Finite Group Databasebeta

← #366402

Subgroups — #366402

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20844C211yesmaximal, characteristic
2#20701C211yesmaximal, characteristic
3C2×#1822C211yesmaximal, characteristic
4#20386C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#11303C212yesmaximal, complemented
7#11290C212yesmaximal, complemented
8#10896C212yesmaximal, complemented
9#10827C212yesmaximal, complemented
10#10727C211yesmaximal, characteristic
11#10724C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2280C2211yescharacteristic
22#1853—22no—
23#1852—22no—
24#1825—22no—
25#1825—22no—
26#1822C418yescomplemented
27#1753—22no—
28#1753—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#341C2212yes—
34C2×#341C2212yes—
35#1736—22no—
36#1736—22no—
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42#1729C2211yescharacteristic
43#1729C2211yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—24no—
49C22×M16—42no—
50C22×M16D412yes—
51C22×M16D412yes—
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117C2×C414yes—
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75#428—24no—
76#428—42no—
77C2×#107—22no—
78C2×#106—22no—
79#407—22no—
80#407—22no—
81#407—22no—
82#407—22no—
83#407C2×C414yes—
84#407C2×C414yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88#405—24no—
89#405—42no—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#381—21no—
95#380D412yes—
96#380D412yes—
97#379—21no—
98C4×C22⋊C4—22no—
99#343—24no—
100#343—42no—
101#343C2×C418yescomplemented
102#343C2×C418yescomplemented
103#341—24no—
104#341—42no—
105#340—24no—
106#340—22no—
107#340—22no—
108#340—24no—
109#340—22no—
110#340—22no—
111#340—24no—
112#336—22no—
113#336—22no—
114C25#4111yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—44no—
122C23×C4—42no—
123C23×C4—84no—
124C23×C4C22×C412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4#4111yescharacteristic
128C23×C4#4112yes—
129C23×C4#4112yes—
130C23×C4#4112yes—
131C23×C4C2411yescharacteristic
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—22no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—48no—
156C2×M16—48no—
157C2×M16C22×C412yes—
158C2×M16C22×C414yes—
159C2×M16C22×C412yes—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8#4114yes—
175#117—28no—
176#117—28no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—44no—
182#117—44no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C22×C414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4#4114yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C42—24no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—42no—
214C2×C42—42no—
215#107—44no—
216#106—44no—
217C22⋊C8—28no—
218C22⋊C8—28no—
219C22⋊C8—44no—
220#97—42no—
221#97—42no—
222C4×C8—42no—
223C4×C8—42no—
224#95—22no—
225#95—22no—
226C24—22no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24#11712yes—
231C24#12112yes—
232C24#12612yes—
233C24C2×#4111yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—48no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4—816no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—84no—
266C22×C4#11712yes—
267C22×C4C4×D414yes—
268C22×C4C4×D414yes—
269C22×C4#12112yes—
270C22×C4#12314yes—
271C22×C4#12414yes—
272C22×C4#12612yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283M16—24no—
284M16—24no—
285M16—24no—
286M16—24no—
287M16—48no—
288M16—48no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16—42no—
296M16—42no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8#12312yes—
321C2×C8#12312yes—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—48no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—48no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—48no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—22no—
348C23—24no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—48no—
358C23—44no—
359C23#38814yes—
360C23C2×#11711yescharacteristic
361C23C2×#12111yescharacteristic
362C23C2×#12611yescharacteristic
363C23#55112yes—
364C23#55212yes—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—28no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—28no—
374C2×C4—24no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—816no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4#38814yes—
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C4×#4112yes—
396C2×C4#52012yes—
397C2×C4C2×#12311yescharacteristic
398C2×C4C2×#12311yescharacteristic
399C2×C4#52511yescharacteristic
400C2×C4C2×#12411yescharacteristic
401C2×C4#52912yes—
402C2×C4#53411yescharacteristic
403C2×C4#53611yescharacteristic
404C2×C4#53912yes—
405C2×C4#54612yes—
406C2×C4#54712yes—
407C2×C4#55112yes—
408C2×C4#55112yes—
409C2×C4#55212yes—
410C2×C4#55212yes—
411C8—22no—
412C8—22no—
413C8—42no—
414C8—44no—
415C8—42no—
416C8—42no—
417C8—82no—
418C22—24no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—48no—
427C22—48no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22C2×#38811yescharacteristic
433C22#234312yes—
434C22#248611yescharacteristic
435C22#249312yes—
436C22#253411yescharacteristic
437C22#306812yes—
438C22#306912yes—
439C4—24no—
440C4—82no—
441C4—84no—
442C4—81no—
443C4—84no—
444C4—81no—
445C4C4×#12311yescharacteristic
446C4#232211yescharacteristic
447C4#235912yes—
448C4#248311yescharacteristic
449C4#248411yescharacteristic
450C4#248812yes—
451C2—22no—
452C2—22no—
453C2—44no—
454C2#1399211yescharacteristic
455C2#2107612yes—
456C2#2767612yes—
457C2#2813711yescharacteristic
458C2#2813811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11303C2semidirect
7. #11290C2semidirect
8. #10896C2semidirect
9. #10827C2semidirect
26. #1822C4semidirect
101. #343C2×C4semidirect
102. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
#208442
#207012
C2×#18222
#203862
C2×#17362
#113032
#112902
#108962
#108272
#107272
#107242