Finite Group Databasebeta

← #200902

Subgroups — #200902

Order 512.

1120 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60864C211yesmaximal, characteristic
2#34388C211yesmaximal, characteristic
3#17731C211yesmaximal, characteristic
4#17584C211yesmaximal, characteristic, complemented
5C2×#1515C211yesmaximal, characteristic, complemented
6#9427C214yesmaximal, complemented
7#9425C214yesmaximal, complemented
8#7978C211yesmaximal, characteristic, complemented
9#7975C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3235—22no—
13#3234—22no—
14#3228C414yescomplemented
15#3227C412yescomplemented
16#3226C412yescomplemented
17#3165C2212yes—
18#3138C2212yes—
19#3094—22no—
20#3077—22no—
21#2920C2211yescharacteristic
22#2920C2211yescharacteristic
23C2×#432C2211yescharacteristic
24#2381C2212yes—
25#2380C2212yes—
26#2364—22no—
27#2353—22no—
28#2339—22no—
29#2337—22no—
30#2325—22no—
31#2318—22no—
32#2283C2211yescharacteristic
33C2×#383C2211yescharacteristic
34#1557—22no—
35#1556C2212yes—
36#1555C2212yes—
37#1540—22no—
38#1518—22no—
39#1515—22no—
40#1515C2214yes—
41#1515C2214yes—
42#1508—22no—
43C2×#328—22no—
44#1288C2214yes—
45#1288C2214yes—
46C22×#41—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yescomplemented
54#533C2312yes—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532C2×C412yes—
60#532C2×C412yes—
61#532D412yes—
62#532D412yes—
63C2×C4⋊Q8C2×C411yescharacteristic, complemented
64C2×C4⋊Q8C2×C411yescharacteristic, complemented
65C2×C4⋊Q8C2311yescharacteristic
66C2×#127C2311yescharacteristic
67C2×#125—21no—
68C2×#124—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×#121—21no—
72C2×#121D412yes—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4—24no—
76C2×C4×D4—21no—
77C2×C4×D4D412yes—
78C2×#117—21no—
79C22×C42C2311yescharacteristic
80#501—24no—
81#500C2×C414yescomplemented
82#498C2×C414yescomplemented
83#490—24no—
84#486C2×C414yescomplemented
85#486C2×C414yescomplemented
86#482—28no—
87#481—24no—
88#480—24no—
89#478—28no—
90#476—24no—
91#474—24no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432C2312yes—
99#432C2312yes—
100#431—21no—
101#431—21no—
102#431C2×C412yescomplemented
103#431C2×C412yescomplemented
104#428—24no—
105#423—21no—
106#423—21no—
107C2×C4⋊C8—21no—
108#418—24no—
109#407—21no—
110#407—21no—
111C2×C22⋊C8—21no—
112C4×M16—21no—
113C4×M16C2312yes—
114C2×#97—21no—
115#391—24no—
116#388—24no—
117#388—24no—
118#387—24no—
119#387—24no—
120#386—24no—
121#386—24no—
122#386—24no—
123#386—24no—
124#385—24no—
125#383—21no—
126#383—21no—
127#383D412yes—
128#383C2312yes—
129#383C2312yes—
130#382D412yes—
131#381—22no—
132#381—21no—
133#380—22no—
134#380—21no—
135C4×C4⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4D412yes—
139#376—21no—
140C2×#95—22no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95D412yes—
144#332—24no—
145#331—24no—
146#328—28no—
147#328—28no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—84no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C2×D412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—22no—
167C2×M16—41no—
168C22×C8—41no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8C22×C412yes—
178C4⋊Q8C22×C412yes—
179C4⋊Q8C22×C412yes—
180C4⋊Q8C2×D412yes—
181#127—21no—
182#127—21no—
183#127C22×C412yes—
184#127C2×D412yes—
185#125—22no—
186#125—22no—
187#125—41no—
188#125—41no—
189#124—22no—
190#124—22no—
191#124—22no—
192#124—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207#121C22⋊C414yes—
208#121C22⋊C414yes—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—88no—
224C4×D4C22⋊C414yes—
225C4×D4C22⋊C414yes—
226#117—22no—
227#117—22no—
228#117—41no—
229#117—41no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C2×D412yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—41no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—41no—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42C2411yescharacteristic
262#107—28no—
263#106—28no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270#102—28no—
271#102—28no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—41no—
277C22⋊C8—41no—
278#97—22no—
279#97—22no—
280#97—22no—
281#97—41no—
282C4×C8—41no—
283C4×C8C22×C414yes—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95C2×D414yes—
295#95C2×D414yes—
296C24—22no—
297C24#12011yescharacteristic
298#41—84no—
299#41—84no—
300#41—84no—
301#41—84no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—48no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4C2×C22⋊C414yes—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—44no—
334C22×C4—81no—
335C22×C4—84no—
336C22×C4—82no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—82no—
340C22×C4—84no—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359M16—44no—
360M16—82no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—82no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—82no—
381C4⋊C4—84no—
382C4⋊C4C2×C22⋊C414yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—44no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C42—42no—
394C42—42no—
395C42—82no—
396C42—84no—
397C42C2×C22⋊C411yescharacteristic
398C42C2×C22⋊C411yescharacteristic
399C42C4×D411yescharacteristic
400C42C4×D411yescharacteristic
401C42C4×D411yescharacteristic
402C42C4×D411yescharacteristic
403C42#12111yescharacteristic
404C42#12111yescharacteristic
405C42#12111yescharacteristic
406C42#12111yescharacteristic
407C42#12311yescharacteristic
408C42#12311yescharacteristic
409C42C23×C411yescharacteristic
410C42C22×D411yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C23—22no—
414C23—22no—
415C23—24no—
416C23—41no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23#38611yescharacteristic
422C23#38611yescharacteristic
423C23C2×#12011yescharacteristic
424Q8—82no—
425Q8—82no—
426Q8—82no—
427Q8—82no—
428Q8—82no—
429Q8—82no—
430D4—44no—
431D4—48no—
432D4—48no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437D4—82no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—164no—
471C2×C4—168no—
472C2×C4#38611yescharacteristic
473C2×C4#38611yescharacteristic
474C2×C4#38611yescharacteristic
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4C22×C22⋊C411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×#12311yescharacteristic
484C8—44no—
485C8—82no—
486C8—82no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—84no—
499C22—84no—
500C22#190412yes—
501C22C2×#38611yescharacteristic
502C22#302212yes—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—44no—
516C4—84no—
517C4—84no—
518C4—82no—
519C4—82no—
520C2—22no—
521C2—22no—
522C2—41no—
523C2—84no—
524C2#2095511yescharacteristic
525C2C2×#190411yescharacteristic
526C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
4. #17584C2semidirect
5. C2×#1515C2semidirect
6. #9427C2semidirect
7. #9425C2semidirect
8. #7978C2semidirect
9. #7975C2semidirect
14. #3228C4semidirect
15. #3227C4semidirect
16. #3226C4semidirect
53. #533C2×C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
64. C2×C4⋊Q8C2×C4semidirect
81. #500C2×C4semidirect
82. #498C2×C4semidirect
84. #486C2×C4semidirect
85. #486C2×C4semidirect
102. #431C2×C4semidirect
103. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
#608642
#343882
#177312
#175842
C2×#15152
#94272
#94252
#79782
#79752