Finite Group Databasebeta

← #410816

Subgroups — #410816

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2#20911C212yesmaximal
3#14139C211yesmaximal, characteristic
4#11296C214yesmaximal, complemented
5#11055C214yesmaximal, complemented
6#10733C212yesmaximal
7#10727C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C2×#419C2214yes—
12C2×#407C2212yes—
13#2316—24no—
14#2316—22no—
15#2316C2214yes—
16C2×#391C2212yes—
17C2×#389—22no—
18C2×#389—22no—
19C2×#380—22no—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1873—24no—
22#1872—22no—
23#1872—22no—
24C2×#343C2214yes—
25#1736C2214yes—
26C2×#339C2218yes—
27#1441—22no—
28C4⋊#95C2212yes—
29C22⋊#95—22no—
30C23×Q8C2311yescharacteristic
31C24×C4D412yes—
32C24×C4C2311yescharacteristic
33C22×M16C2314yes—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8Q814yes—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—48no—
42C2×#117D414yes—
43C2×#117C2314yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—28no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C42—22no—
59C22×C42C2312yes—
60#419—48no—
61C2×#103—24no—
62C2×#103—24no—
63#407—24no—
64#407—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67#391—28no—
68#389—28no—
69#389—28no—
70#389—28no—
71#389—28no—
72#380—28no—
73#380—28no—
74C4×C22⋊C4—24no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—21no—
78C2×#95—24no—
79#343—28no—
80#340—24no—
81#340—24no—
82#339—48no—
83C25C2×D411yescharacteristic
84C22×Q8—21no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—24no—
88C22×Q8—24no—
89C22×Q8C2×Q812yes—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—28no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—44no—
101C23×C4—82no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D414yes—
106C23×C4C2×Q812yes—
107C23×C4#4114yes—
108C23×C4#4112yes—
109C23×C4C2411yescharacteristic
110C2×M16—28no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—28no—
114C22×C8—24no—
115C22⋊Q8—28no—
116C22⋊Q8—28no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119#117—28no—
120#117—28no—
121#117—44no—
122#117—44no—
123#117—48no—
124#117—48no—
125C2×C4⋊C4—216no—
126C2×C4⋊C4—216no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—84no—
144C2×C4⋊C4—84no—
145C2×C4⋊C4C2×D414yes—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×Q812yes—
148C2×C4⋊C4#4118yes—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—48no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—48no—
165#103—48no—
166#103—48no—
167C22⋊C8—44no—
168C22⋊C8—44no—
169#95—216no—
170#95—48no—
171#95—42no—
172#95—84no—
173C24—22no—
174C24—22no—
175C24—22no—
176C24—22no—
177C24—41no—
178C24—41no—
179C24#12011yescharacteristic
180C24#12011yescharacteristic
181C24C22⋊Q812yes—
182C24#12412yes—
183C24C22×D411yescharacteristic
184C2×Q8—24no—
185C2×Q8—28no—
186C2×Q8—42no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—48no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—28no—
198C22×C4—216no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—816no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4#12012yes—
237C22×C4C22⋊Q814yes—
238C22×C4C22⋊Q814yes—
239C22×C4C22⋊Q812yes—
240C22×C4#12318yes—
241C22×C4#12414yes—
242C22×C4#12414yes—
243C22×C4#12412yes—
244C22×C4C22×D412yes—
245C22×C4C22×Q811yescharacteristic
246C22×C4C2×#4112yes—
247C22×C4C2×#4114yes—
248C22×C4C2×#4111yescharacteristic
249C22×C4H(5,𝔽2)14yes—
250M16—48no—
251M16—48no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C4⋊C4—216no—
256C4⋊C4—216no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—48no—
266C4⋊C4—88no—
267C4⋊C4—88no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—48no—
274C42—48no—
275C42—48no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—24no—
287C23—21no—
288C23—24no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—24no—
293C23—24no—
294C23—42no—
295C23—42no—
296C23—44no—
297C23—44no—
298C23—44no—
299C23—44no—
300C23#39312yes—
301C23#39312yes—
302C23#45512yes—
303C23#45612yes—
304C23C2×#12011yescharacteristic
305C23C2×C22⋊Q811yescharacteristic
306C23C2×#12312yes—
307C23C2×#12411yescharacteristic
308C23#53512yes—
309Q8—48no—
310Q8—84no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—416no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—84no—
338C2×C4—88no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—88no—
344C2×C4—88no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4#39314yes—
352C2×C4C2×C22⋊Q812yes—
353C2×C4#52514yes—
354C2×C4C2×#12412yes—
355C2×C4#53414yes—
356C2×C4#55318yes—
357C2×C4#55314yes—
358C2×C4#55414yes—
359C8—84no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—48no—
378C22—48no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22#202814yes—
384C22C2×#39311yescharacteristic
385C22#248712yes—
386C22#250412yes—
387C22#261311yescharacteristic
388C22#284911yescharacteristic
389C22C2×#45511yescharacteristic
390C22C2×#45611yescharacteristic
391C22#301911yescharacteristic
392C22#302011yescharacteristic
393C4—24no—
394C4—48no—
395C4—84no—
396C4—82no—
397C4—84no—
398C4—84no—
399C4—82no—
400C4#257914yes—
401C4#259114yes—
402C2—22no—
403C2—22no—
404C2—24no—
405C2—42no—
406C2#1509111yescharacteristic
407C2C2×#202811yescharacteristic
408C2#2346211yescharacteristic
409C2#2961811yescharacteristic
410C2#2961911yescharacteristic
411C2#3299311yescharacteristic
412C2#3299411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11296C2semidirect
5. #11055C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#209112
#141392
#112962
#110552
#107332
#107272