Finite Group Databasebeta

← #192068

Subgroups — #192068

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34048C211yesmaximal, characteristic
2#21359C211yesmaximal, characteristic, complemented
3#21358C211yesmaximal, characteristic, complemented
4C2×#1458C211yesmaximal, characteristic
5#13852C211yesmaximal, characteristic
6#8858C214yesmaximal
7#8855C214yesmaximal, complemented
8#8014C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C22×#117C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2314—22no—
14#2314—22no—
15#2313—21no—
16#2313C2212yes—
17#2312—21no—
18#2312C2212yes—
19#2298—22no—
20#2298—22no—
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#2280—21no—
24C4×#117—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4—21no—
28#1948—22no—
29#1947—22no—
30#1853C2214yes—
31#1852C2214yes—
32C4×#107—22no—
33C4×#106—22no—
34#1458C418yescomplemented
35C2×#324C2212yes—
36C2×#323C2212yes—
37#1441—21no—
38#1441C2212yes—
39#1439—21no—
40#1439C2212yes—
41C4×#95—21no—
42#1294C2214yes—
43#1294C2214yes—
44C24×C4D411yescharacteristic
45C2×#117—21no—
46C2×#117—21no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—21no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117C2×C414yes—
54C2×#117Q814yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42—21no—
63C22×C42—21no—
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431C2×C414yes—
69#431C2×C414yes—
70C2×#107—24no—
71C2×#106—24no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2312yes—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#388—24no—
79#388—24no—
80#388—24no—
81#388—24no—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#384D412yes—
87#384D412yes—
88#382—21no—
89#382—21no—
90#381—22no—
91#380—22no—
92#380—22no—
93#380—21no—
94#380—21no—
95#380C2312yes—
96#380C2312yes—
97#379—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—44no—
115#376—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95C2312yes—
121C43—22no—
122#340—24no—
123#340—24no—
124#324C2×C418yes—
125#323C2×C418yes—
126C25C2×D411yescharacteristic
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C4⋊C414yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145#117—24no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—24no—
151#117—24no—
152#117—24no—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
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—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4C4⋊C414yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4C4⋊C414yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C2×Q812yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C4⋊C414yes—
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q812yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218#107—48no—
219#106—48no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8C22×C414yes—
225C22⋊C8C22×C414yes—
226C4×C8—24no—
227C4×C8—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—41no—
242#95—41no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24C2×C4⋊C412yes—
252C24C22×D411yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C2×C4⋊C412yes—
294C22×C4C2×C4⋊C412yes—
295C22×C4C2×C4⋊C412yes—
296C22×C4C2×C4⋊C412yes—
297C22×C4C2×C4⋊C412yes—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#13512yes—
305C22×C4#13512yes—
306C22×C4AGL(1,ℤ/8)12yes—
307C22×C4#13712yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—48no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—416no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C42—24no—
349C42—24no—
350C42—44no—
351C42—44no—
352C42—84no—
353C42—84no—
354C42—82no—
355C42—82no—
356C42—84no—
357C42C4×D412yes—
358C42C4×D412yes—
359C42C4×D412yes—
360C42C4×D412yes—
361C42C22⋊Q812yes—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42C22⋊Q812yes—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—42no—
371C23—42no—
372C23—44no—
373C23—44no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—81no—
381C23—81no—
382C23—82no—
383C23—82no—
384C23C22×C4⋊C411yescharacteristic
385C23C2×#12011yescharacteristic
386C23C2×#12311yescharacteristic
387C23C2×#13511yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×AGL(1,ℤ/8)11yescharacteristic
390C23C2×#13711yescharacteristic
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#38512yes—
439C2×C4#38512yes—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4#42714yes—
443C2×C4#42814yes—
444C2×C4#44912yes—
445C2×C4#44912yes—
446C2×C4#45212yes—
447C2×C4#45212yes—
448C2×C4#48214yes—
449C2×C4#48514yes—
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×C22⋊Q811yescharacteristic
454C8—44no—
455C8—44no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22C2×#38511yescharacteristic
474C22C2×#42711yescharacteristic
475C22C2×#42811yescharacteristic
476C22C2×#44911yescharacteristic
477C22C2×#45211yescharacteristic
478C22#301812yes—
479C22#302512yes—
480C22C2×#48211yescharacteristic
481C22C2×#48511yescharacteristic
482C4—22no—
483C4—22no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—84no—
492C4—84no—
493C4—84no—
494C4—84no—
495C4#185414yes—
496C4#185414yes—
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2—82no—
502C2#1812612yes—
503C2#1812712yes—
504C2C2×#185411yescharacteristic
505C2#2085511yescharacteristic
506C2#2088811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21359C2semidirect
3. #21358C2semidirect
7. #8855C2semidirect
8. #8014C2semidirect
9. #8014C2semidirect
34. #1458C4semidirect

Maximal subgroups

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

subgroupindex
#340482
#213592
#213582
C2×#14582
#138522
#88582
#88552
#80142