Finite Group Databasebeta

← #379384

Subgroups — #379384

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2060C211yesmaximal, characteristic
2#21326C212yesmaximal
3#21160C212yesmaximal
4#11340C212yesmaximal
5#11210C211yesmaximal, characteristic
6#10896C212yesmaximal
7#10896C212yesmaximal
8#10841C211yesmaximal, characteristic
9#10706C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#428C2212yes—
12C2×#428C2212yes—
13C2×#405C2212yes—
14C2×#405C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2317C2212yes—
17#2316C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
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⋊C4—41no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#428—24no—
58#428—24no—
59#428—42no—
60#428C2×C418yes—
61C2×#107—22no—
62C2×#107—22no—
63C2×#106—22no—
64C2×#106—22no—
65#405—24no—
66#405—24no—
67#405—42no—
68#405—42no—
69C2×#97—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—41no—
74C2×C4×C8—41no—
75C2×C4×C8Q814yescomplemented
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#385—42no—
81#384—22no—
82#384—22no—
83#384—41no—
84#384—41no—
85#384C2×C414yes—
86#382—22no—
87#380—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—44no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42C22×C412yes—
186C2×C42C2×Q812yes—
187C2×C42#4112yes—
188#107—28no—
189#107—44no—
190#106—28no—
191#106—44no—
192#97—44no—
193#97—44no—
194C4×C8—44no—
195C4×C8—44no—
196C4×C8—44no—
197#95—42no—
198#95—42no—
199#95—42no—
200C24H(5,𝔽2)11yescharacteristic
201C2×Q8—44no—
202C2×Q8—161no—
203C2×Q8—161no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4C23×C411yescharacteristic
231C22×C4C22×D412yes—
232C22×C4C22×Q811yescharacteristic
233C22×C4C22×Q812yes—
234C22×C4C2×#4112yes—
235C22×C4C2×#4111yescharacteristic
236C22×C4C2×#4112yes—
237C22×C4H(5,𝔽2)11yescharacteristic
238C22×C4H(5,𝔽2)11yescharacteristic
239C22×C42−1+411yescharacteristic
240M16—24no—
241M16—24no—
242M16—24no—
243M16—24no—
244M16—24no—
245M16—42no—
246M16—42no—
247M16C4×Q818yes—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8C4⋊Q814yes—
263C2×C8C4⋊Q814yes—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—88no—
281C4⋊C4—88no—
282C4⋊C4—162no—
283C4⋊C4—162no—
284C4⋊C4—161no—
285C4⋊C4—161no—
286C4⋊C4C4×D414yes—
287C4⋊C4C4×D414yes—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C42—24no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—24no—
297C42—42no—
298C42—42no—
299C42C4×Q814yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23#51811yescharacteristic
305C23#53811yescharacteristic
306C23#54411yescharacteristic
307C23D4211yescharacteristic
308C23#55111yescharacteristic
309C23#55511yescharacteristic
310C23Q8211yescharacteristic
311Q8—164no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
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—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—84no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—161no—
344C2×C4—161no—
345C2×C4—162no—
346C2×C4C2×C4×D412yes—
347C2×C4C2×C4×Q811yescharacteristic
348C2×C4C2×C4×Q812yes—
349C2×C4#52011yescharacteristic
350C2×C4C2×C4⋊Q812yes—
351C2×C4#53212yes—
352C2×C4#53312yes—
353C2×C4#53312yes—
354C2×C4#53911yescharacteristic
355C2×C4#54311yescharacteristic
356C2×C4#54612yes—
357C2×C4#54612yes—
358C2×C4D4×Q812yes—
359C2×C4#55212yes—
360C2×C4#55412yes—
361C2×C4#55412yes—
362C8—28no—
363C8—24no—
364C8—24no—
365C8—44no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22#232911yescharacteristic
379C22#235811yescharacteristic
380C22#236911yescharacteristic
381C22#255111yescharacteristic
382C22#255511yescharacteristic
383C22#258611yescharacteristic
384C22#261211yescharacteristic
385C22#329111yescharacteristic
386C22#329111yescharacteristic
387C22#329211yescharacteristic
388C22#329311yescharacteristic
389C4—28no—
390C4—44no—
391C4—44no—
392C4—82no—
393C4—82no—
394C4—82no—
395C4—82no—
396C4—162no—
397C4C4×C4⋊Q812yes—
398C4#236012yes—
399C4#254912yes—
400C4#255612yes—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2#1423611yescharacteristic
406C2#2289411yescharacteristic
407C2#2289711yescharacteristic
408C2#2882411yescharacteristic
409C2#2882511yescharacteristic
410C2#2893811yescharacteristic
411C2#2893911yescharacteristic
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
75. C2×C4×C8Q8semidirect
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
#113402
#112102
#108962
#108412
#107062