Finite Group Databasebeta

← #193394

Subgroups — #193394

Order 512.

938 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60236C211yesmaximal, characteristic
2#21815C211yesmaximal, characteristic
3#17819C211yesmaximal, characteristic, complemented
4#17806C211yesmaximal, characteristic, complemented
5C2×#1543C211yesmaximal, characteristic
6#8935C214yesmaximal, complemented
7#8934C214yesmaximal, complemented
8#7869C211yesmaximal, characteristic, complemented
9#7866C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11#2981—22no—
12#2979C2214yes—
13#2975—21no—
14#2974—22no—
15#2972—21no—
16#2935—22no—
17#2931—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2880—22no—
24C4×#117C2211yescharacteristic
25#1980—22no—
26#1962C2214yes—
27#1789—22no—
28#1574—22no—
29#1574C2212yescomplemented
30#1574C2212yescomplemented
31#1574C2214yescomplemented
32#1573—22no—
33#1568—22no—
34#1564—21no—
35#1564—21no—
36#1543C412yescomplemented
37#1543C412yescomplemented
38#1543C2212yes—
39#1543C2212yes—
40C2×#333C411yescharacteristic, complemented
41C2×#333C411yescharacteristic, complemented
42C2×#331C411yescharacteristic, complemented
43C2×#331C411yescharacteristic, complemented
44#1276C2214yescomplemented
45#1276C2212yescomplemented
46#1276C2212yescomplemented
47C22×#41D412yes—
48C2×#131—24no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532C2×C412yes—
54#532D412yes—
55#532Q812yes—
56#532C2312yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#124Q812yes—
60C2×#121—21no—
61C2×#121—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×#117—21no—
65C22×C42C2311yescharacteristic
66#436—24no—
67#436—24no—
68#436—24no—
69#435—28no—
70C8×D4—24no—
71C8×D4—24no—
72C8×D4—24no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432D412yes—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431C2×C411yescharacteristic
87#431C2×C411yescharacteristic
88#431C2×C412yes—
89#431D411yescharacteristic
90#431D411yescharacteristic
91#431C2311yescharacteristic
92#431C2312yes—
93#431C2311yescharacteristic
94#431C2312yes—
95#431C2312yes—
96#423D412yes—
97#423C2312yes—
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8C2×C411yescharacteristic
102C2×C4⋊C8D412yes—
103C2×#104—28no—
104#408—24no—
105#408—28no—
106#407—24no—
107#407—21no—
108#407—21no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C4×M16—21no—
114C4×M16—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122#376—21no—
123C43C2312yes—
124#335—22no—
125#335—22no—
126#335—24no—
127#334—22no—
128#334—22no—
129#334—24no—
130#333C2×C412yes—
131#333C2×C412yes—
132#333C2×C412yes—
133#333C2×C412yes—
134#331C2×C412yes—
135#331C2×C412yes—
136#331C2×C412yes—
137#331C2×C412yes—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—84no—
141C2×#41C4⋊C414yes—
142C2×#41C2×D414yes—
143C22×Q8C4⋊C412yes—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C4⋊C412yes—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151C23×C4#4111yescharacteristic
152#131—88no—
153C2×M16—22no—
154C2×M16—28no—
155C2×M16—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—28no—
160C22×C8—21no—
161C22×C8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C4⋊C412yes—
165C4⋊Q8C22×C412yes—
166#127—21no—
167#127—21no—
168#127C4⋊C412yes—
169#127C22×C412yes—
170#124—22no—
171#124—22no—
172#124C4⋊C414yes—
173#124C2×Q814yes—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190#117—24no—
191#117—42no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—41no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×Q811yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8C22×C411yescharacteristic
222C4⋊C8C22×C411yescharacteristic
223C4⋊C8C22×C411yescharacteristic
224C4⋊C8C22×C411yescharacteristic
225C4⋊C8C22×C412yes—
226C4⋊C8C22×C412yes—
227C4⋊C8C2×D412yes—
228C4⋊C8C2×D412yes—
229C4⋊C8C2×D414yes—
230#104—88no—
231C22⋊C8—28no—
232C22⋊C8—28no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—22no—
236C22⋊C8—21no—
237C22⋊C8—22no—
238C22⋊C8—21no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—22no—
244C22⋊C8—41no—
245C22⋊C8—41no—
246#97—22no—
247#97—41no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—22no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—41no—
259#95—41no—
260#95—41no—
261C24—22no—
262C24#12311yescharacteristic
263#41—48no—
264#41—48no—
265#41—84no—
266#41—84no—
267C2×Q8—22no—
268C2×Q8—82no—
269C2×Q8C2×C4⋊C414yes—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—81no—
280C2×D4—81no—
281C2×D4—82no—
282C2×D4—84no—
283C2×D4C2×C4⋊C414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—84no—
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312M16—44no—
313M16—44no—
314M16—88no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—44no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—81no—
329C2×C8—81no—
330C2×C8—88no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—44no—
341C42—24no—
342C42—42no—
343C42—44no—
344C42—42no—
345C42—42no—
346C42C2×C4⋊C411yescharacteristic
347C42C2×C4⋊C411yescharacteristic
348C42C4×D411yescharacteristic
349C42C4×D411yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42#12011yescharacteristic
353C42#12011yescharacteristic
354C42C22⋊Q811yescharacteristic
355C42C22⋊Q811yescharacteristic
356C42C22⋊Q811yescharacteristic
357C42C22⋊Q811yescharacteristic
358C42C23×C411yescharacteristic
359C42C22×D411yescharacteristic
360C42C22×D411yescharacteristic
361C42C22×Q811yescharacteristic
362C23—22no—
363C23—24no—
364C23—41no—
365C23—41no—
366C23—44no—
367C23—82no—
368C23—82no—
369C23#38511yescharacteristic
370C23#38511yescharacteristic
371C23C2×#12311yescharacteristic
372Q8—44no—
373Q8—44no—
374D4—44no—
375D4—44no—
376D4—48no—
377D4—44no—
378D4—42no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—82no—
383D4—84no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#38511yescharacteristic
414C2×C4#38511yescharacteristic
415C2×C4#38511yescharacteristic
416C2×C4#38511yescharacteristic
417C2×C4#38511yescharacteristic
418C2×C4#38511yescharacteristic
419C2×C4#44312yes—
420C2×C4#44412yes—
421C2×C4C22×C4⋊C411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×C4×D411yescharacteristic
424C2×C4C2×#12011yescharacteristic
425C2×C4C2×C22⋊Q811yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C8—41no—
428C8—41no—
429C8—41no—
430C8—41no—
431C8—42no—
432C8—42no—
433C8—42no—
434C8—42no—
435C8—44no—
436C8—82no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—82no—
449C22—164no—
450C22#186612yes—
451C22C2×#38511yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—44no—
461C4—84no—
462C4#186612yes—
463C4C2×#44311yescharacteristic
464C4C2×#44411yescharacteristic
465C4#295711yescharacteristic
466C4#295711yescharacteristic
467C4#301912yes—
468C2—22no—
469C2—41no—
470C2—41no—
471C2—84no—
472C2#2085611yescharacteristic
473C2C2×#186611yescharacteristic
474C2#2088611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17819C2semidirect
4. #17806C2semidirect
6. #8935C2semidirect
7. #8934C2semidirect
8. #7869C2semidirect
9. #7866C2semidirect
29. #1574C22semidirect
30. #1574C22semidirect
31. #1574C22semidirect
36. #1543C4semidirect
37. #1543C4semidirect
40. C2×#333C4semidirect
41. C2×#333C4semidirect
42. C2×#331C4semidirect
43. C2×#331C4semidirect
44. #1276C22semidirect
45. #1276C22semidirect
46. #1276C22semidirect

Maximal subgroups

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

subgroupindex
#602362
#218152
#178192
#178062
C2×#15432
#89352
#89342
#78692
#78662