Finite Group Databasebeta

← #115782

Subgroups — #115782

Order 512.

938 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1520C212yesmaximal
2#14347C211yesmaximal, characteristic
3#14281C211yesmaximal, characteristic
4C2×#1289C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C22×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2211yescharacteristic
9C2×#380C2212yes—
10C22×#95C2211yescharacteristic
11#1520—24no—
12#1520—24no—
13C2×#328—24no—
14#1446C412yescomplemented
15#1445C412yescomplemented
16#1442—24no—
17#1441—22no—
18C22⋊#95—22no—
19C22⋊#95—22no—
20#1289—24no—
21#1280—22no—
22C23×Q8C2×C411yescharacteristic
23C24×C4D412yes—
24C24×C4C2311yescharacteristic
25C23×C8—22no—
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8—21no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8D412yes—
33C2×C22⋊Q8Q812yes—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C22⋊C4—21no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—21no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8D414yes—
47#380—22no—
48#380—22no—
49#380—22no—
50#380—22no—
51#380—22no—
52#380—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—24no—
56C2×#95—24no—
57C2×#95—24no—
58C2×#95—24no—
59C2×#95C2×C412yes—
60C2×#95C2×C412yes—
61C2×#95D414yes—
62C2×#95D412yes—
63#328—216no—
64C25C2×D411yescharacteristic
65C22×Q8—21no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—82no—
69C22×Q8C4⋊C412yes—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—22no—
77C23×C4—24no—
78C23×C4—82no—
79C23×C4—82no—
80C23×C4—82no—
81C23×C4C22⋊C414yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C23×C4C2×Q811yescharacteristic
88C23×C4#4112yes—
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4112yes—
92C22×C8—24no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—84no—
96C22⋊Q8—24no—
97C22⋊Q8—24no—
98C22⋊Q8—24no—
99C22⋊Q8—24no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—82no—
115C2×C4⋊C4—84no—
116C2×C4⋊C4C4⋊C412yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—44no—
129C2×C22⋊C4—44no—
130C22⋊C8—24no—
131C22⋊C8—24no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—84no—
140#95—84no—
141#95—84no—
142#95—84no—
143C24—41no—
144C24—42no—
145C24—41no—
146C24C2×C22⋊C411yescharacteristic
147C24#12011yescharacteristic
148C24#12111yescharacteristic
149C24C22⋊Q811yescharacteristic
150C24#12311yescharacteristic
151C24#12411yescharacteristic
152C24#12711yescharacteristic
153C24#13514yes—
154C24#13714yes—
155C2×Q8—24no—
156C2×Q8—42no—
157C2×Q8—84no—
158C2×Q8—82no—
159C2×Q8—82no—
160C2×Q8—84no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—48no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—48no—
193C22×C4—81no—
194C22×C4—82no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—82no—
198C22×C4—81no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—88no—
202C22×C4—88no—
203C22×C4C2×C4⋊C411yescharacteristic
204C22×C4#11711yescharacteristic
205C22×C4C4×D412yes—
206C22×C4C4×D412yes—
207C22×C4#12012yes—
208C22×C4#12112yes—
209C22×C4#12112yes—
210C22×C4#12111yescharacteristic
211C22×C4#12112yes—
212C22×C4C22⋊Q812yes—
213C22×C4C22⋊Q812yes—
214C22×C4C22⋊Q811yescharacteristic
215C22×C4#12312yes—
216C22×C4#12312yes—
217C22×C4#12412yes—
218C22×C4#12612yes—
219C2×C8—44no—
220C2×C8—84no—
221C2×C8—84no—
222C2×C8—84no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—84no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—44no—
253C23—44no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—44no—
259C23—44no—
260C23#38011yescharacteristic
261C23#38611yescharacteristic
262C23#39011yescharacteristic
263C23#39211yescharacteristic
264C23#39311yescharacteristic
265C23#39711yescharacteristic
266C23#39911yescharacteristic
267C23#41612yes—
268C23#41912yes—
269C23#45214yes—
270C23#46414yes—
271C23#47914yes—
272C23#48514yes—
273C23#48812yes—
274C23#49112yes—
275C23#49512yes—
276C23#49712yes—
277Q8—84no—
278Q8—84no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—88no—
305C2×C4—82no—
306C2×C4—88no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—84no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—88no—
317C2×C4—84no—
318C2×C4—88no—
319C2×C4—82no—
320C2×C4—84no—
321C2×C4#38512yes—
322C2×C4#38812yes—
323C2×C4#39412yes—
324C2×C4#39612yes—
325C8—84no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22—22no—
343C22—24no—
344C22—22no—
345C22—24no—
346C22—44no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22#143911yescharacteristic
353C22#180512yes—
354C22#187312yes—
355C22#191111yescharacteristic
356C22#191211yescharacteristic
357C22#191312yes—
358C22#195811yescharacteristic
359C22#196011yescharacteristic
360C22#201512yes—
361C22#202612yes—
362C22#207412yes—
363C22#209712yes—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—82no—
369C4—82no—
370C4—84no—
371C4—162no—
372C4—164no—
373C2—21no—
374C2—22no—
375C2—22no—
376C2—22no—
377C2—22no—
378C2—21no—
379C2—22no—
380C2#920112yes—
381C2#920212yes—
382C2#1104611yescharacteristic
383C2#1105211yescharacteristic
384C2#1107211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ 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
14. #1446C4semidirect
15. #1445C4semidirect

Maximal subgroups

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

subgroupindex
C2×#15202
#143472
#142812
C2×#12892
C2×#12802