Finite Group Databasebeta

← #365863

Subgroups — #365863

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1951C211yesmaximal, characteristic
2#21035C212yesmaximal
3#20802C211yesmaximal, characteristic
4#20454C211yesmaximal, characteristic
5#11290C212yesmaximal
6#11217C212yesmaximal
7#10899C212yesmaximal
8#10869C211yesmaximal, characteristic
9#10861C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#427C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2317C2212yes—
22#2274C2212yes—
23#1951C418yescomplemented
24#1921—22no—
25#1920—22no—
26#1919—22no—
27#1918—22no—
28#1847—22no—
29#1846—21no—
30#1844—21no—
31#1776—22no—
32C4×#106—21no—
33C4×#106—21no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2211yescharacteristic
37#1736C2212yes—
38#1736C2212yes—
39#1735C2211yescharacteristic
40C2×#340C2212yes—
41C2×#339C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2211yescharacteristic
44#1729C2212yes—
45C4⋊#95C2212yes—
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117C2312yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57C2×#108—22no—
58C2×#108—22no—
59C2×#108C2×C414yes—
60#428—24no—
61#428—24no—
62#428—42no—
63#428C2×C418yes—
64#427—22no—
65#427—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107D412yes—
69C2×#106—21no—
70C2×#106—21no—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106C2×C414yes—
79C2×#106D411yescharacteristic
80C2×#106D411yescharacteristic
81#405—22no—
82#405—22no—
83#405C2×C414yes—
84#405Q814yes—
85C2×#97—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92#389—22no—
93#389—22no—
94#385—42no—
95#384—22no—
96#382—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C22⋊C4—42no—
102C2×#95—42no—
103#343—42no—
104#343C2×C418yes—
105#341—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340D412yes—
113#340D412yes—
114#339—22no—
115#339—42no—
116#339Q814yes—
117#336—22no—
118#336—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2411yescharacteristic
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—21no—
127C2×M16—22no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—22no—
135C2×M16—21no—
136C2×M16C22×C414yes—
137C2×M16C22×C412yes—
138C2×M16C2×Q812yes—
139C2×M16C2×Q812yes—
140C2×M16#4112yes—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C22×C412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×Q812yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
158C22×C8#4112yes—
159#117—21no—
160#117—21no—
161#117—21no—
162#117—21no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117C22×C412yes—
168#117C2×Q812yes—
169#117C2×Q812yes—
170#117#4112yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42C2×Q812yes—
204#108—28no—
205#107—24no—
206#107—28no—
207#107—44no—
208#106—24no—
209#106—24no—
210#106—22no—
211#106—22no—
212#106—28no—
213#106—44no—
214#106—42no—
215#106—42no—
216#97—24no—
217#97—24no—
218C4×C8—24no—
219C4×C8—24no—
220C4×C8—44no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—84no—
226C24H(5,𝔽2)11yescharacteristic
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×Q811yescharacteristic
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4H(5,𝔽2)11yescharacteristic
259C22×C42−1+411yescharacteristic
260C22×C42−1+411yescharacteristic
261M16—22no—
262M16—22no—
263M16—22no—
264M16—24no—
265M16—24no—
266M16—22no—
267M16—22no—
268M16—42no—
269M16—41no—
270M16—42no—
271M16—41no—
272M16C4×Q814yes—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8#11714yes—
290C2×C8C4×D412yes—
291C2×C8C4×D412yes—
292C2×C8C4×Q814yes—
293C2×C8C22⋊Q812yes—
294C2×C8C22⋊Q812yes—
295C2×C8#12312yes—
296C2×C8#12312yes—
297C2×C8#12514yes—
298C2×C8#12614yes—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—162no—
315C4⋊C4—162no—
316C4⋊C4C4×D414yes—
317C4⋊C4C4⋊Q812yes—
318C4⋊C4C4⋊Q812yes—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—41no—
332C42—42no—
333C42—42no—
334C42—41no—
335C42—162no—
336C42—162no—
337C23—22no—
338C23—22no—
339C23#52011yescharacteristic
340C23#54011yescharacteristic
341C23#54411yescharacteristic
342C23D4211yescharacteristic
343C23#55311yescharacteristic
344C23#55411yescharacteristic
345C23#55911yescharacteristic
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—162no—
373C2×C4—162no—
374C2×C4C2×#11711yescharacteristic
375C2×C4C2×C4×D411yescharacteristic
376C2×C4C2×C4×D411yescharacteristic
377C2×C4C2×C4×Q811yescharacteristic
378C2×C4C2×C4×Q811yescharacteristic
379C2×C4#52011yescharacteristic
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×#12311yescharacteristic
382C2×C4C2×#12511yescharacteristic
383C2×C4C2×#12611yescharacteristic
384C2×C4C2×C4⋊Q811yescharacteristic
385C2×C4#53211yescharacteristic
386C2×C4#53311yescharacteristic
387C2×C4#53311yescharacteristic
388C2×C4#54111yescharacteristic
389C2×C4#54411yescharacteristic
390C2×C4#54611yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4#54711yescharacteristic
393C2×C4#54811yescharacteristic
394C2×C4D4×Q811yescharacteristic
395C2×C4#55011yescharacteristic
396C2×C4#55211yescharacteristic
397C2×C4#55411yescharacteristic
398C2×C4#55511yescharacteristic
399C2×C4#55611yescharacteristic
400C2×C4#55611yescharacteristic
401C2×C4#55711yescharacteristic
402C8—24no—
403C8—44no—
404C8—42no—
405C8—42no—
406C8#38714yes—
407C8#38714yes—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22#233011yescharacteristic
417C22#235511yescharacteristic
418C22#236111yescharacteristic
419C22#244011yescharacteristic
420C22#244511yescharacteristic
421C22#249711yescharacteristic
422C22#257011yescharacteristic
423C22#329012yes—
424C22#329112yes—
425C4—44no—
426C4—44no—
427C4—42no—
428C4—42no—
429C4—84no—
430C4—82no—
431C4—82no—
432C4—162no—
433C4C2×#38711yescharacteristic
434C4C4×C4⋊Q811yescharacteristic
435C4#235411yescharacteristic
436C4#236011yescharacteristic
437C4#243911yescharacteristic
438C4#244611yescharacteristic
439C4#249611yescharacteristic
440C4#255211yescharacteristic
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2#1398111yescharacteristic
446C2#2304412yes—
447C2#2608011yescharacteristic
448C2#2608111yescharacteristic
449C2#2627012yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1951C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19512
#210352
#208022
#204542
#112902
#112172
#108992
#108692
#108612
#108592
#107062