Finite Group Databasebeta

← #106137

Subgroups — #106137

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#10727C212yesmaximal
4#7909C212yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C23×C42C411yescharacteristic
10C2×#407C2214yes—
11C2×#389—24no—
12C2×#389—24no—
13C2×#389—24no—
14C22×#95C411yescharacteristic
15C22×#95C2211yescharacteristic
16C2×#343—24no—
17#1463—24no—
18#1463C418yescomplemented
19C2×#324—24no—
20#1280—24no—
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2311yescharacteristic
28C22×M16—24no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—44no—
42C22×C4⋊C4—44no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4C2×C414yes—
45C22×C4⋊C4D414yes—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D414yes—
48C22×C4⋊C4Q812yes—
49C22×C42—22no—
50C22×C42—22no—
51C22×C42—22no—
52C22×C42—44no—
53#407—24no—
54#407—24no—
55#407—24no—
56#407C2×C418yes—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—24no—
59#389—232no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95D414yes—
66#343—48no—
67#324—216no—
68C25C22×C411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4C4212yes—
104C23×C4C4212yes—
105C23×C4C22⋊C412yes—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C4⋊C412yes—
112C23×C4C4⋊C412yes—
113C23×C4C22×C412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×Q811yescharacteristic
119C23×C4#4112yes—
120C23×C4#4112yes—
121C2×M16—28no—
122C2×M16—44no—
123C2×M16—48no—
124C2×M16—48no—
125C22×C8—44no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4C22⋊C418yes—
150C2×C4⋊C4C4⋊C418yes—
151C2×C42—28no—
152C2×C42—48no—
153C2×C42—48no—
154C2×C42—44no—
155C2×C42—44no—
156C22⋊C8—28no—
157C22⋊C8—28no—
158C22⋊C8—44no—
159C22⋊C8—44no—
160#95—28no—
161#95—28no—
162C24—22no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24C2×C4211yescharacteristic
169C24C2×C22⋊C411yescharacteristic
170C24C2×C22⋊C411yescharacteristic
171C24C2×C22⋊C411yescharacteristic
172C24C2×C4⋊C411yescharacteristic
173C24#11711yescharacteristic
174C24#11711yescharacteristic
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—28no—
192C22×C4—28no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—416no—
224C22×C4—416no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4#9514yes—
263C22×C4#9514yes—
264C22×C4#9914yes—
265C22×C4#10014yes—
266C22×C4#10114yes—
267C22×C4#10114yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C4⋊C412yes—
270C22×C4C4×D414yes—
271C22×C4C4×D414yes—
272C22×C4#12014yes—
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4C22⋊Q812yes—
276C22×C4C22⋊Q812yes—
277C22×C4#12314yes—
278M16—416no—
279M16—88no—
280C2×C8—48no—
281C2×C8—84no—
282C2×C8—84no—
283C4⋊C4—216no—
284C4⋊C4—216no—
285C4⋊C4—416no—
286C4⋊C4—416no—
287C4⋊C4—416no—
288C42—416no—
289C23—28no—
290C23—28no—
291C23—28no—
292C23—28no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23C2×#9511yescharacteristic
318C23C4×C22⋊C411yescharacteristic
319C23C4×C22⋊C411yescharacteristic
320C23#37911yescharacteristic
321C23#38011yescharacteristic
322C23#38011yescharacteristic
323C23#38111yescharacteristic
324C23C2×#9912yes—
325C23#41012yes—
326C23C2×#10012yes—
327C23C2×#10112yes—
328C23C2×#10112yes—
329C23#41312yes—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4#34318yescomplemented
387C2×C4#38514yes—
388C2×C4#38614yes—
389C8—88no—
390C22—28no—
391C22—28no—
392C22—28no—
393C22—28no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22C22⋊#9511yescharacteristic
419C22C2×#34312yes—
420C22C4×#9912yes—
421C22C4×#10112yes—
422C22#178111yescharacteristic
423C22#178211yescharacteristic
424C22#178311yescharacteristic
425C22#178411yescharacteristic
426C22#178411yescharacteristic
427C22#178511yescharacteristic
428C22#179912yes—
429C22#180112yes—
430C22#182112yes—
431C4—24no—
432C4—24no—
433C4—28no—
434C4—44no—
435C4—44no—
436C4—44no—
437C4—82no—
438C4—82no—
439C2—22no—
440C2—22no—
441C2—22no—
442C2—22no—
443C2—22no—
444C2—22no—
445C2#854911yescharacteristic
446C2#855111yescharacteristic
447C2#855411yescharacteristic
448C2#855511yescharacteristic
449C2#1072711yescharacteristic
450C2#1076111yescharacteristic
451C2#1076311yescharacteristic
subgroupG/H
center, socleC23#379
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
18. #1463C4semidirect
386. C2×C4#343semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#14632
#107272
#79092