Finite Group Databasebeta

← #360609

Subgroups — #360609

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2#20661C212yesmaximal
3C22×#343C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#11342C212yesmaximal, complemented
6#11296C212yesmaximal, complemented
7#10899C214yesmaximal, complemented
8#10732C212yesmaximal
9C23×M16C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C2×#428C2214yes—
12#2345—24no—
13#2344—22no—
14#2344—22no—
15#2316C2212yes—
16#2313—22no—
17#2280—22no—
18#2280C2212yes—
19C2×#381—22no—
20C2×#380C2211yescharacteristic
21#2274C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1809—24no—
24#1808—22no—
25#1807—22no—
26C2×#343C418yescomplemented
27C2×#343C2211yescharacteristic
28C2×#343C2211yescharacteristic
29C2×#343C2212yes—
30#1736C2214yes—
31C2×#339C2214yes—
32C2×#339C2214yes—
33#1445—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C24×C4D412yes—
37C24×C4C2311yescharacteristic
38C22×M16—22no—
39C22×M16—24no—
40C22×M16—28no—
41C22×M16—44no—
42C22×M16C2×C418yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117C2314yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4Q814yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C42C2312yes—
61#428—28no—
62#428—44no—
63C2×#107—24no—
64C2×#106—24no—
65#389—24no—
66#389—24no—
67#387—28no—
68#387—28no—
69#381—24no—
70#381—22no—
71#381—48no—
72#380—24no—
73#380—24no—
74#380—22no—
75#380—22no—
76#380D412yes—
77#380D412yes—
78#380D414yes—
79#379—22no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—24no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—21no—
90C2×#95—21no—
91#343C2×C414yes—
92#343C2×C418yes—
93#343C2×C414yes—
94#340—24no—
95#340—24no—
96#339—28no—
97#339—44no—
98C25C2×D411yescharacteristic
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—44no—
108C23×C4—44no—
109C23×C4C4⋊C418yes—
110C23×C4C22×C412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×Q812yes—
117C23×C4#4114yes—
118C23×C4C2411yescharacteristic
119C2×M16—22no—
120C2×M16—24no—
121C2×M16—28no—
122C2×M16—22no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—216no—
126C2×M16—48no—
127C2×M16—48no—
128C2×M16—416no—
129C2×M16C22×C414yes—
130C2×M16C22×C414yes—
131C2×M16#4118yes—
132C22×C8—22no—
133C22×C8—28no—
134C22×C8—22no—
135C22×C8—48no—
136#117—24no—
137#117—24no—
138#117—24no—
139#117—24no—
140#117—84no—
141#117—84no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4C4⋊C418yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×Q812yes—
164C2×C4⋊C4C2×Q812yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—24no—
182#107—48no—
183#106—48no—
184#95—28no—
185#95—28no—
186#95—24no—
187#95—22no—
188#95—22no—
189#95—48no—
190#95—48no—
191#95—42no—
192#95—42no—
193C24—22no—
194C24—22no—
195C24—22no—
196C24—22no—
197C24—42no—
198C24C2×C4⋊C412yes—
199C24#12011yescharacteristic
200C24#12011yescharacteristic
201C24#12312yes—
202C24C22×D411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—28no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—28no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—416no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—88no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C414yes—
248C22×C4C2×C4⋊C414yes—
249C22×C4C4×D414yes—
250C22×C4C4×D414yes—
251C22×C4#12012yes—
252C22×C4C22⋊Q814yes—
253C22×C4C22⋊Q814yes—
254C22×C4#12312yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D412yes—
257C22×C4C22×Q811yescharacteristic
258C22×C4C2×#4114yes—
259C22×C4C2×#4112yes—
260C22×C4H(5,𝔽2)12yes—
261C22×C42−1+412yes—
262M16—24no—
263M16—28no—
264M16—24no—
265M16—24no—
266M16—416no—
267M16—44no—
268M16#11718yescomplemented
269C2×C8—24no—
270C2×C8—28no—
271C2×C8—24no—
272C2×C8—416no—
273C2×C8—44no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—216no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C42—44no—
296C42—44no—
297C42—44no—
298C42—44no—
299C42—84no—
300C42—84no—
301C23—22no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—24no—
308C23—24no—
309C23—22no—
310C23—24no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23#38512yes—
322C23#38512yes—
323C23#45312yes—
324C23#45512yes—
325C23C22×C4⋊C411yescharacteristic
326C23C2×#12011yescharacteristic
327C23C2×#12311yescharacteristic
328C23#53212yes—
329C23#53712yes—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—416no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—88no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4#38514yes—
371C2×C4C2×#11712yes—
372C2×C4C2×C4×D412yes—
373C2×C4#51811yescharacteristic
374C2×C4#51911yescharacteristic
375C2×C4C2×C22⋊Q812yes—
376C2×C4#53312yes—
377C2×C4#54412yes—
378C2×C4#54614yes—
379C2×C4#54714yes—
380C2×C4#55114yes—
381C2×C4#55214yes—
382C8—28no—
383C8—44no—
384C22—22no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—24no—
393C22—22no—
394C22—22no—
395C22—24no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—48no—
400C22—44no—
401C22—44no—
402C22—42no—
403C22—42no—
404C22#185614yes—
405C22#228511yescharacteristic
406C22C2×#38511yescharacteristic
407C22#231611yescharacteristic
408C22#243012yes—
409C22#249812yes—
410C22C2×#45311yescharacteristic
411C22C2×#45511yescharacteristic
412C22#302012yes—
413C4—24no—
414C4—44no—
415C4—44no—
416C4—84no—
417C4—84no—
418C4—84no—
419C4—162no—
420C4#234212yes—
421C4#234412yes—
422C4#243814yes—
423C2—22no—
424C2—21no—
425C2—24no—
426C2—21no—
427C2—42no—
428C2#1387811yescharacteristic
429C2C2×#185611yescharacteristic
430C2#2085711yescharacteristic
431C2#2551512yes—
432C2#2551712yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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. #11342C2semidirect
6. #11296C2semidirect
7. #10899C2semidirect
26. C2×#343C4semidirect
268. M16#117semidirect

Maximal subgroups

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

subgroupindex
#341782
#206612
C22×#3432
#139042
#113422
#112962
#108992
#107322