Finite Group Databasebeta

← #205998

Subgroups — #205998

Order 512.

1137 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3138C211yesmaximal, characteristic
2#21879C211yesmaximal, characteristic, complemented
3#21797C211yesmaximal, characteristic
4#17796C211yesmaximal, characteristic, complemented
5#17715C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic
7#17327C211yesmaximal, characteristic, complemented
8#9812C212yesmaximal, complemented
9#9804C212yesmaximal, complemented
10#9800C212yesmaximal, complemented
11#9796C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C22×#127C2211yescharacteristic
14#3138—22no—
15#3138—22no—
16#3138—21no—
17#3138—21no—
18#3138—22no—
19C2×#490C414yes—
20C2×#486C2212yes—
21C2×#486C2212yes—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#418C414yes—
27C2×C4×M16C2211yescharacteristic
28#1997C2212yescomplemented
29#1966C2212yescomplemented
30#1965—22no—
31#1965C2212yes—
32#1965C2212yes—
33#1901—22no—
34#1801—22no—
35#1783—22no—
36#1569—22no—
37#1567C2212yescomplemented
38#1566C2212yescomplemented
39#1565—22no—
40#1552—22no—
41#1549C2211yescharacteristic, complemented
42#1549C2211yescharacteristic, complemented
43#1546C2211yescharacteristic, complemented
44#1546C2211yescharacteristic, complemented
45#1545—22no—
46#1545C2212yes—
47#1545C2212yes—
48#1544—22no—
49C2×#331—22no—
50#1453—22no—
51#1451—21no—
52#1451C2212yescomplemented
53C8×M16—21no—
54C8×M16C2212yescomplemented
55C23×D4—21no—
56C23×D4—21no—
57C23×D4C2×C412yes—
58C22×M16—21no—
59C22×M16C2×C412yes—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533—21no—
67#533—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127—21no—
71C2×#127—21no—
72C2×#127—21no—
73C2×#127—21no—
74C2×#127—21no—
75C2×#127—21no—
76C2×#127—21no—
77C2×#127—21no—
78C2×#127—21no—
79C2×#127—21no—
80C2×#127C2×C412yes—
81C2×#127C2312yes—
82C2×#125C2×C412yes—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×#117C2×C412yes—
88C22×C42C2311yescharacteristic
89#490—28no—
90#490—24no—
91#490—24no—
92#486—22no—
93#486—22no—
94#486—22no—
95#486—22no—
96#486—24no—
97#486—24no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431D412yes—
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111#423—21no—
112#423—21no—
113#423—21no—
114#423D412yes—
115#423C2312yes—
116#423C2311yescharacteristic
117#423C2311yescharacteristic
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8D412yes—
121#418—24no—
122#418—24no—
123#418—28no—
124C2×#102C2×C418yes—
125C2×#102C2×C418yes—
126C2×#101—24no—
127C2×#100—24no—
128C2×#100—24no—
129#407—21no—
130#407D412yes—
131C2×C22⋊C8—21no—
132C4×M16—21no—
133C4×M16D412yes—
134C4×M16D412yes—
135C4×M16D411yescharacteristic
136C4×M16D411yescharacteristic
137C4×M16D412yes—
138C4×M16C2312yes—
139C2×#97C2×C412yes—
140C2×C4×C8—21no—
141C2×C4×C8C2312yes—
142#343—24no—
143#335—24no—
144#334—24no—
145#332—22no—
146#332—22no—
147#332—22no—
148#332—22no—
149#331—24no—
150#331—24no—
151#322—22no—
152#322—22no—
153#322—22no—
154C82—22no—
155C25—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—48no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—82no—
177C22×D4—82no—
178C22×D4—84no—
179C22×D4C22×C414yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—44no—
187C2×M16—41no—
188C2×M16—82no—
189C2×M16C22⋊C412yes—
190C2×M16C22⋊C412yes—
191C2×M16C22⋊C414yes—
192C2×M16C2×D414yes—
193C22×C8—22no—
194C22×C8—41no—
195C22×C8C22×C414yes—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204#127—22no—
205#127—22no—
206#127—22no—
207#127—22no—
208#127—22no—
209#127—22no—
210#127—22no—
211#127—22no—
212#127—41no—
213#127—41no—
214#127—41no—
215#127—41no—
216#127—41no—
217#127—41no—
218#127—41no—
219#127—41no—
220#125—22no—
221#125—22no—
222#125—22no—
223#125—22no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C22⋊Q8—41no—
231C22⋊Q8—41no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—41no—
239C4×Q8—41no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C22⋊C4—22no—
251C2×C42—22no—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—22no—
272C4⋊C8—21no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8C2×D412yes—
276C4⋊C8C2×D412yes—
277#102—28no—
278#102—28no—
279#102—28no—
280#102—28no—
281#101—84no—
282#100—84no—
283#100—84no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291#97—22no—
292#97—41no—
293#97C22⋊C414yes—
294#97C22⋊C412yes—
295#97C22⋊C412yes—
296C4×C8—22no—
297C4×C8—21no—
298C4×C8—22no—
299C4×C8—21no—
300C4×C8—41no—
301C4×C8C2×D412yes—
302C4×C8C2×D412yes—
303C4×C8C2×D414yes—
304C24—24no—
305C24—42no—
306C24—42no—
307C24—84no—
308C24—84no—
309C24—84no—
310C24C4×D411yescharacteristic
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—82no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—48no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12411yescharacteristic
369C22×C4#12411yescharacteristic
370C22×C4#12711yescharacteristic
371C22×C4#12711yescharacteristic
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375M16—24no—
376M16—24no—
377M16—24no—
378M16—24no—
379M16—44no—
380M16—42no—
381M16—44no—
382M16—42no—
383M16—82no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—44no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—82no—
392C2×C8—82no—
393C2×C8C2×C22⋊C414yes—
394C2×C8C2×C22⋊C414yes—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C42—42no—
410C42—42no—
411C42C2×C22⋊C411yescharacteristic
412C42C2×C22⋊C411yescharacteristic
413C42C4×D411yescharacteristic
414C42C4×D411yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12111yescharacteristic
418C42#12111yescharacteristic
419C42#12411yescharacteristic
420C42#12411yescharacteristic
421C42#12711yescharacteristic
422C42#12711yescharacteristic
423C42C22×D411yescharacteristic
424C42C22×D411yescharacteristic
425C42C22×D411yescharacteristic
426C42C2×#4111yescharacteristic
427C23—22no—
428C23—22no—
429C23—41no—
430C23—44no—
431C23—48no—
432C23—84no—
433C23—84no—
434C23—84no—
435C23—84no—
436C23—84no—
437C23—84no—
438C23—84no—
439C23—84no—
440C23—1616no—
441C23#39011yescharacteristic
442C23#39011yescharacteristic
443C23C2×C4×D411yescharacteristic
444Q8—82no—
445Q8—81no—
446Q8—81no—
447Q8—82no—
448Q8—82no—
449D4—48no—
450D4—84no—
451D4—84no—
452D4—84no—
453D4—84no—
454D4—84no—
455D4—84no—
456D4—84no—
457D4—82no—
458D4—82no—
459D4—84no—
460D4—84no—
461D4—84no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—82no—
484C2×C4#39011yescharacteristic
485C2×C4#39011yescharacteristic
486C2×C4#39011yescharacteristic
487C2×C4#39011yescharacteristic
488C2×C4#39011yescharacteristic
489C2×C4#39011yescharacteristic
490C2×C4#41314yes—
491C2×C4C22×C22⋊C411yescharacteristic
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×#12411yescharacteristic
496C2×C4C2×#12711yescharacteristic
497C2×C4#57612yes—
498C2×C4#57712yes—
499C8—24no—
500C8—24no—
501C8—44no—
502C8—42no—
503C8—42no—
504C8—81no—
505C8—82no—
506C8—81no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—44no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—168no—
519C22—168no—
520C22—168no—
521C22#196712yes—
522C22C2×#39011yescharacteristic
523C22#294612yes—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—82no—
532C4—82no—
533C4#196712yes—
534C4C2×#41312yes—
535C4#306212yes—
536C4#306812yes—
537C4#315411yescharacteristic
538C4#315511yescharacteristic
539C2—22no—
540C2—22no—
541C2—41no—
542C2—164no—
543C2C2×#196711yescharacteristic
544C2#2123511yescharacteristic
545C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ 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
2. #21879C2semidirect
4. #17796C2semidirect
5. #17715C2semidirect
7. #17327C2semidirect
8. #9812C2semidirect
9. #9804C2semidirect
10. #9800C2semidirect
11. #9796C2semidirect
28. #1997C22semidirect
29. #1966C22semidirect
37. #1567C22semidirect
38. #1566C22semidirect
41. #1549C22semidirect
42. #1549C22semidirect
43. #1546C22semidirect
44. #1546C22semidirect
52. #1451C22semidirect
54. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31382
#218792
#217972
#177962
#177152
C2×#15452
#173272
#98122
#98042
#98002
#97962