Finite Group Databasebeta

← #115739

Subgroups — #115739

Order 512.

1168 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1823C212yesmaximal
2#14347C211yesmaximal, characteristic
3C2×#1445C211yesmaximal, characteristic
4C2×#1294C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2212yes—
9C22×#95—22no—
10C22×#95C2211yescharacteristic
11C22×#95C2211yescharacteristic
12#1823—24no—
13#1823—24no—
14C2×#340—24no—
15#1445—22no—
16#1445C414yescomplemented
17#1442—24no—
18C22⋊#95—22no—
19#1294—24no—
20#1280—22no—
21C24×C4—21no—
22C24×C4—21no—
23C24×C4D412yes—
24C24×C4C2311yescharacteristic
25C23×C8—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D412yes—
35C22×C4⋊C4D414yes—
36C22×C4⋊C4Q812yes—
37C22×C22⋊C4—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8D414yes—
44#380—22no—
45#380—22no—
46#380—22no—
47#380—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—28no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—24no—
56C2×#95—24no—
57C2×#95—48no—
58C2×#95C2×C412yes—
59C2×#95C2×C412yes—
60C2×#95D412yes—
61#340—216no—
62C25C2×D411yescharacteristic
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—24no—
72C23×C4—24no—
73C23×C4—44no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—42no—
77C23×C4—44no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—44no—
85C23×C4—44no—
86C23×C4—44no—
87C23×C4C22⋊C414yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×Q811yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4112yes—
97C23×C4#4112yes—
98C22×C8—24no—
99C22×C8—48no—
100C22×C8—42no—
101C22×C8—42no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—48no—
120C2×C4⋊C4C4⋊C414yes—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#95—24no—
130#95—24no—
131#95—24no—
132#95—416no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—84no—
137#95—84no—
138C24—22no—
139C24—22no—
140C24—24no—
141C24C2×C22⋊C411yescharacteristic
142C24#12011yescharacteristic
143C24#12111yescharacteristic
144C24C22⋊Q811yescharacteristic
145C24#12311yescharacteristic
146C24#12411yescharacteristic
147C24#12711yescharacteristic
148C24#13514yes—
149C24AGL(1,ℤ/8)14yes—
150C22×C4—28no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—416no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—416no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—48no—
187C22×C4—48no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—48no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—88no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4—88no—
208C22×C4C2×C4⋊C411yescharacteristic
209C22×C4#11711yescharacteristic
210C22×C4C4×D412yes—
211C22×C4C4×D412yes—
212C22×C4#12012yes—
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4#12111yescharacteristic
216C22×C4#12112yes—
217C22×C4C22⋊Q812yes—
218C22×C4C22⋊Q812yes—
219C22×C4C22⋊Q811yescharacteristic
220C22×C4#12312yes—
221C22×C4#12312yes—
222C22×C4#12412yes—
223C22×C4#12612yes—
224C2×C8—48no—
225C2×C8—44no—
226C2×C8—84no—
227C2×C8—84no—
228C4⋊C4—28no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—84no—
234C22⋊C4—84no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C23—28no—
238C23—28no—
239C23—28no—
240C23—28no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23#38011yescharacteristic
256C23#38611yescharacteristic
257C23#39011yescharacteristic
258C23#39211yescharacteristic
259C23#39311yescharacteristic
260C23#39711yescharacteristic
261C23#39911yescharacteristic
262C23#41612yes—
263C23#41812yes—
264C23#44914yes—
265C23#45512yes—
266C23#45612yes—
267C23#45814yes—
268C23#46314yes—
269C23#47814yes—
270C23#48214yes—
271C23#48812yes—
272C23#49012yes—
273C23#49512yes—
274C23#49612yes—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—82no—
300C2×C4—84no—
301C2×C4—88no—
302C2×C4—88no—
303C2×C4—88no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—88no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—88no—
315C2×C4—88no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4#38512yes—
320C2×C4#38812yes—
321C2×C4#39412yes—
322C2×C4#39612yes—
323C8—84no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—22no—
334C22—24no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—24no—
339C22—28no—
340C22—24no—
341C22—24no—
342C22—28no—
343C22—24no—
344C22#143911yescharacteristic
345C22#160918yes—
346C22#161218yes—
347C22#180412yes—
348C22#187512yes—
349C22#188511yescharacteristic
350C22#188611yescharacteristic
351C22#189012yes—
352C22#195811yescharacteristic
353C22#196111yescharacteristic
354C22#200712yes—
355C22#201212yes—
356C22#202212yes—
357C22#202512yes—
358C22#202812yes—
359C22#203012yes—
360C22#207312yes—
361C22#209212yes—
362C4—22no—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—84no—
367C4—84no—
368C4—84no—
369C4—164no—
370C2—21no—
371C2—22no—
372C2—22no—
373C2—22no—
374C2—21no—
375C2#902612yes—
376C2#902812yes—
377C2#997114yes—
378C2#1034614yes—
379C2#1105111yescharacteristic
380C2#1105511yescharacteristic
381C2#1107011yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1445C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18232
#143472
C2×#14452
C2×#12942
C2×#12802