Finite Group Databasebeta

← #390608

Subgroups — #390608

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23638C211yesmaximal, characteristic
2#21050C212yesmaximal
3#11291C212yesmaximal, complemented
4#11258C212yesmaximal, complemented
5#11097C211yesmaximal, characteristic
6#10909C212yesmaximal, complemented
7#10887C212yesmaximal, complemented
8#10719C212yesmaximal
9#10712C211yesmaximal, characteristic
10C23×C42C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#424C2212yes—
14C2×#407C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#405C2212yes—
18#2280C2212yes—
19#2077C412yescomplemented
20#2077C412yescomplemented
21#2076C411yescharacteristic, complemented
22#2076C411yescharacteristic, complemented
23#2075C411yescharacteristic, complemented
24#2075C411yescharacteristic, complemented
25#1924—22no—
26#1924—22no—
27#1923—22no—
28#1923—22no—
29C2×#341C2212yes—
30C2×#341C2211yescharacteristic
31#1735C2211yescharacteristic
32#1735C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37#1729C2212yes—
38#1729C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×#117C2312yes—
49C2×#117C2312yes—
50C22×C42—24no—
51C22×C42—21no—
52C22×C42—41no—
53C22×C42—41no—
54C22×C42—42no—
55C22×C42C2312yes—
56C2×#108—42no—
57C2×#108—41no—
58#427—41no—
59#427—41no—
60C2×#107C2×C411yescharacteristic, complemented
61C2×#107C2×C411yescharacteristic, complemented
62C2×#106C2×C412yescomplemented
63#424—24no—
64#424—42no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67#407—24no—
68#407—42no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8—41no—
75C2×C22⋊C8C2×C411yescharacteristic
76C2×C22⋊C8C2×C411yescharacteristic
77C2×C22⋊C8C2×C411yescharacteristic
78C2×C22⋊C8C2×C411yescharacteristic
79C2×C22⋊C8D411yescharacteristic
80C2×C22⋊C8D412yes—
81C2×C22⋊C8D411yescharacteristic
82#405—42no—
83#405—42no—
84C2×#97—22no—
85C2×C4×C8—22no—
86#381—22no—
87#380C2×C412yes—
88#380C2×C412yes—
89#380C2×C412yes—
90#380C2×C412yes—
91#379—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94#341—24no—
95#341—41no—
96#340C2×C411yescharacteristic, complemented
97#340C2×C411yescharacteristic, complemented
98#340C2×C412yescomplemented
99#340C2×C412yescomplemented
100#340C2×C412yescomplemented
101#340C2×C412yescomplemented
102#340C2×C412yescomplemented
103#339—24no—
104#339—42no—
105#339—42no—
106#339—42no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—24no—
112#336—42no—
113C25#4111yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C23×C4C2411yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16#4114yes—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—44no—
194C2×C42—48no—
195C2×C42—44no—
196C2×C42—42no—
197C2×C42—44no—
198C2×C42#4114yes—
199#108—44no—
200#107—41no—
201#107—41no—
202#106—42no—
203C4⋊C8—44no—
204C4⋊C8—44no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—42no—
211C22⋊C8—41no—
212C22⋊C8—44no—
213C22⋊C8—44no—
214#97—44no—
215C4×C8—44no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220C24—21no—
221C24—21no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—42no—
227C24C4×D411yescharacteristic
228C24C4×D411yescharacteristic
229C24C2×#4111yescharacteristic
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—416no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—416no—
263C22×C4—48no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4C4×D411yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4C2×#4112yes—
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C42−1+411yescharacteristic
287C22×C42−1+412yes—
288M16—24no—
289M16—24no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16—42no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—24no—
343C42—24no—
344C42—48no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—22no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—22no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—42no—
372C23—41no—
373C23#44312yes—
374C23#44412yes—
375C23C2×C4×D411yescharacteristic
376C23#53811yescharacteristic
377C23#53911yescharacteristic
378C23#55112yes—
379C23#55212yes—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—432no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
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×C4C4×#4112yes—
420C2×C4#51811yescharacteristic
421C2×C4#52011yescharacteristic
422C2×C4#52012yes—
423C2×C4#53411yescharacteristic
424C2×C4#53711yescharacteristic
425C2×C4#53812yes—
426C2×C4#53912yes—
427C2×C4#54112yes—
428C2×C4#54212yes—
429C2×C4#54612yes—
430C2×C4#54712yes—
431C2×C4#55111yescharacteristic
432C2×C4#55312yes—
433C2×C4#55911yescharacteristic
434C2×C4H(3,𝔽4)11yescharacteristic
435C2×C4#56212yes—
436C2×C4#56311yescharacteristic
437C8—41no—
438C8—41no—
439C8—41no—
440C8—41no—
441C8—42no—
442C8—42no—
443C8—44no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—24no—
455C22—22no—
456C22—22no—
457C22—44no—
458C22—48no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22#232511yescharacteristic
467C22#235912yes—
468C22#273611yescharacteristic
469C22#273912yes—
470C22#277111yescharacteristic
471C22C2×#44311yescharacteristic
472C22C2×#44411yescharacteristic
473C22#295612yes—
474C4—48no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4#233412yes—
482C4#237411yescharacteristic
483C4#237911yescharacteristic
484C4#270812yes—
485C4#271211yescharacteristic
486C4#272211yescharacteristic
487C2—21no—
488C2—22no—
489C2—21no—
490C2—42no—
491C2—42no—
492C2#1457711yescharacteristic
493C2#2220411yescharacteristic
494C2#2220711yescharacteristic
495C2#2294812yes—
496C2#2295012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11291C2semidirect
4. #11258C2semidirect
6. #10909C2semidirect
7. #10887C2semidirect
19. #2077C4semidirect
20. #2077C4semidirect
21. #2076C4semidirect
22. #2076C4semidirect
23. #2075C4semidirect
24. #2075C4semidirect
60. C2×#107C2×C4semidirect
61. C2×#107C2×C4semidirect
62. C2×#106C2×C4semidirect
96. #340C2×C4semidirect
97. #340C2×C4semidirect
98. #340C2×C4semidirect
99. #340C2×C4semidirect
100. #340C2×C4semidirect
101. #340C2×C4semidirect
102. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#236382
#210502
#112912
#112582
#110972
#109092
#108872
#107192
#107122