Finite Group Databasebeta

← #389272

Subgroups — #389272

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 405 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#10891C212yesmaximal
8#10882C212yesmaximal
9#10841C211yesmaximal, characteristic
10C2×#544C2211yescharacteristic
11C2×#428C2212yes—
12C2×#424C2212yes—
13C22×C4×C8C2211yescharacteristic
14#2316C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
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⋊C4—41no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57#428—24no—
58#428C2×C418yes—
59C2×#107—22no—
60C2×#106—22no—
61#424—24no—
62#424—42no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4×C8—22no—
66C2×C4×C8—41no—
67C2×C4×C8—41no—
68#389—22no—
69#389—22no—
70#384—41no—
71#384—41no—
72#384C2×C414yes—
73#384Q814yescomplemented
74#380—41no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C22⋊C4—42no—
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—22no—
106C23×C4—21no—
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—44no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×Q814yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C22×C412yes—
182C2×C42C2×Q812yes—
183C2×C42#4112yes—
184#107—28no—
185#106—28no—
186C4⋊C8—44no—
187C4⋊C8—44no—
188C4×C8—44no—
189#95—42no—
190#95—42no—
191#95—42no—
192C24H(5,𝔽2)11yescharacteristic
193C2×Q8—44no—
194C2×Q8—161no—
195C2×Q8—161no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D412yes—
224C22×C4C22×Q811yescharacteristic
225C22×C4C2×#4112yes—
226C22×C4C2×#4111yescharacteristic
227C22×C4H(5,𝔽2)11yescharacteristic
228C22×C4H(5,𝔽2)11yescharacteristic
229C22×C42−1+412yes—
230C22×C42−1+412yes—
231C22×C42−1+411yescharacteristic
232M16—24no—
233M16—24no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240C2×C8—24no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—88no—
273C4⋊C4—88no—
274C4⋊C4—161no—
275C4⋊C4—161no—
276C4⋊C4C4×D414yes—
277C4⋊C4C4×D414yes—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—24no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—24no—
287C42—42no—
288C42—42no—
289C42—162no—
290C42—162no—
291C42C4×Q814yes—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23#51811yescharacteristic
297C23#53911yescharacteristic
298C23#54311yescharacteristic
299C23D4211yescharacteristic
300C23#55111yescharacteristic
301C23#56311yescharacteristic
302C23#56311yescharacteristic
303Q8—164no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—84no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—162no—
336C2×C4—161no—
337C2×C4—161no—
338C2×C4C2×C4×D412yes—
339C2×C4C2×C4×Q811yescharacteristic
340C2×C4#51912yes—
341C2×C4#52011yescharacteristic
342C2×C4#53612yes—
343C2×C4#53712yes—
344C2×C4#53811yescharacteristic
345C2×C4#54112yes—
346C2×C4#54212yes—
347C2×C4#54411yescharacteristic
348C2×C4#54612yes—
349C2×C4#54712yes—
350C2×C4D4×Q812yes—
351C2×C4#55612yes—
352C2×C4#55712yes—
353C2×C4#56312yes—
354C8—44no—
355C8—42no—
356C8—42no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22#232911yescharacteristic
371C22#235811yescharacteristic
372C22#238111yescharacteristic
373C22#268011yescharacteristic
374C22#268311yescharacteristic
375C22#278911yescharacteristic
376C22#279911yescharacteristic
377C22#329111yescharacteristic
378C22#329111yescharacteristic
379C22#329211yescharacteristic
380C22#329311yescharacteristic
381C4—28no—
382C4—44no—
383C4—44no—
384C4—84no—
385C4—82no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4#232312yes—
390C4#236512yes—
391C4#271312yes—
392C4#271512yes—
393C2—21no—
394C2—21no—
395C2—21no—
396C2—21no—
397C2#1453811yescharacteristic
398C2#2289411yescharacteristic
399C2#2289711yescharacteristic
400C2#3111011yescharacteristic
401C2#3111211yescharacteristic
402C2#3125411yescharacteristic
403C2#3125611yescharacteristic
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
73. #384Q8semidirect

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
#108912
#108822
#108412