Finite Group Databasebeta

← #454446

Subgroups — #454446

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic
3C2×#2043C212yesmaximal, complemented
4C2×#2026C212yesmaximal, complemented
5#11208C212yesmaximal, complemented
6#11050C212yesmaximal, complemented
7#10970C214yesmaximal
8#10713C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×#125C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#117C2211yescharacteristic
13C22×#103C2214yes—
14C22×#103C2214yes—
15C22×C22⋊C8C2212yes—
16#2629—28no—
17#2621—28no—
18C2×#398—22no—
19C2×#398—22no—
20C2×#397—22no—
21C2×#396—22no—
22C2×#391C2212yes—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27#2043—28no—
28#2026—28no—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C23×Q8C2312yes—
34C24×C4C2311yescharacteristic
35C23×C8C2314yes—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8—42no—
39C2×#125—24no—
40C2×#125—24no—
41C2×#125—42no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—28no—
45C2×C22⋊Q8—44no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—42no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#103—28no—
64C2×#103—28no—
65C2×#103—28no—
66C2×#103—44no—
67C2×#103—44no—
68C2×#103—44no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—42no—
72#398—216no—
73#398—216no—
74#397—216no—
75#396—216no—
76#391—24no—
77#391D418yes—
78#389—24no—
79#389D418yes—
80#384—24no—
81#382—24no—
82#380—24no—
83#380—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—24no—
87C2×#95—22no—
88#340—28no—
89#340—44no—
90#336—24no—
91#336—24no—
92C25C2×D411yescharacteristic
93C22×Q8—22no—
94C22×Q8—24no—
95C22×Q8—24no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8C2×D414yes—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4#4114yes—
118C23×C4C2411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—24no—
123C22×C8—24no—
124C22×C8—24no—
125C22×C8—24no—
126C4⋊Q8—416no—
127#125—416no—
128C22⋊Q8—416no—
129C22⋊Q8—416no—
130#117—416no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—84no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C2×D414yes—
174C2×C42C2×D414yes—
175#103—416no—
176#103—416no—
177C22⋊C8—48no—
178#95—28no—
179#95—28no—
180#95—28no—
181#95—216no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—24no—
189C24—24no—
190C24#12412yes—
191C24C22×D411yescharacteristic
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—28no—
195C2×Q8—28no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—44no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4#12014yes—
255C22×C4#12014yes—
256C22×C4#12412yes—
257C22×C4C22×D412yes—
258C22×C4C22×D412yes—
259C22×C4C2×#4112yes—
260C22×C4H(5,𝔽2)11yescharacteristic
261C22×C4H(5,𝔽2)12yes—
262C22×C4H(5,𝔽2)12yes—
263C22×C42−1+411yescharacteristic
264C22×C42−1+412yes—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—44no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—48no—
277C4⋊C4—416no—
278C4⋊C4—416no—
279C4⋊C4—416no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—88no—
285C4⋊C4—88no—
286C22⋊C4—48no—
287C22⋊C4—416no—
288C22⋊C4—88no—
289C42—28no—
290C42—28no—
291C23—28no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—44no—
308C23C2×#12011yescharacteristic
309C23C2×#12011yescharacteristic
310C23C2×#12411yescharacteristic
311C23#53411yescharacteristic
312C23#53511yescharacteristic
313C23#53611yescharacteristic
314C23#53711yescharacteristic
315C23#57714yes—
316C23#57814yes—
317Q8—48no—
318Q8—48no—
319Q8—84no—
320Q8—84no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—28no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4#53512yes—
363C2×C4#53612yes—
364C2×C4#53712yes—
365C2×C4#53712yes—
366C2×C4#53812yes—
367C2×C4#53912yes—
368C2×C4D4214yes—
369C2×C4#54614yes—
370C2×C4#54814yes—
371C2×C4#56214yes—
372C8—84no—
373C22—44no—
374C22—44no—
375C22—42no—
376C22—44no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22#240311yescharacteristic
383C22#243511yescharacteristic
384C22#243611yescharacteristic
385C22#261511yescharacteristic
386C22#261611yescharacteristic
387C22#263911yescharacteristic
388C22#281611yescharacteristic
389C22#300912yes—
390C22#301112yes—
391C22#301314yes—
392C22#313714yes—
393C22#320114yes—
394C22#322112yes—
395C22#322312yes—
396C22#324214yes—
397C22#326012yes—
398C22#326112yes—
399C4—44no—
400C4—44no—
401C4—42no—
402C4—84no—
403C4—84no—
404C4—84no—
405C4—84no—
406C4#267614yes—
407C4#268114yes—
408C2—82no—
409C2#1625311yescharacteristic
410C2#2392712yes—
411C2#2586812yes—
412C2#2591412yes—
413C2#2967311yescharacteristic
414C2#2967411yescharacteristic
415C2#2970112yes—
416C2#3019612yes—
417C2#3278412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2043C2semidirect
4. C2×#2026C2semidirect
5. #11208C2semidirect
6. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#26292
C2×#20432
C2×#20262
#112082
#110502
#109702
#107132