Finite Group Databasebeta

← #192783

Subgroups — #192783

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21982C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8890C214yesmaximal, complemented
7#8882C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9#7893C211yesmaximal, characteristic, complemented
10C2×#518—21no—
11C22×C4×D4C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102—21no—
16#2349—22no—
17#2349—22no—
18D4×C4⋊C4—21no—
19D4×C4⋊C4—21no—
20D4×C4⋊C4C2212yes—
21D4×C4⋊C4C2212yes—
22#2316—21no—
23#2315—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#380—22no—
28#1947—22no—
29#1947C2214yescomplemented
30#1852—22no—
31#1852—22no—
32#1845—21no—
33#1845C2212yescomplemented
34#1834—21no—
35#1834C2212yescomplemented
36#1802—22no—
37#1802C2214yes—
38C2×#340—21no—
39#1478C418yescomplemented
40C2×D4⋊C8C2214yes—
41#1296C2212yescomplemented
42#1296C2212yescomplemented
43#1279C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C23×C8—21no—
47#518—44no—
48#518—44no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4D412yes—
59C2×C4×D4Q814yes—
60C2×C4×D4C2312yes—
61C2×#117—21no—
62C2×#117—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—24no—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431C2×C414yescomplemented
75#431C2×C414yescomplemented
76C2×#107—24no—
77C2×#107—24no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×#102—24no—
85C2×#102—44no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#385—24no—
95#385—24no—
96#385—24no—
97#385—24no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#384C2312yes—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#382—21no—
107#382—21no—
108#380—24no—
109#380—24no—
110#380—21no—
111#380—21no—
112#380—22no—
113#380—22no—
114#380D412yes—
115#380D412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4C2312yes—
120C2×#95—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124D4⋊C8C2×C4116yescomplemented
125C25—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4C4⋊C414yes—
131C22×D4D814yes—
132C22×D4Q1614yes—
133C22×D4C2×D412yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—41no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—42no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—48no—
163C4×D4—48no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4C4⋊C418yes—
167C4×D4C22×C414yes—
168C4×D4C2×Q814yes—
169#117—42no—
170#117—42no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C4⋊C414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—416no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222#107—84no—
223#107—84no—
224C4⋊C8—24no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—42no—
229C4⋊C8C22×C414yes—
230#102—48no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C4×C8—24no—
234C4×C8—22no—
235C4×C8C22×C414yes—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242C24—22no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—81no—
247C24—81no—
248C24—81no—
249C24—81no—
250C24#12011yescharacteristic
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—28no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—28no—
259C2×D4—28no—
260C2×D4#107116yescomplemented
261C2×D4C2×C4⋊C414yes—
262C2×D4C2×D814yes—
263C2×D4C2×Q1614yes—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—88no—
283C22×C4—88no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4C2×C4⋊C412yes—
296C22×C4C4×D412yes—
297C22×C4C4×D412yes—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4C22⋊Q812yes—
302C22×C4C22⋊Q812yes—
303C22×C4#12312yes—
304C22×C4C2×D812yes—
305C22×C4C2×Q1612yes—
306C22×C4AGL(1,ℤ/8)12yes—
307C22×C4#13712yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—84no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C4⋊C4—82no—
346C4⋊C4C2×C4⋊C414yes—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—28no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C42—42no—
355C42—42no—
356C42—44no—
357C42C2×C4⋊C412yes—
358C42C4×D412yes—
359C42C4×D412yes—
360C42C22⋊Q812yes—
361C42C22⋊Q812yes—
362C42#12312yes—
363C42C23×C411yescharacteristic
364C42C22×Q811yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—216no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—84no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23#38512yes—
386C23#44712yes—
387C23#45112yes—
388C23C2×#12011yescharacteristic
389D4—24no—
390D4—24no—
391D4—216no—
392D4—24no—
393D4C2×#10718yes—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—168no—
427C2×C4—168no—
428C2×C4—164no—
429C2×C4#38512yes—
430C2×C4#38512yes—
431C2×C4#38512yes—
432C2×C4C2×#10714yes—
433C2×C4#42814yes—
434C2×C4#44712yes—
435C2×C4#45112yes—
436C2×C4#48014yes—
437C2×C4#48414yes—
438C2×C4C22×C4⋊C411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×C22⋊Q811yescharacteristic
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4C22×D811yescharacteristic
445C2×C4C22×Q1611yescharacteristic
446C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C2×C4#57612yes—
449C2×C4#57712yes—
450C8—44no—
451C8—44no—
452C8—82no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—28no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—816no—
467C22#185214yes—
468C22C2×#38511yescharacteristic
469C22C2×#44711yescharacteristic
470C22C2×#45111yescharacteristic
471C22#300811yescharacteristic
472C22#300911yescharacteristic
473C22#302111yescharacteristic
474C22#302211yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—44no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4—162no—
485C4#185214yes—
486C4C22×#10711yescharacteristic
487C4C2×#42811yescharacteristic
488C4#294612yes—
489C4#308112yes—
490C4C2×#48011yescharacteristic
491C4C2×#48411yescharacteristic
492C2—24no—
493C2—48no—
494C2#1889211yescharacteristic
495C2#1889511yescharacteristic
496C2#1890211yescharacteristic
497C2#1890511yescharacteristic
498C2C2×#185211yescharacteristic
499C2#2086511yescharacteristic
500C2#2087611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
3. #21982C2semidirect
6. #8890C2semidirect
7. #8882C2semidirect
8. #8039C2semidirect
9. #7893C2semidirect
29. #1947C22semidirect
33. #1845C22semidirect
35. #1834C22semidirect
39. #1478C4semidirect
41. #1296C22semidirect
42. #1296C22semidirect
43. #1279C22semidirect
74. #431C2×C4semidirect
75. #431C2×C4semidirect
124. D4⋊C8C2×C4semidirect
260. C2×D4#107semidirect

Maximal subgroups

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

subgroupindex
#341812
#219822
C2×#18022
C2×#14782
#88902
#88822
#80392
#78932