#187812yes—
498C22#187812yes—
499C22C2×#38611yescharacteristic
500C22C2×#46211yescharacteristic
501C22C2×#46411yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—44no—
511C4—44no—
512C4—82no—
513C4—82no—
514C4—162no—
515C4#187812yes—
516C4#187812yes—
517C4C2×C4×Q1611yescharacteristic
518C4C2×#44111yescharacteristic
519C4C2×#45111yescharacteristic
520C4C2×#45211yescharacteristic
521C2—21no—
522C2—22no—
523C2—22no—
524C2—21no—
525C2C2×#187811yescharacteristic
526C2#2099511yescharacteristic
527C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9290C2semidirect
7. #9288C2semidirect
13. #3230C4semidirect
15. #3209C4semidirect
18. #3170C4semidirect
85. #500C2×C4semidirect
86. #494C2×C4semidirect
87. #487C2×C4semidirect
88. #487C2×C4semidirect
170. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
#601922
#213232
#177082
C2×#15182
#92902
#92882
C2×#12862
#78602