Finite Group Databasebeta

← #192167

Subgroups — #192167

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34186C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21981C211yesmaximal, characteristic, complemented
4C2×#1460C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6#8858C218yesmaximal, complemented
7#8014C211yesmaximal, characteristic, complemented
8#8014C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2357—22no—
13#2357—22no—
14#2354—21no—
15#2354—21no—
16#2354—21no—
17#2354—21no—
18#2312C2214yes—
19#2309—21no—
20C2×#390—21no—
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2280—22no—
25C2×#380—22no—
26#2274—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1948—22no—
29#1947—22no—
30#1853C2214yes—
31#1852C2214yes—
32#1845—21no—
33#1844—21no—
34#1834—21no—
35#1834—21no—
36#1460C418yescomplemented
37C2×#323C2214yes—
38#1439C2214yes—
39C4⋊#95—21no—
40#1294C2214yes—
41#1294C2214yes—
42C23×D4D411yescharacteristic
43C24×C4D412yes—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4C2×C414yes—
51C2×C4×D4Q814yes—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—22no—
55C2×#117—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—24no—
62C22×C22⋊C4C2312yes—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431C2×C414yes—
68#431C2×C414yes—
69C2×#107—24no—
70C2×#106—24no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#390—22no—
78#390—22no—
79#390—22no—
80#390—22no—
81#390—44no—
82#388—24no—
83#388—24no—
84#388—24no—
85#388—24no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384D411yescharacteristic
91#384D411yescharacteristic
92#384D411yescharacteristic
93#384D411yescharacteristic
94#382—21no—
95#382—21no—
96#381—24no—
97#380—22no—
98#380—22no—
99#380—21no—
100#380—21no—
101#380—48no—
102#380C2312yes—
103#380C2312yes—
104#379—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—24no—
117C2×#95—24no—
118C2×#95C2312yes—
119#340—24no—
120#340—24no—
121#323C2×C4116yes—
122C25C2×D412yes—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4C4⋊C414yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4C4⋊C418yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157#117—84no—
158#117—84no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C4⋊C414yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4—48no—
200C2×C22⋊C4—84no—
201C2×C22⋊C4C22×C414yes—
202C2×C22⋊C4C2×Q814yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
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—84no—
219#106—84no—
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—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—48no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24C2×C4⋊C414yes—
248C24C22×D411yescharacteristic
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—44no—
258C2×D4—44no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—416no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—88no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C4⋊C412yes—
292C22×C4C2×C4⋊C414yes—
293C22×C4C2×C4⋊C412yes—
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12312yes—
299C22×C4#12312yes—
300C22×C4#13512yes—
301C22×C4#13512yes—
302C22×C4AGL(1,ℤ/8)12yes—
303C22×C4#13712yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—48no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C22⋊C4—816no—
350C42—44no—
351C42—44no—
352C42—84no—
353C42—84no—
354C42—84no—
355C42C4×D412yes—
356C42C4×D412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42C22⋊Q812yes—
360C42C22⋊Q812yes—
361C42C22⋊Q812yes—
362C42C22⋊Q812yes—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23C22×C4⋊C411yescharacteristic
391C23C2×#12011yescharacteristic
392C23C2×#12311yescharacteristic
393C23C2×#13511yescharacteristic
394C23C2×#13511yescharacteristic
395C23C2×AGL(1,ℤ/8)11yescharacteristic
396C23C2×#13711yescharacteristic
397D4—48no—
398D4—48no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—416no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
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—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—88no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#38512yes—
443C2×C4#38512yes—
444C2×C4#38512yes—
445C2×C4#38512yes—
446C2×C4#42714yes—
447C2×C4#42814yes—
448C2×C4#44912yes—
449C2×C4#44912yes—
450C2×C4#45212yes—
451C2×C4#45212yes—
452C2×C4#45312yes—
453C2×C4#45512yes—
454C2×C4#48214yes—
455C2×C4#48514yes—
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C22⋊Q811yescharacteristic
459C2×C4C2×C22⋊Q811yescharacteristic
460C8—44no—
461C8—44no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—48no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22C2×#38511yescharacteristic
486C22C2×#42711yescharacteristic
487C22C2×#42811yescharacteristic
488C22C2×#44911yescharacteristic
489C22C2×#45211yescharacteristic
490C22C2×#45311yescharacteristic
491C22C2×#45511yescharacteristic
492C22#302711yescharacteristic
493C22#302811yescharacteristic
494C22C2×#48211yescharacteristic
495C22C2×#48511yescharacteristic
496C4—22no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—44no—
502C4—84no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—162no—
508C4#185414yes—
509C4#185414yes—
510C4#185614yes—
511C2—21no—
512C2—21no—
513C2—21no—
514C2—21no—
515C2—84no—
516C2#1840511yescharacteristic
517C2#1840611yescharacteristic
518C2#1841211yescharacteristic
519C2#1841311yescharacteristic
520C2C2×#185411yescharacteristic
521C2C2×#185611yescharacteristic
522C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
3. #21981C2semidirect
6. #8858C2semidirect
7. #8014C2semidirect
8. #8014C2semidirect
36. #1460C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#219822
#219812
C2×#14602
#139062
#88582
#80142