Finite Group Databasebeta

← #110032

Subgroups — #110032

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20524C211yesmaximal, characteristic
2#14130C211yesmaximal, characteristic
3#10727C212yesmaximal
4C2×#1305C212yesmaximal
5#7911C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9C2×#389—21no—
10C2×#385C412yes—
11C2×#380C2211yescharacteristic
12C2×C4×C22⋊C4C2211yescharacteristic
13#1753—24no—
14#1749—21no—
15C4×C22⋊C8—21no—
16C2×#343—24no—
17#1444—21no—
18#1442—21no—
19C4⋊#95—21no—
20C22⋊#95C412yes—
21#1305C418yescomplemented
22#1280—21no—
23#1280—21no—
24C24×C4C2×C412yes—
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C22×M16D414yes—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—44no—
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2×C412yes—
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407C2×C414yes—
46#407C2×C414yes—
47#407Q814yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×#97—22no—
53C2×C4×C8—22no—
54#389—44no—
55#385—44no—
56#380—22no—
57#380C2×C414yes—
58C4×C22⋊C4—22no—
59C4×C22⋊C4—22no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95D411yescharacteristic
65C2×#95D411yescharacteristic
66#343—48no—
67#336—22no—
68#336—22no—
69C25C22×C411yescharacteristic
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—24no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—44no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—82no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C412yes—
84C23×C4C2×D412yes—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×Q812yes—
88C23×C4#4112yes—
89C23×C4#4112yes—
90C23×C4#4112yes—
91C2×M16—24no—
92C2×M16—24no—
93C2×M16—42no—
94C2×M16—44no—
95C2×M16—44no—
96C2×M16—44no—
97C2×M16—44no—
98C2×M16C4⋊C418yescomplemented
99C22×C8—24no—
100C22×C8—42no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—48no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—82no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4C4214yes—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—42no—
131C2×C42—42no—
132C22⋊C8—24no—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140#97—82no—
141C4×C8—82no—
142#95—24no—
143#95—22no—
144#95—22no—
145#95—44no—
146#95—44no—
147#95—82no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24C2×C4211yescharacteristic
157C24C2×C22⋊C411yescharacteristic
158C24#11711yescharacteristic
159C24C4×D412yes—
160C24C4×Q812yes—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—28no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—88no—
208C22×C4—88no—
209C22×C4—84no—
210C22×C4—84no—
211C22×C4#10012yes—
212C22×C4#10112yes—
213C22×C4C2×C4⋊C412yes—
214C22×C4#11712yes—
215C22×C4C4×D412yes—
216C22×C4C4×D412yes—
217C22×C4C4×D412yes—
218C22×C4C4×Q812yes—
219C22×C4#12011yescharacteristic
220C22×C4#12112yes—
221C22×C4C22⋊Q812yes—
222C22×C4C22⋊Q812yes—
223C22×C4#12312yes—
224C22×C4#12311yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12512yes—
227C22×C4#12612yes—
228C22×C4C4⋊Q811yescharacteristic
229M16—48no—
230M16—48no—
231M16—84no—
232M16—84no—
233C2×C8—28no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—82no—
238C2×C8—82no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—48no—
242C4⋊C4—44no—
243C4⋊C4—88no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C42—42no—
249C42—42no—
250C42—81no—
251C42—81no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23C4×C22⋊C411yescharacteristic
283C23C4×C4⋊C412yes—
284C23#38611yescharacteristic
285C23#38712yes—
286C23#39111yescharacteristic
287C23C2×#10011yescharacteristic
288C23C2×#10111yescharacteristic
289C23#41311yescharacteristic
290C23#41311yescharacteristic
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—88no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—88no—
344C2×C4#38512yes—
345C2×C4#38712yes—
346C2×C4#38812yes—
347C2×C4#38912yes—
348C2×C4B(4,𝔽2)12yes—
349C2×C4#45812yes—
350C8—48no—
351C8—84no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—44no—
373C22—44no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22#143611yescharacteristic
381C22C4×#10011yescharacteristic
382C22C4×#10111yescharacteristic
383C22#188911yescharacteristic
384C22#189011yescharacteristic
385C22#190011yescharacteristic
386C22#190111yescharacteristic
387C22#191411yescharacteristic
388C22#191511yescharacteristic
389C22#199711yescharacteristic
390C22#199811yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—84no—
397C4—81no—
398C4—84no—
399C4—81no—
400C4#186714yes—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—42no—
406C2—42no—
407C2#950411yescharacteristic
408C2#950611yescharacteristic
409C2#950711yescharacteristic
410C2#950911yescharacteristic
411C2#1091311yescharacteristic
412C2#1091611yescharacteristic
413C2#1091811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1305C4semidirect
98. C2×M16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205242
#141302
#107272
C2×#13052
#79112