Finite Group Databasebeta

← #378608

Subgroups — #378608

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2061C211yesmaximal, characteristic
2#21315C212yesmaximal
3#20453C212yesmaximal
4#11372C211yesmaximal, characteristic
5#11216C212yesmaximal
6#10899C212yesmaximal
7#10896C212yesmaximal
8#10892C211yesmaximal, characteristic
9#10887C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#428C2212yes—
12C2×#428C2212yes—
13C2×#424C2212yes—
14C2×#405C2212yes—
15C2×#382C2211yescharacteristic
16#2274C2212yes—
17#2274C2211yescharacteristic
18#2061C418yescomplemented
19#1989—22no—
20#1989—22no—
21#1988—22no—
22#1987—22no—
23#1776—22no—
24#1776—22no—
25C4×#107—22no—
26C4×#106—22no—
27C2×#343C2212yes—
28C2×#343C2211yescharacteristic
29C2×#343C2211yescharacteristic
30C2×#341C2212yes—
31#1736C2212yes—
32#1736C2212yes—
33#1735C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38C2×#336C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C22×M16C2312yes—
41C22×M16C2312yes—
42C23×C8C2312yes—
43#544—24no—
44#544—41no—
45#544C2×C414yes—
46C2×C4⋊Q8—22no—
47C2×#125—22no—
48C2×C22⋊Q8—41no—
49C2×#117C2312yes—
50C2×#117C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55#428—24no—
56#428—24no—
57#428C2×C418yes—
58#428Q818yescomplemented
59C2×#107—22no—
60C2×#107—22no—
61C2×#106—22no—
62C2×#106—22no—
63#424—24no—
64#424—42no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67#405—24no—
68#405—24no—
69C2×#97—22no—
70C2×C4×C8—22no—
71#382—41no—
72#382C2×C414yes—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C22⋊C4—42no—
80C4×C22⋊C4—41no—
81C2×#95—41no—
82C2×#95—41no—
83#343—24no—
84#343—41no—
85#343C2×C414yes—
86#341—24no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340D412yes—
96#340D412yes—
97#340D412yes—
98#340D412yes—
99#339—24no—
100#339C2×C418yes—
101#336—41no—
102C22×Q8—41no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4C2411yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16C22×C414yes—
118C2×M16C2×Q814yes—
119C2×M16#4114yes—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C2×D412yes—
129C22×C8C2×D412yes—
130C4⋊Q8—28no—
131C4⋊Q8—44no—
132#125—28no—
133#125—44no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117C22×C414yes—
141#117C2×Q814yes—
142#117#4114yes—
143#117#4114yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×Q814yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42C22×C412yes—
175#107—28no—
176#107—28no—
177#106—28no—
178#106—28no—
179C4⋊C8—44no—
180C4⋊C8—44no—
181#97—28no—
182C4×C8—28no—
183#95—44no—
184#95—41no—
185#95—41no—
186#95—42no—
187#95—42no—
188#95—82no—
189C24H(5,𝔽2)11yescharacteristic
190C2×Q8—44no—
191C2×Q8—161no—
192C2×Q8—161no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—82no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D412yes—
218C22×C4C22×Q812yes—
219C22×C4C2×#4112yes—
220C22×C4C2×#4112yes—
221C22×C4C2×#4112yes—
222C22×C4H(5,𝔽2)11yescharacteristic
223C22×C42−1+412yes—
224M16—24no—
225M16—24no—
226M16—24no—
227M16—42no—
228M16—42no—
229M16—42no—
230M16—42no—
231M16C4×Q818yes—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4C4×D414yes—
263C4⋊C4C4×D414yes—
264C4⋊C4C22⋊Q814yes—
265C4⋊C4C22⋊Q814yes—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C42—24no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—22no—
275C42—22no—
276C42—42no—
277C42—162no—
278C42—162no—
279C42—162no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23#51811yescharacteristic
284C23#53912yes—
285C23D4211yescharacteristic
286C23#55111yescharacteristic
287C23#55412yes—
288Q8—164no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—162no—
318C2×C4—161no—
319C2×C4—161no—
320C2×C4C2×C4×D412yes—
321C2×C4C2×C4×Q812yes—
322C2×C4C4×#4111yescharacteristic
323C2×C4#51911yescharacteristic
324C2×C4C2×C22⋊Q812yes—
325C2×C4#52512yes—
326C2×C4#52912yes—
327C2×C4#53312yes—
328C2×C4#54212yes—
329C2×C4#54612yes—
330C2×C4#54712yes—
331C2×C4D4×Q812yes—
332C2×C4#55511yescharacteristic
333C2×C4#55612yes—
334C2×C4#55712yes—
335C2×C4Q8211yescharacteristic
336C8—28no—
337C8—42no—
338C8—42no—
339C8—82no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22#233211yescharacteristic
349C22#235811yescharacteristic
350C22#236911yescharacteristic
351C22#244912yes—
352C22#258812yes—
353C22#328811yescharacteristic
354C22#328911yescharacteristic
355C22#329012yes—
356C4—44no—
357C4—44no—
358C4—44no—
359C4—44no—
360C4—82no—
361C4—84no—
362C4—82no—
363C4—82no—
364C4C4×C22⋊Q812yes—
365C4#236512yes—
366C4#245112yes—
367C4#247412yes—
368C2—21no—
369C2—22no—
370C2—21no—
371C2#1421911yescharacteristic
372C2#2289011yescharacteristic
373C2#2289511yescharacteristic
374C2#2640212yes—
375C2#2640312yes—
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
18. #2061C4semidirect
58. #428Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20612
#213152
#204532
#113722
#112162
#108992
#108962
#108922
#108872