Finite Group Databasebeta

← #378607

Subgroups — #378607

Order 512.

883 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23560C211yesmaximal, characteristic
2#21307C212yesmaximal
3#20445C212yesmaximal
4#11370C211yesmaximal, characteristic
5#11215C212yesmaximal
6#10901C212yesmaximal
7#10897C212yesmaximal
8#10893C211yesmaximal, characteristic
9#10886C212yesmaximal
10#3458C2211yescharacteristic
11C2×#430C2212yes—
12#2911C2212yes—
13#2873C2212yes—
14#2873C2211yescharacteristic
15C2×C4×M16C2212yes—
16#2296C2212yes—
17#2285C2211yescharacteristic
18#2061C412yescomplemented
19#2061C412yescomplemented
20#2060C412yescomplemented
21#2060C412yescomplemented
22#1986—24no—
23#1986—24no—
24#1773—24no—
25#1772—24no—
26C2×#344C2212yes—
27#1742C2211yescharacteristic
28#1740C2211yescharacteristic
29#1733C2212yes—
30#1733C2212yes—
31C2×#339C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37#1728C2211yescharacteristic
38C4×#95C2211yescharacteristic
39C22×M16C2312yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42#544—22no—
43#544—22no—
44#544C2×C412yes—
45#544C2×C412yes—
46#543—41no—
47#533—24no—
48#533—41no—
49C2×#117C2312yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C2×#117C2312yes—
53C22×C42C2311yescharacteristic
54C22×C42C2312yes—
55#430—24no—
56#430Q818yescomplemented
57C2×#108—24no—
58#428—22no—
59#428—22no—
60#428C2×C414yes—
61#428C2×C414yes—
62#427—24no—
63#405—24no—
64#405—24no—
65#405—42no—
66C4×M16—24no—
67C4×M16—24no—
68C4×M16—24no—
69C4×M16—41no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#389—22no—
73#389—22no—
74#385—42no—
75#384C2×C411yescharacteristic
76#384C2×C411yescharacteristic
77#382C2×C411yescharacteristic
78#382C2×C411yescharacteristic
79#380—41no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—41no—
83C2×#95—41no—
84#344—24no—
85#343C2×C412yes—
86#343C2×C412yes—
87#342—41no—
88#339—24no—
89#339—24no—
90#339—24no—
91#339—24no—
92#339—22no—
93#339—22no—
94#339—22no—
95#339—22no—
96#339C2×C414yes—
97#339C2×C414yes—
98#339D414yes—
99#339D414yes—
100#336—41no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4C2411yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16C22×C414yes—
116C2×M16C2×D414yes—
117C2×M16C2×Q814yes—
118C2×M16C2×Q814yes—
119C2×M16C2×Q814yes—
120C2×M16#4114yes—
121C2×M16#4114yes—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C4⋊Q8—24no—
126C4⋊Q8—24no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129#125—24no—
130#125—24no—
131#125—42no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C4×Q8—42no—
137C4×Q8—42no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117C22×C414yes—
149#117C2×D414yes—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4#4112yes—
160C2×C4⋊C4#4112yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181#108—216no—
182#107—24no—
183#107—24no—
184#106—24no—
185#106—24no—
186#97—28no—
187#97—44no—
188#97—42no—
189C4×C8—28no—
190C4×C8—44no—
191C4×C8—42no—
192#95—41no—
193#95—41no—
194#95—42no—
195#95—42no—
196#95—44no—
197C24H(5,𝔽2)11yescharacteristic
198C2×Q8—42no—
199C2×Q8—42no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4—82no—
218C22×C4C23×C411yescharacteristic
219C22×C4C22×D412yes—
220C22×C4C22×Q812yes—
221C22×C4C2×#4112yes—
222C22×C4C2×#4112yes—
223C22×C4C2×#4112yes—
224C22×C4H(5,𝔽2)11yescharacteristic
225C22×C42−1+412yes—
226M16—24no—
227M16—24no—
228M16—24no—
229M16—24no—
230M16—24no—
231M16—24no—
232M16—24no—
233M16—24no—
234M16—42no—
235M16—42no—
236M16C4×Q818yes—
237M16C22⋊Q818yes—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—84no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—84no—
263C4⋊C4C4×D414yes—
264C4⋊C4C4×D414yes—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—24no—
272C42—24no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—42no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—42no—
288C42—42no—
289C42—42no—
290C23—41no—
291C23—41no—
292C23#51811yescharacteristic
293C23#54212yes—
294C23#54612yes—
295C23#55511yescharacteristic
296C23Q8211yescharacteristic
297Q8—162no—
298Q8—162no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C2×C4×D412yes—
330C2×C4C2×C4×Q812yes—
331C2×C4C4×#4111yescharacteristic
332C2×C4#51911yescharacteristic
333C2×C4C2×C22⋊Q812yes—
334C2×C4#52512yes—
335C2×C4#52912yes—
336C2×C4#53312yes—
337C2×C4#53912yes—
338C2×C4D4211yescharacteristic
339C2×C4#54712yes—
340C2×C4D4×Q812yes—
341C2×C4#55111yescharacteristic
342C2×C4#55412yes—
343C2×C4#55612yes—
344C2×C4#55712yes—
345C8—28no—
346C8—28no—
347C8—44no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22#233211yescharacteristic
356C22#235811yescharacteristic
357C22#236911yescharacteristic
358C22#245112yes—
359C22#247412yes—
360C4—44no—
361C4—42no—
362C4—44no—
363C4—44no—
364C4—82no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—162no—
371C4C4×C22⋊Q812yes—
372C4#236512yes—
373C4#244912yes—
374C4#258812yes—
375C2—22no—
376C2—21no—
377C2—21no—
378C2—22no—
379C2#1421911yescharacteristic
380C2#2289811yescharacteristic
381C2#2289911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2358
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #2061C4semidirect
19. #2061C4semidirect
20. #2060C4semidirect
21. #2060C4semidirect
56. #430Q8semidirect

Maximal subgroups

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

subgroupindex
#235602
#213072
#204452
#113702
#112152
#109012
#108972
#108932
#108862