Finite Group Databasebeta

← #389274

Subgroups — #389274

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2060C211yesmaximal, characteristic
2#21315C212yesmaximal
3#21036C212yesmaximal
4#11303C212yesmaximal
5#11216C212yesmaximal
6#11210C211yesmaximal, characteristic
7#10892C212yesmaximal
8#10883C212yesmaximal
9#10843C211yesmaximal, characteristic
10C2×#544C2211yescharacteristic
11C2×#428C2212yes—
12C2×#424C2212yes—
13C22×#97C2211yescharacteristic
14#2316C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#2274C2212yes—
18#2060C418yescomplemented
19#1989—22no—
20#1989—22no—
21#1988—22no—
22#1987—22no—
23#1921—22no—
24#1920—22no—
25#1919—22no—
26#1918—22no—
27C2×#343C2212yes—
28C2×#343C2211yescharacteristic
29C2×#343C2211yescharacteristic
30C2×#341C2212yes—
31#1736C2212yes—
32#1736C2212yes—
33#1736C2212yes—
34#1735C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39#1729C2212yes—
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⋊C4—41no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56#428—24no—
57#428C2×C418yes—
58C2×#107—22no—
59C2×#106—22no—
60#424—24no—
61#424—42no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×#97—22no—
65C2×#97—41no—
66C2×#97—41no—
67#389—22no—
68#389—22no—
69#384—41no—
70#384C2×C414yes—
71#382—41no—
72#382Q814yescomplemented
73#380—41no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C22⋊C4—42no—
79C2×#95—41no—
80#343—42no—
81#343—41no—
82#343C2×C414yes—
83#341—42no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340D412yes—
95#340D412yes—
96#340D412yes—
97#340D412yes—
98#339—24no—
99#339C2×C418yes—
100#336—22no—
101#336—22no—
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×M16—22no—
118C2×M16—22no—
119C2×M16C22×C414yes—
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#117—22no—
141#117—22no—
142#117C22×C414yes—
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—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×Q814yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C22×C412yes—
177C2×C42C2×Q812yes—
178C2×C42#4112yes—
179#107—28no—
180#106—28no—
181C4⋊C8—44no—
182C4⋊C8—44no—
183#97—44no—
184#95—44no—
185#95—42no—
186#95—42no—
187#95—41no—
188#95—41no—
189C24H(5,𝔽2)11yescharacteristic
190C2×Q8—44no—
191C2×Q8—162no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D412yes—
216C22×C4C22×Q811yescharacteristic
217C22×C4C2×#4112yes—
218C22×C4C2×#4111yescharacteristic
219C22×C4H(5,𝔽2)11yescharacteristic
220C22×C4H(5,𝔽2)11yescharacteristic
221C22×C42−1+412yes—
222C22×C42−1+412yes—
223C22×C42−1+411yescharacteristic
224M16—24no—
225M16—24no—
226M16—42no—
227M16—42no—
228M16—42no—
229M16—42no—
230M16—42no—
231M16—42no—
232C2×C8—24no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—42no—
240C2×C8—42no—
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—24no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—88no—
265C4⋊C4—162no—
266C4⋊C4C4×D414yes—
267C4⋊C4C4×D414yes—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—24no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—24no—
277C42—42no—
278C42—42no—
279C42—162no—
280C42—162no—
281C42C4×Q814yes—
282C23—22no—
283C23—22no—
284C23#51811yescharacteristic
285C23#53911yescharacteristic
286C23#54311yescharacteristic
287C23D4211yescharacteristic
288C23#55111yescharacteristic
289C23#56311yescharacteristic
290C23#56311yescharacteristic
291Q8—164no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—162no—
320C2×C4—161no—
321C2×C4—161no—
322C2×C4C2×C4×D412yes—
323C2×C4C2×C4×Q811yescharacteristic
324C2×C4#51912yes—
325C2×C4#52011yescharacteristic
326C2×C4#53612yes—
327C2×C4#53712yes—
328C2×C4#53811yescharacteristic
329C2×C4#54112yes—
330C2×C4#54212yes—
331C2×C4#54411yescharacteristic
332C2×C4#54612yes—
333C2×C4#54712yes—
334C2×C4D4×Q812yes—
335C2×C4#55612yes—
336C2×C4#55712yes—
337C2×C4#56312yes—
338C8—44no—
339C8—42no—
340C8—42no—
341C8—82no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22#232911yescharacteristic
351C22#235811yescharacteristic
352C22#238111yescharacteristic
353C22#268011yescharacteristic
354C22#268311yescharacteristic
355C22#278911yescharacteristic
356C22#279911yescharacteristic
357C22#328812yes—
358C22#328912yes—
359C4—44no—
360C4—44no—
361C4—44no—
362C4—84no—
363C4—82no—
364C4—82no—
365C4—82no—
366C4—82no—
367C4#232312yes—
368C4#236512yes—
369C4#271312yes—
370C4#271512yes—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2#1453811yescharacteristic
376C2#2289111yescharacteristic
377C2#2289211yescharacteristic
378C2#3111312yes—
379C2#3125712yes—
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. #2060C4semidirect
72. #382Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20602
#213152
#210362
#113032
#112162
#112102
#108922
#108832
#108432