Finite Group Databasebeta

← #106142

Subgroups — #106142

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#387C211yesmaximal, characteristic
2C2×#1821C212yesmaximal
3#7909C212yesmaximal
4#7909C212yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C23×C42C411yescharacteristic
8C2×#407C2214yes—
9C2×#387—24no—
10C2×#387—24no—
11C2×#387—24no—
12C22×#95C411yescharacteristic
13C22×#95C411yescharacteristic
14C22×#95C2211yescharacteristic
15C22×#95C2211yescharacteristic
16#1821—24no—
17#1821C418yescomplemented
18C2×#343—24no—
19#1280—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—44no—
37C22×C4⋊C4C2×C414yes—
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D414yes—
40C22×C4⋊C4Q812yes—
41C22×C42—22no—
42C22×C42—22no—
43C22×C42—22no—
44C22×C42—44no—
45#407—24no—
46#407—24no—
47#407—24no—
48#407C2×C418yes—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51#387—232no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95D414yes—
63C2×#95D414yes—
64#343—216no—
65C25C22×C411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
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×C4C4212yes—
101C23×C4C4212yes—
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C412yes—
104C23×C4C22⋊C412yes—
105C23×C4C22⋊C412yes—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4C4⋊C412yes—
109C23×C4C4⋊C412yes—
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×Q811yescharacteristic
116C23×C4#4112yes—
117C23×C4#4112yes—
118C2×M16—28no—
119C2×M16—44no—
120C2×M16—48no—
121C2×M16—48no—
122C22×C8—44no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—48no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4C22⋊C418yes—
137C2×C4⋊C4C4⋊C418yes—
138C2×C42—28no—
139C2×C42—48no—
140C2×C42—48no—
141C2×C42—44no—
142C2×C42—44no—
143C22⋊C8—28no—
144C22⋊C8—28no—
145C22⋊C8—44no—
146C22⋊C8—44no—
147#95—28no—
148#95—28no—
149#95—28no—
150#95—28no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—22no—
157C24C2×C4211yescharacteristic
158C24C2×C22⋊C411yescharacteristic
159C24C2×C22⋊C411yescharacteristic
160C24C2×C22⋊C411yescharacteristic
161C24C2×C4⋊C411yescharacteristic
162C24#11711yescharacteristic
163C24#11711yescharacteristic
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—416no—
203C22×C4—416no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
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—42no—
249C22×C4—88no—
250C22×C4—88no—
251C22×C4#9514yes—
252C22×C4#9514yes—
253C22×C4#10014yes—
254C22×C4#10114yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C4⋊C412yes—
257C22×C4C4×D414yes—
258C22×C4C4×D414yes—
259C22×C4#12014yes—
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4C22⋊Q812yes—
263C22×C4C22⋊Q812yes—
264C22×C4#12314yes—
265M16—416no—
266M16—88no—
267C2×C8—48no—
268C2×C8—84no—
269C2×C8—84no—
270C4⋊C4—216no—
271C4⋊C4—216no—
272C4⋊C4—416no—
273C42—416no—
274C23—28no—
275C23—28no—
276C23—28no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×#9511yescharacteristic
303C23C4×C22⋊C411yescharacteristic
304C23C4×C22⋊C411yescharacteristic
305C23#37911yescharacteristic
306C23#38011yescharacteristic
307C23#38011yescharacteristic
308C23#38111yescharacteristic
309C23#41012yes—
310C23#41012yes—
311C23C2×#10012yes—
312C23C2×#10112yes—
313C23#41312yes—
314C23#41312yes—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4#38514yes—
371C2×C4#38614yes—
372C8—88no—
373C22—28no—
374C22—28no—
375C22—28no—
376C22—28no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22C22⋊#9511yescharacteristic
402C22#174012yes—
403C22C4×#10012yes—
404C22C4×#10112yes—
405C22#178211yescharacteristic
406C22#178211yescharacteristic
407C22#178311yescharacteristic
408C22#178411yescharacteristic
409C22#178511yescharacteristic
410C22#178511yescharacteristic
411C22#180012yes—
412C22#180112yes—
413C22#182012yes—
414C4—24no—
415C4—24no—
416C4—44no—
417C4—44no—
418C4—44no—
419C4—44no—
420C4—82no—
421C4—82no—
422C2—22no—
423C2—22no—
424C2—22no—
425C2—22no—
426C2—22no—
427C2—22no—
428C2#855311yescharacteristic
429C2#855411yescharacteristic
430C2#855511yescharacteristic
431C2#855611yescharacteristic
432C2#1072611yescharacteristic
433C2#1076211yescharacteristic
434C2#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
17. #1821C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3872
C2×#18212
#79092