Finite Group Databasebeta

← #362159

Subgroups — #362159

Order 512.

1146 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#20661C212yesmaximal
2C2×#1809C211yesmaximal, characteristic
3#20386C211yesmaximal, characteristic
4C2×#1738C211yesmaximal, characteristic
5#11331C211yesmaximal, characteristic, complemented
6#11330C211yesmaximal, characteristic, complemented
7#11271C211yesmaximal, characteristic, complemented
8#11270C211yesmaximal, characteristic, complemented
9#10896C212yesmaximal, complemented
10#10896C212yesmaximal, complemented
11#10739C212yesmaximal
12C23×M16C2212yes—
13C23×M16C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2212yes—
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2280C2211yescharacteristic
22C2×#380C2212yes—
23#1809—22no—
24#1809—22no—
25#1809—22no—
26#1809—22no—
27#1808—22no—
28#1807—22no—
29#1753—22no—
30#1753—22no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35#1738C418yescomplemented
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41C2×#340C2212yes—
42#1729C2211yescharacteristic
43#1729C2211yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—24no—
47C22×M16—24no—
48C22×M16—24no—
49C22×M16—42no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16—42no—
53C22×M16C2×C418yes—
54C22×M16C2×C414yes—
55C22×M16D412yes—
56C22×M16D412yes—
57C22×M16Q814yes—
58C22×M16C2312yes—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C23×C8—22no—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70#428—24no—
71#428—24no—
72#428—42no—
73#428—42no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—42no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—42no—
82#407—22no—
83#407Q814yes—
84C2×C22⋊C8—22no—
85#405—42no—
86#405—42no—
87C2×#97—21no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#381—21no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380D412yes—
97#380D412yes—
98#379—21no—
99C4×C22⋊C4—22no—
100C2×#95—22no—
101#343—24no—
102#343—24no—
103#343—42no—
104#343—42no—
105#341C2×C414yes—
106#341C2×C418yes—
107#341C2×C414yes—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#336—22no—
115#336—22no—
116C25#4111yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×Q812yes—
127C23×C4C2×Q812yes—
128C23×C4#4112yes—
129C23×C4#4111yescharacteristic
130C23×C4C2411yescharacteristic
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—28no—
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—216no—
147C2×M16—48no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C2×M16—48no—
153C2×M16—44no—
154C2×M16—44no—
155C2×M16—48no—
156C2×M16—48no—
157C2×M16—48no—
158C2×M16C4⋊C418yes—
159C2×M16C4⋊C418yes—
160C2×M16C22×C414yes—
161C2×M16C22×C414yes—
162C2×M16C2×Q812yes—
163C2×M16C2×Q812yes—
164C2×M16#4114yes—
165C2×M16#4114yes—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—28no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8C22×C412yes—
178C22×C8C22×C412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×Q812yes—
183C22×C8C2×Q812yes—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—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—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—84no—
210C2×C42—21no—
211C2×C42—21no—
212#107—44no—
213#107—44no—
214#107—44no—
215#106—44no—
216#106—44no—
217#106—44no—
218C22⋊C8—28no—
219#97—42no—
220#97—42no—
221C4×C8—42no—
222C4×C8—42no—
223#95—22no—
224#95—22no—
225#95—42no—
226#95—42no—
227C24—22no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24C4×Q812yes—
232C24#12112yes—
233C24#12512yes—
234C24C2×#4111yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C2×C4⋊C414yes—
267C22×C4C4×D414yes—
268C22×C4C4×Q812yes—
269C22×C4#12112yes—
270C22×C4C22⋊Q814yes—
271C22×C4#12512yes—
272C22×C4C4⋊Q814yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×Q811yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C4H(5,𝔽2)12yes—
283M16—24no—
284M16—24no—
285M16—24no—
286M16—24no—
287M16—24no—
288M16—24no—
289M16—24no—
290M16—48no—
291M16—48no—
292M16—416no—
293M16—42no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16#11718yes—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—416no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8C2×C4⋊C414yes—
322C2×C8C2×C4⋊C412yes—
323C2×C8C2×C4⋊C412yes—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—44no—
357C23—44no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23#38914yes—
363C23C2×C4×Q811yescharacteristic
364C23C2×#12111yescharacteristic
365C23C2×#12511yescharacteristic
366C23#55312yes—
367C23#55312yes—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—162no—
394C2×C4—162no—
395C2×C4—162no—
396C2×C4—162no—
397C2×C4#38914yes—
398C2×C4C22×C4⋊C411yescharacteristic
399C2×C4C2×#11712yes—
400C2×C4C2×C4×D411yescharacteristic
401C2×C4#51812yes—
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4C2×C4⋊Q811yescharacteristic
404C2×C4#53211yescharacteristic
405C2×C4#53211yescharacteristic
406C2×C4#53312yes—
407C2×C4#53411yescharacteristic
408C2×C4#53411yescharacteristic
409C2×C4#54312yes—
410C2×C4#54612yes—
411C2×C4#54612yes—
412C2×C4#55112yes—
413C2×C4#55112yes—
414C2×C4#55412yes—
415C2×C4#55412yes—
416C8—22no—
417C8—22no—
418C8—24no—
419C8—28no—
420C8—42no—
421C8—42no—
422C8—44no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—84no—
437C22—84no—
438C22C2×#38911yescharacteristic
439C22#234512yes—
440C22#254311yescharacteristic
441C22#254311yescharacteristic
442C22#256912yes—
443C22#306812yes—
444C22#306912yes—
445C4—24no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4—162no—
450C4—162no—
451C4#228511yescharacteristic
452C4#231511yescharacteristic
453C4#234212yes—
454C4#254912yes—
455C4#256411yescharacteristic
456C4#256411yescharacteristic
457C2—22no—
458C2—22no—
459C2—44no—
460C2#1390811yescharacteristic
461C2#2118212yes—
462C2#2859411yescharacteristic
463C2#2859511yescharacteristic
464C2#2859612yes—
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
5. #11331C2semidirect
6. #11330C2semidirect
7. #11271C2semidirect
8. #11270C2semidirect
9. #10896C2semidirect
10. #10896C2semidirect
35. #1738C4semidirect

Maximal subgroups

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

subgroupindex
#206612
C2×#18092
#203862
C2×#17382
#113312
#113302
#112712
#112702
#108962
#107392