Finite Group Databasebeta

← #198124

Subgroups — #198124

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic, complemented
3#21844C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic
5#17384C211yesmaximal, characteristic
6#9242C218yesmaximal, complemented
7#8007C211yesmaximal, characteristic, complemented
8#7924C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3228—22no—
12#3228—22no—
13#3228—22no—
14#3228—22no—
15C2×#500C411yescharacteristic, complemented
16C2×#498C411yescharacteristic, complemented
17C2×#486C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#482C2214yes—
20C2×#478C414yescomplemented
21#2986—22no—
22#2985—22no—
23#2984—22no—
24#2983—21no—
25#2982—21no—
26#2962—22no—
27#2960—22no—
28#2924C2214yes—
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1994—22no—
35#1986—22no—
36#1826C2214yes—
37#1507C2214yes—
38#1505—22no—
39#1502—22no—
40#1471C2214yes—
41#1469—22no—
42#1460—21no—
43#1460—21no—
44#1293C2214yes—
45#1282C2214yes—
46C22×#41—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533D412yes—
52#533D412yes—
53#533D412yes—
54#533D412yes—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×#127C2×C411yescharacteristic
67C2×#125C2×C412yes—
68C2×#124—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×#121—21no—
72C2×#121C2312yes—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C412yes—
77C2×#117—21no—
78C2×#117C2312yes—
79C22×C42C2311yescharacteristic
80#500—22no—
81#500—22no—
82#498—22no—
83#498—22no—
84#486—22no—
85#486—22no—
86#486—22no—
87#486—22no—
88#482—28no—
89#482—28no—
90#478—28no—
91#478—28no—
92#446—24no—
93#446—24no—
94#446—24no—
95#446—28no—
96C8×Q8—24no—
97C8×Q8—24no—
98C8×Q8—24no—
99#433—28no—
100#432—21no—
101#432—24no—
102#432—21no—
103#432—21no—
104#432D412yes—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—24no—
113#431—21no—
114#431—21no—
115#431D411yescharacteristic
116#431D411yescharacteristic
117#431D412yes—
118#431D412yes—
119C2×#107C2×C414yes—
120C2×#106C2×C414yes—
121#424—24no—
122#423—21no—
123#423—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8C2×C412yes—
126C2×#102C2×C414yes—
127C2×#102C2×C414yes—
128#407C2312yes—
129#407C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132#405—24no—
133C4×M16—21no—
134C4×M16—21no—
135C2×C4×C8—21no—
136C2×C4×C8C2×C411yescharacteristic
137C2×C4×C8C2×C411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4C2312yes—
142#376—21no—
143C43—21no—
144#339—28no—
145#327—28no—
146#323—24no—
147#323—24no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—41no—
151C2×#41—41no—
152C2×#41—84no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4#4111yescharacteristic
162C23×C4#4111yescharacteristic
163C23×C4C2411yescharacteristic
164C2×M16—22no—
165C2×M16—22no—
166C22×C8—22no—
167C22×C8C22×C412yes—
168C22×C8C22×C412yes—
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⋊Q8C22⋊C412yes—
181#127—21no—
182#127—21no—
183#127—21no—
184#127—21no—
185#125—22no—
186#125—22no—
187#125C22⋊C414yes—
188#125C22⋊C414yes—
189#124—41no—
190#124—41no—
191#124—41no—
192#124—41no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—41no—
206#121—41no—
207#121—41no—
208#121—41no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225#117—22no—
226#117—22no—
227#117—24no—
228#117—42no—
229#117C2×D414yes—
230#117C2×D414yes—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260#107—28no—
261#106—28no—
262C4⋊C8—28no—
263C4⋊C8—28no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273#102—28no—
274#102—28no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8C2×D412yes—
282C22⋊C8C2×D412yes—
283C22⋊C8C2×D414yes—
284#97—28no—
285#97—22no—
286#97—41no—
287C4×C8—28no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—22no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8—41no—
297#95—22no—
298#95—41no—
299C24—22no—
300C24C2×#4111yescharacteristic
301#41—84no—
302#41—84no—
303#41—84no—
304#41—84no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—82no—
327C2×D4—84no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—84no—
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12311yescharacteristic
348C22×C4#12311yescharacteristic
349C22×C4#12311yescharacteristic
350C22×C4#12311yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357M16—44no—
358M16—44no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4C2×C22⋊C414yes—
384C4⋊C4C2×C22⋊C414yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C42—24no—
394C42—24no—
395C42—44no—
396C42—44no—
397C42—82no—
398C42C2×C22⋊C411yescharacteristic
399C42C2×C22⋊C411yescharacteristic
400C42C2×C22⋊C411yescharacteristic
401C42C2×C22⋊C411yescharacteristic
402C42C4×D411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42#12011yescharacteristic
407C42#12011yescharacteristic
408C42#12011yescharacteristic
409C42#12011yescharacteristic
410C42#12111yescharacteristic
411C42#12111yescharacteristic
412C42#12111yescharacteristic
413C42#12111yescharacteristic
414C23—21no—
415C23—21no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—82no—
422C23—82no—
423C23C2×C4×D411yescharacteristic
424C23C2×#12111yescharacteristic
425C23C2×#12311yescharacteristic
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—82no—
430Q8—82no—
431Q8—82no—
432D4—44no—
433D4—82no—
434D4—82no—
435D4—84no—
436D4—84no—
437D4—82no—
438D4—82no—
439D4—82no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#38611yescharacteristic
471C2×C4#38611yescharacteristic
472C2×C4#38611yescharacteristic
473C2×C4#38611yescharacteristic
474C2×C4#38611yescharacteristic
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#44312yes—
479C2×C4#44412yes—
480C2×C4C22×C22⋊C411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×#12011yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C8—42no—
485C8—42no—
486C8—44no—
487C8—42no—
488C8—42no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—162no—
504C22—162no—
505C22C2×#38611yescharacteristic
506C22C2×#44311yescharacteristic
507C22C2×#44411yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—44no—
518C4—84no—
519C4—82no—
520C4#187612yes—
521C4#187612yes—
522C4#190312yes—
523C4#288412yes—
524C2—21no—
525C2—21no—
526C2—41no—
527C2—41no—
528C2—162no—
529C2C2×#187611yescharacteristic
530C2#2098611yescharacteristic
531C2C2×#190311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60250C2semidirect
3. #21844C2semidirect
6. #9242C2semidirect
7. #8007C2semidirect
8. #7924C2semidirect
15. C2×#500C4semidirect
16. C2×#498C4semidirect
17. C2×#486C4semidirect
18. C2×#486C4semidirect
20. C2×#478C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#602502
#218442
#175442
#173842
#92422
#80072
#79242