Finite Group Databasebeta

← #375025

Subgroups — #375025

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20845C211yesmaximal, characteristic
2#20844C211yesmaximal, characteristic
3C2×#1811C211yesmaximal, characteristic
4#20386C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#11249C212yesmaximal, complemented
7#11249C212yesmaximal, complemented
8#10896C212yesmaximal, complemented
9#10896C212yesmaximal, complemented
10#10728C211yesmaximal, characteristic
11#10727C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428C2212yes—
17C2×#428C2212yes—
18C2×#407C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C2×#405C2211yescharacteristic
23#2280C2211yescharacteristic
24#1854—22no—
25#1854—22no—
26#1853—22no—
27#1852—22no—
28#1811C418yescomplemented
29#1753—22no—
30#1753—22no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#341C2212yes—
36#1736—22no—
37#1736—22no—
38#1736C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42#1729C2211yescharacteristic
43#1729C2211yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C22×M16—42no—
47C22×M16—42no—
48C22×M16C2×C414yes—
49C22×M16D412yes—
50C22×M16D412yes—
51C22×M16Q814yes—
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—24no—
61C2×#117—42no—
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C2×#108—42no—
74C2×#108—42no—
75C2×#108C2×C418yescomplemented
76#428—24no—
77#428—24no—
78#428—42no—
79#428—42no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#106—22no—
83C2×#106—22no—
84#407—22no—
85#407—22no—
86#407—22no—
87#407C2×C414yes—
88#407C2×C414yes—
89#407Q814yes—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93#405—42no—
94#405—42no—
95C2×#97—21no—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99#381—21no—
100#380D412yes—
101#380D412yes—
102#379—21no—
103C4×C22⋊C4—22no—
104#343—24no—
105#343—24no—
106#343—42no—
107#343—42no—
108#341C2×C418yescomplemented
109#340—24no—
110#340—22no—
111#340—22no—
112#340—24no—
113#340—22no—
114#340—22no—
115#336—22no—
116#336—22no—
117C25#4111yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—44no—
125C23×C4—42no—
126C23×C4—84no—
127C23×C4C22×C412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×Q812yes—
131C23×C4C2×Q812yes—
132C23×C4#4112yes—
133C23×C4#4111yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
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—48no—
152C2×M16—42no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—42no—
156C2×M16—48no—
157C2×M16C4⋊C418yes—
158C2×M16C4⋊C418yes—
159C2×M16C22×C414yes—
160C2×M16C22×C414yes—
161C2×M16C2×Q812yes—
162C2×M16C2×Q812yes—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C22×C412yes—
172C22×C8C22×C412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×Q812yes—
177C22×C8C2×Q812yes—
178#117—28no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—48no—
184#117—44no—
185C2×C4⋊C4—28no—
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—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4#4114yes—
204C2×C4⋊C4#4114yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—44no—
214C2×C42—42no—
215#108—48no—
216#107—44no—
217#107—44no—
218#106—44no—
219#106—44no—
220C22⋊C8—28no—
221C22⋊C8—28no—
222C22⋊C8—28no—
223#97—42no—
224#97—42no—
225C4×C8—42no—
226C4×C8—42no—
227#95—22no—
228#95—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—42no—
233C24C4×Q812yes—
234C24#12112yes—
235C24#12512yes—
236C24C2×#4111yescharacteristic
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—816no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—84no—
269C22×C4C2×C4⋊C414yes—
270C22×C4C4×D414yes—
271C22×C4C4×Q812yes—
272C22×C4#12112yes—
273C22×C4C22⋊Q814yes—
274C22×C4#12512yes—
275C22×C4C4⋊Q814yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×Q811yescharacteristic
280C22×C4C22×Q811yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)12yes—
285C22×C4H(5,𝔽2)12yes—
286M16—24no—
287M16—24no—
288M16—24no—
289M16—24no—
290M16—24no—
291M16—24no—
292M16—48no—
293M16—48no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8C2×C4⋊C412yes—
323C2×C8C2×C4⋊C412yes—
324C2×C8C2×C4⋊C414yes—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C22⋊C4—48no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—48no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23#38914yes—
365C23C2×C4×Q811yescharacteristic
366C23C2×#12111yescharacteristic
367C23C2×#12511yescharacteristic
368C23#55112yes—
369C23#55112yes—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—24no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—816no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4#38914yes—
398C2×C4C22×C4⋊C411yescharacteristic
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C4×#4112yes—
401C2×C4#51812yes—
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4#52511yescharacteristic
404C2×C4#52511yescharacteristic
405C2×C4C2×C4⋊Q811yescharacteristic
406C2×C4#53312yes—
407C2×C4#53411yescharacteristic
408C2×C4#53411yescharacteristic
409C2×C4#54312yes—
410C2×C4#54612yes—
411C2×C4#54612yes—
412C2×C4#55312yes—
413C2×C4#55312yes—
414C2×C4#55412yes—
415C2×C4#55412yes—
416C8—24no—
417C8—22no—
418C8—22no—
419C8—42no—
420C8—42no—
421C8—44no—
422C8—44no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—48no—
433C22—48no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22C2×#38911yescharacteristic
439C22#236812yes—
440C22#253411yescharacteristic
441C22#253411yescharacteristic
442C22#256912yes—
443C22#306812yes—
444C22#306912yes—
445C4—24no—
446C4—81no—
447C4—84no—
448C4—81no—
449C4—82no—
450C4—84no—
451C4#229611yescharacteristic
452C4#231511yescharacteristic
453C4#236412yes—
454C4#254912yes—
455C4#257911yescharacteristic
456C4#257911yescharacteristic
457C2—22no—
458C2—22no—
459C2—44no—
460C2#1414911yescharacteristic
461C2#2118212yes—
462C2#2813711yescharacteristic
463C2#2813811yescharacteristic
464C2#2813912yes—
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. #11249C2semidirect
7. #11249C2semidirect
8. #10896C2semidirect
9. #10896C2semidirect
28. #1811C4semidirect
75. C2×#108C2×C4semidirect
108. #341C2×C4semidirect

Maximal subgroups

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

subgroupindex
#208452
#208442
C2×#18112
#203862
C2×#17362
#112492
#108962
#107282
#107272