Finite Group Databasebeta

← #379386

Subgroups — #379386

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2061C211yesmaximal, characteristic
2#21326C212yesmaximal
3#21160C212yesmaximal
4#11340C212yesmaximal
5#11211C211yesmaximal, 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#2061C418yescomplemented
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—22no—
82#384—41no—
83#382—22no—
84#382—41no—
85#382C2×C414yes—
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—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42C22×C412yes—
183C2×C42C2×Q812yes—
184C2×C42#4112yes—
185#107—28no—
186#107—44no—
187#106—28no—
188#106—44no—
189#97—44no—
190#97—44no—
191#97—44no—
192C4×C8—44no—
193C4×C8—44no—
194#95—44no—
195#95—42no—
196#95—42no—
197#95—42no—
198C24H(5,𝔽2)11yescharacteristic
199C2×Q8—44no—
200C2×Q8—161no—
201C2×Q8—161no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4C23×C411yescharacteristic
229C22×C4C22×D412yes—
230C22×C4C22×Q811yescharacteristic
231C22×C4C22×Q812yes—
232C22×C4C2×#4112yes—
233C22×C4C2×#4111yescharacteristic
234C22×C4C2×#4112yes—
235C22×C4H(5,𝔽2)11yescharacteristic
236C22×C4H(5,𝔽2)11yescharacteristic
237C22×C42−1+411yescharacteristic
238M16—24no—
239M16—24no—
240M16—24no—
241M16—24no—
242M16—24no—
243M16—42no—
244M16—42no—
245M16C4×Q818yes—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—24no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8C4⋊Q814yes—
261C2×C8C4⋊Q814yes—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—88no—
279C4⋊C4—162no—
280C4⋊C4—162no—
281C4⋊C4—161no—
282C4⋊C4—161no—
283C4⋊C4C4×D414yes—
284C4⋊C4C4×D414yes—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—24no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—42no—
295C42—42no—
296C42C4×Q814yes—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23#51811yescharacteristic
302C23#53811yescharacteristic
303C23#54411yescharacteristic
304C23D4211yescharacteristic
305C23#55111yescharacteristic
306C23#55511yescharacteristic
307C23Q8211yescharacteristic
308Q8—164no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—42no—
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—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—161no—
341C2×C4—161no—
342C2×C4—162no—
343C2×C4C2×C4×D412yes—
344C2×C4C2×C4×Q811yescharacteristic
345C2×C4C2×C4×Q812yes—
346C2×C4#52011yescharacteristic
347C2×C4C2×C4⋊Q812yes—
348C2×C4#53212yes—
349C2×C4#53312yes—
350C2×C4#53312yes—
351C2×C4#53911yescharacteristic
352C2×C4#54311yescharacteristic
353C2×C4#54612yes—
354C2×C4#54612yes—
355C2×C4D4×Q812yes—
356C2×C4#55212yes—
357C2×C4#55412yes—
358C2×C4#55412yes—
359C8—28no—
360C8—24no—
361C8—24no—
362C8—44no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22#232911yescharacteristic
376C22#235811yescharacteristic
377C22#236911yescharacteristic
378C22#255111yescharacteristic
379C22#255511yescharacteristic
380C22#258611yescharacteristic
381C22#261211yescharacteristic
382C22#328811yescharacteristic
383C22#328811yescharacteristic
384C22#328911yescharacteristic
385C22#328911yescharacteristic
386C4—44no—
387C4—44no—
388C4—44no—
389C4—82no—
390C4—82no—
391C4—82no—
392C4—82no—
393C4—162no—
394C4C4×C4⋊Q812yes—
395C4#236012yes—
396C4#254912yes—
397C4#255612yes—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#1423611yescharacteristic
403C2#2289011yescharacteristic
404C2#2289311yescharacteristic
405C2#2882611yescharacteristic
406C2#2882611yescharacteristic
407C2#2894011yescharacteristic
408C2#2894111yescharacteristic
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. #2061C4semidirect
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×#20612
#213262
#211602
#113402
#112112
#108962
#108432
#107072