Finite Group Databasebeta

← #189043

Subgroups — #189043

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60236C211yesmaximal, characteristic
2#17817C211yesmaximal, characteristic, complemented
3#17815C211yesmaximal, characteristic, complemented
4#17750C211yesmaximal, characteristic
5#17738C211yesmaximal, characteristic
6#8640C212yesmaximal, complemented
7#8640C212yesmaximal, complemented
8#8638C212yesmaximal, complemented
9#8636C212yesmaximal, complemented
10#8130C211yesmaximal, characteristic, complemented
11#8128C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2981—21no—
14#2981—21no—
15#2979—22no—
16#2975—21no—
17#2974—22no—
18#2972—21no—
19#2935C414yes—
20#2931C2212yes—
21#2931C2212yes—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25#2920C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28#2880C414yes—
29#1574—22no—
30#1573—22no—
31#1572—21no—
32#1572—21no—
33#1572—21no—
34#1572—21no—
35#1567C2212yescomplemented
36#1567C2211yescharacteristic, complemented
37#1567C2211yescharacteristic, complemented
38#1566C2212yescomplemented
39#1566C2211yescharacteristic, complemented
40#1566C2211yescharacteristic, complemented
41#1560C2212yes—
42#1560C2212yes—
43#1560C2212yes—
44#1560C2212yes—
45#1558—21no—
46#1555—21no—
47#1551—22no—
48#1548—21no—
49#1545—21no—
50#1543—22no—
51#1309C2212yescomplemented
52#1309C2212yescomplemented
53#1309C2212yescomplemented
54#1309C2212yescomplemented
55C22×#41C2×C412yes—
56C2×#131C2×C414yes—
57C2×#131C2×C414yes—
58C22×M16—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532C2×C412yes—
64#532D412yes—
65#532Q812yes—
66#532C2312yes—
67C2×C4⋊Q8D411yescharacteristic
68C2×#127D411yescharacteristic
69C2×#124Q812yes—
70C2×#121—21no—
71C2×#121—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C22×C42C2311yescharacteristic
75#436—24no—
76#436—22no—
77#436—22no—
78#436—22no—
79#436—22no—
80#435—24no—
81#435—24no—
82C8×D4—24no—
83C8×D4—22no—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—22no—
87#432—21no—
88#432—21no—
89#432—21no—
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431D411yescharacteristic
104#431D411yescharacteristic
105#431D411yescharacteristic
106#431D411yescharacteristic
107#431D412yes—
108#431D412yes—
109#423—21no—
110#423—21no—
111#423—21no—
112#423C2311yescharacteristic
113#423C2311yescharacteristic
114#423C2311yescharacteristic
115#423C2312yes—
116#423C2312yes—
117#423C2311yescharacteristic
118#423C2312yes—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8C2×C412yes—
122#408—24no—
123#408—24no—
124#408C2×C414yes—
125#408C2×C414yes—
126#407—24no—
127#407—21no—
128#407—21no—
129#407—21no—
130#407C2×C412yes—
131C2×C22⋊C8—24no—
132C2×C22⋊C8C2×C412yes—
133C4×M16—21no—
134C4×M16—21no—
135C4×M16—21no—
136C4×M16C2×C412yes—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#335—22no—
140#335—22no—
141#335—22no—
142#335—22no—
143#334—22no—
144#334—22no—
145#334—22no—
146#334—22no—
147#333—22no—
148#333—22no—
149#333—22no—
150#333—22no—
151#331—22no—
152#331—22no—
153#331—22no—
154#331—22no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—84no—
158C2×#41C4⋊C414yes—
159C2×#41C22×C414yes—
160C22×Q8C4⋊C412yes—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C4⋊C412yes—
165C23×C4—22no—
166C23×C4C22×C411yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4#4111yescharacteristic
169#131—84no—
170#131—84no—
171C2×M16—24no—
172C2×M16—24no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—21no—
179C2×M16—41no—
180C2×M16—82no—
181C2×M16C22×C414yes—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—41no—
185C22×C8C22×C412yes—
186C22×C8C22×C412yes—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8C4⋊C412yes—
190C4⋊Q8C2×D412yes—
191#127—21no—
192#127—21no—
193#127C4⋊C412yes—
194#127C2×D412yes—
195#124—22no—
196#124—22no—
197#124C4⋊C414yes—
198#124C2×Q814yes—
199#121—22no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×Q811yescharacteristic
227C2×C42C2×Q811yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42C2411yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—22no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8C22⋊C412yes—
248C4⋊C8C22⋊C412yes—
249C4⋊C8C22⋊C412yes—
250C4⋊C8C22⋊C412yes—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—24no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—24no—
257C22⋊C8—41no—
258C22⋊C8—41no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8C22⋊C412yes—
264C22⋊C8C22⋊C412yes—
265C22⋊C8C22⋊C412yes—
266C22⋊C8C22⋊C412yes—
267#97—22no—
268#97—41no—
269#97—41no—
270#97—41no—
271C4×C8—22no—
272C4×C8—41no—
273C4×C8—41no—
274C4×C8—41no—
275C4×C8C2×D411yescharacteristic
276C4×C8C2×D411yescharacteristic
277C4×C8C2×D412yes—
278C4×C8C2×D412yes—
279C4×C8C2×D411yescharacteristic
280C4×C8C2×D411yescharacteristic
281C24—22no—
282C24#11711yescharacteristic
283#41—44no—
284#41—44no—
285#41—84no—
286#41—84no—
287#41—84no—
288C2×Q8—22no—
289C2×Q8—82no—
290C2×Q8C2×C4⋊C414yes—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—44no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4—82no—
302C2×D4—84no—
303C2×D4C2×C4⋊C414yes—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—84no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C4⋊C411yescharacteristic
315C22×C4C2×C4⋊C411yescharacteristic
316C22×C4#11711yescharacteristic
317C22×C4#11711yescharacteristic
318C22×C4#11711yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4C22⋊Q811yescharacteristic
324C22×C4C22⋊Q811yescharacteristic
325C22×C4C22⋊Q811yescharacteristic
326C22×C4C22⋊Q811yescharacteristic
327C22×C4C23×C411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330M16—44no—
331M16—84no—
332M16—84no—
333M16—82no—
334M16—82no—
335M16—82no—
336M16—81no—
337M16—81no—
338M16—81no—
339M16—81no—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—84no—
350C2×C8—84no—
351C2×C8—82no—
352C2×C8C2×C22⋊C412yes—
353C2×C8C2×C22⋊C412yes—
354C2×C8C2×C22⋊C412yes—
355C2×C8C2×C22⋊C412yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C42C2×C22⋊C411yescharacteristic
365C42C2×C22⋊C411yescharacteristic
366C42C2×C4⋊C411yescharacteristic
367C42C2×C4⋊C411yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42C22⋊Q811yescharacteristic
373C42C22⋊Q811yescharacteristic
374C42C22⋊Q811yescharacteristic
375C42C22⋊Q811yescharacteristic
376C42C22×D411yescharacteristic
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C42C22×Q811yescharacteristic
380C23—24no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—44no—
385C23—82no—
386C23—82no—
387C23#38011yescharacteristic
388C23#38011yescharacteristic
389C23C2×#11711yescharacteristic
390Q8—44no—
391Q8—82no—
392D4—44no—
393D4—44no—
394D4—48no—
395D4—44no—
396D4—42no—
397D4—42no—
398D4—82no—
399D4—82no—
400D4—84no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4#38011yescharacteristic
428C2×C4#38011yescharacteristic
429C2×C4#38011yescharacteristic
430C2×C4#38011yescharacteristic
431C2×C4#38011yescharacteristic
432C2×C4#38011yescharacteristic
433C2×C4C22×C22⋊C411yescharacteristic
434C2×C4C22×C4⋊C411yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C8—22no—
440C8—22no—
441C8—22no—
442C8—22no—
443C8—44no—
444C8—82no—
445C8—82no—
446C8—81no—
447C8—81no—
448C8—81no—
449C8—81no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—82no—
463C22—164no—
464C22C2×#38011yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—44no—
475C4#288812yes—
476C4#306712yes—
477C4#306811yescharacteristic
478C4#306911yescharacteristic
479C2—41no—
480C2—41no—
481C2—41no—
482C2—84no—
483C2#2063811yescharacteristic
484C2#2067811yescharacteristic
485C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. #17817C2semidirect
3. #17815C2semidirect
6. #8640C2semidirect
7. #8640C2semidirect
8. #8638C2semidirect
9. #8636C2semidirect
10. #8130C2semidirect
11. #8128C2semidirect
35. #1567C22semidirect
36. #1567C22semidirect
37. #1567C22semidirect
38. #1566C22semidirect
39. #1566C22semidirect
40. #1566C22semidirect
51. #1309C22semidirect
52. #1309C22semidirect
53. #1309C22semidirect
54. #1309C22semidirect

Maximal subgroups

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

subgroupindex
#602362
#178172
#178152
#177502
#177382
#86402
#86382
#86362
#81302
#81282