Finite Group Databasebeta

← #379385

Subgroups — #379385

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2060C211yesmaximal, characteristic
2#21326C212yesmaximal
3#21160C212yesmaximal
4#11339C212yesmaximal
5#11210C211yesmaximal, characteristic
6#10896C212yesmaximal
7#10896C212yesmaximal
8#10843C211yesmaximal, characteristic
9#10707C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#428C2212yes—
12C2×#428C2212yes—
13C2×#405C2212yes—
14C2×#405C2212yes—
15C22×#97C2211yescharacteristic
16#2317C2212yes—
17#2316C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20#2060C418yescomplemented
21#1993—22no—
22#1992—22no—
23#1991—22no—
24#1990—22no—
25#1944—22no—
26#1943—22no—
27#1942—22no—
28#1941—22no—
29C2×#343C2212yes—
30C2×#343C2211yescharacteristic
31C2×#343C2211yescharacteristic
32C2×#341C2212yes—
33#1736C2212yes—
34#1736C2212yes—
35#1735C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
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⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42C2311yescharacteristic
56#428—24no—
57#428—24no—
58#428—42no—
59#428C2×C418yes—
60C2×#107—22no—
61C2×#107—22no—
62C2×#106—22no—
63C2×#106—22no—
64#405—24no—
65#405—24no—
66#405—42no—
67#405—42no—
68C2×#97—22no—
69C2×#97—22no—
70C2×#97—41no—
71C2×#97—41no—
72C2×#97Q814yescomplemented
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#385—42no—
80#384—22no—
81#384—41no—
82#384C2×C414yes—
83#382—22no—
84#382—22no—
85#382—41no—
86#380—41no—
87C2×#95—41no—
88#343—24no—
89#343—41no—
90#343C2×C414yes—
91#341Q818yescomplemented
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340D412yes—
101#340D412yes—
102#340D412yes—
103#340D412yes—
104#339—24no—
105#339C2×C418yes—
106C22×Q8—41no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C2411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16C22×C414yes—
121C2×M16C2×Q814yes—
122C2×M16C2×Q814yes—
123C2×M16#4114yes—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×D412yes—
131C22×C8C2×D412yes—
132C22×C8C2×Q814yes—
133C22×C8C2×Q814yes—
134C4⋊Q8—28no—
135C4⋊Q8—44no—
136#125—28no—
137#125—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117C22×C414yes—
147#117#4114yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42C22×C412yes—
181C2×C42C2×Q812yes—
182C2×C42#4112yes—
183#107—28no—
184#107—44no—
185#106—28no—
186#106—44no—
187#97—44no—
188#97—44no—
189#97—44no—
190C4×C8—44no—
191C4×C8—44no—
192#95—41no—
193#95—41no—
194#95—42no—
195#95—42no—
196#95—44no—
197C24H(5,𝔽2)11yescharacteristic
198C2×Q8—44no—
199C2×Q8—162no—
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—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D412yes—
224C22×C4C22×Q811yescharacteristic
225C22×C4C22×Q812yes—
226C22×C4C2×#4112yes—
227C22×C4C2×#4111yescharacteristic
228C22×C4C2×#4112yes—
229C22×C4H(5,𝔽2)11yescharacteristic
230C22×C4H(5,𝔽2)11yescharacteristic
231C22×C42−1+411yescharacteristic
232M16—24no—
233M16—24no—
234M16—24no—
235M16—24no—
236M16—24no—
237M16—42no—
238M16—42no—
239M16C4×Q818yes—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8C4⋊Q814yes—
255C2×C8C4⋊Q814yes—
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—44no—
265C4⋊C4—48no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—88no—
273C4⋊C4—162no—
274C4⋊C4—162no—
275C4⋊C4—162no—
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—
289C42C4×Q814yes—
290C23—22no—
291C23—22no—
292C23#51811yescharacteristic
293C23#53811yescharacteristic
294C23#54411yescharacteristic
295C23D4211yescharacteristic
296C23#55111yescharacteristic
297C23#55511yescharacteristic
298C23Q8211yescharacteristic
299Q8—164no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—161no—
328C2×C4—161no—
329C2×C4—162no—
330C2×C4C2×C4×D412yes—
331C2×C4C2×C4×Q811yescharacteristic
332C2×C4C2×C4×Q812yes—
333C2×C4#52011yescharacteristic
334C2×C4C2×C4⋊Q812yes—
335C2×C4#53212yes—
336C2×C4#53312yes—
337C2×C4#53312yes—
338C2×C4#53911yescharacteristic
339C2×C4#54311yescharacteristic
340C2×C4#54612yes—
341C2×C4#54612yes—
342C2×C4D4×Q812yes—
343C2×C4#55212yes—
344C2×C4#55412yes—
345C2×C4#55412yes—
346C8—28no—
347C8—24no—
348C8—24no—
349C8—44no—
350C22—22no—
351C22—22no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22#232911yescharacteristic
359C22#235811yescharacteristic
360C22#236911yescharacteristic
361C22#255111yescharacteristic
362C22#255511yescharacteristic
363C22#258611yescharacteristic
364C22#261211yescharacteristic
365C22#328812yes—
366C22#328912yes—
367C4—44no—
368C4—44no—
369C4—44no—
370C4—82no—
371C4—82no—
372C4—82no—
373C4—82no—
374C4—162no—
375C4C4×C4⋊Q812yes—
376C4#236012yes—
377C4#254912yes—
378C4#255612yes—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2#1423611yescharacteristic
384C2#2289111yescharacteristic
385C2#2289211yescharacteristic
386C2#2882612yes—
387C2#2894212yes—
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
20. #2060C4semidirect
72. C2×#97Q8semidirect
91. #341Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20602
#213262
#211602
#113392
#112102
#108962
#108432
#107072