Finite Group Databasebeta

← #190050

Subgroups — #190050

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34109C211yesmaximal, characteristic
2C2×#1822C211yesmaximal, characteristic
3#20688C211yesmaximal, characteristic
4#8741C214yesmaximal, complemented
5#8740C214yesmaximal, complemented
6#7946C212yesmaximal
7#7942C212yesmaximal
8C22×#117C2211yescharacteristic
9C2×#407C2212yes—
10#2347—21no—
11#2346—21no—
12#2345—21no—
13#2344—21no—
14#2343—22no—
15#2342—22no—
16#2298C2212yes—
17#2298C2212yes—
18#2296C411yescharacteristic, complemented
19#2296C411yescharacteristic, complemented
20#2283C412yescomplemented
21#2280C2212yes—
22#2280C2211yescharacteristic
23C4×#117C412yescomplemented
24C4×#117C411yescharacteristic, complemented
25C4×#117C411yescharacteristic, complemented
26C2×C4×C22⋊C4C2211yescharacteristic
27#1822—22no—
28#1822C2214yes—
29#1820C2214yes—
30#1819—22no—
31C2×#343—21no—
32C2×#343—21no—
33#1740—21no—
34#1740—21no—
35#1284C2214yes—
36#1284C2214yes—
37#1284C2218yes—
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C2×#117D412yes—
51C2×#117D411yescharacteristic
52C2×#117D411yescharacteristic
53C2×#117C2312yes—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4D412yes—
58C22×C42C2×C412yes—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61#407—22no—
62#407—22no—
63#407—22no—
64#407C2314yes—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#388—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#385—22no—
80#385—22no—
81#383—22no—
82#382—22no—
83#381—22no—
84#381—21no—
85#381C2314yes—
86#380—22no—
87#380—21no—
88#380—21no—
89#380—22no—
90#380—22no—
91#379—21no—
92#379—22no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4C2×C411yescharacteristic
110C4×C22⋊C4C2×C411yescharacteristic
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4C2311yescharacteristic
113C4×C22⋊C4C2311yescharacteristic
114C4×C22⋊C4C2312yes—
115#376—22no—
116#376C2×C412yes—
117#376C2×C412yes—
118C2×#95—21no—
119C43C2×C412yes—
120C43C2×C412yes—
121#343—24no—
122#343—24no—
123#342—24no—
124#342—24no—
125C25C2×D411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C23×C4C2411yescharacteristic
142C2×M16—24no—
143C2×M16—42no—
144C2×M16—44no—
145C2×M16—44no—
146C22×C8—42no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—22no—
154#117—22no—
155#117—24no—
156#117—24no—
157#117—24no—
158#117—24no—
159#117—48no—
160#117—42no—
161#117—42no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C22⋊C412yes—
213C2×C42C22⋊C412yes—
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C22⋊C8—24no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225C22⋊C8—42no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—24no—
241#95—24no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24—81no—
246C24C2×C22⋊C411yescharacteristic
247C24C2×C22⋊C411yescharacteristic
248C24#12312yes—
249C24#12312yes—
250C24C22×D411yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—82no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4#11712yes—
293C22×C4#11712yes—
294C22×C4#11711yescharacteristic
295C22×C4#11711yescharacteristic
296C22×C4#11711yescharacteristic
297C22×C4#11711yescharacteristic
298C22×C4#12312yes—
299C22×C4#12312yes—
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308M16—44no—
309M16—44no—
310M16—84no—
311C2×C8—44no—
312C2×C8—84no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—84no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—42no—
355C42—42no—
356C42—44no—
357C42—44no—
358C42—42no—
359C42—42no—
360C42—84no—
361C42—82no—
362C42—82no—
363C23—22no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—24no—
369C23—21no—
370C23—21no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—84no—
382C23—84no—
383C23#38112yes—
384C23#38111yescharacteristic
385C23#38111yescharacteristic
386C23C22×C22⋊C411yescharacteristic
387C23C2×#12312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4#38111yescharacteristic
444C2×C4#38111yescharacteristic
445C2×C4#38112yes—
446C2×C4#41012yes—
447C2×C4#41012yes—
448C2×C4C2×#11711yescharacteristic
449C2×C4C2×#11711yescharacteristic
450C2×C4C2×#12312yes—
451C8—84no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—21no—
458C22—22no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—21no—
464C22—22no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—81no—
470C22—81no—
471C22—84no—
472C22—84no—
473C22#182712yes—
474C22C2×#38111yescharacteristic
475C22C2×#41011yescharacteristic
476C22C2×#41011yescharacteristic
477C4—22no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—82no—
489C4—82no—
490C4—81no—
491C4—81no—
492C4—164no—
493C4#182014yes—
494C2—21no—
495C2—21no—
496C2—22no—
497C2—42no—
498C2—41no—
499C2—42no—
500C2C2×#182011yescharacteristic
501C2C2×#182711yescharacteristic
502C2#2070811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8741C2semidirect
5. #8740C2semidirect
18. #2296C4semidirect
19. #2296C4semidirect
20. #2283C4semidirect
23. C4×#117C4semidirect
24. C4×#117C4semidirect
25. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#341092
C2×#18222
#206882
#87412
#87402
#79462
#79422