Finite Group Databasebeta

← #121823

Subgroups — #121823

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13965C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3C2×#1294C212yesmaximal
4C2×#1289C212yesmaximal
5#7907C211yesmaximal, characteristic
6C22×C22⋊C8C2212yes—
7C2×#385—21no—
8C2×#385—21no—
9C2×#380C2212yes—
10C2×#380C2211yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12C22×#95C2211yescharacteristic
13C2×#342—21no—
14C2×#342—21no—
15#1444—22no—
16C4⋊#95C412yescomplemented
17C4⋊#95C412yescomplemented
18#1294—24no—
19#1289—24no—
20#1280—22no—
21C24×C4—21no—
22C24×C4C2311yescharacteristic
23C23×C8—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42C2×C412yes—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8Q814yes—
39#385—44no—
40#385—42no—
41#385—42no—
42#380—22no—
43#380—22no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380D414yes—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95C2×C412yes—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D412yes—
60#342—42no—
61#342—42no—
62C25C2×D411yescharacteristic
63C23×C4—22no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—44no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4—42no—
72C23×C4—82no—
73C23×C4—82no—
74C23×C4C22⋊C412yes—
75C23×C4C22⋊C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C23×C4C2×Q812yes—
80C23×C4#4112yes—
81C23×C4#4112yes—
82C23×C4#4112yes—
83C23×C4#4112yes—
84C22×C8—24no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—84no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—82no—
97C2×C4⋊C4—82no—
98C2×C4⋊C4—82no—
99C2×C4⋊C4—84no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4D812yes—
115C2×C22⋊C4SD1612yes—
116C2×C22⋊C4SD1612yes—
117C2×C22⋊C4Q1612yes—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—82no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#95—24no—
126#95—24no—
127#95—22no—
128#95—22no—
129#95—24no—
130#95—44no—
131#95—44no—
132#95—41no—
133#95—42no—
134#95—41no—
135C24—42no—
136C24—42no—
137C24C2×C22⋊C411yescharacteristic
138C24#12112yes—
139C24#12312yes—
140C24#12411yescharacteristic
141C24C4⋊Q811yescharacteristic
142C24C2×D811yescharacteristic
143C24C2×SD1611yescharacteristic
144C24C2×SD1611yescharacteristic
145C24C2×Q1611yescharacteristic
146C24AGL(1,ℤ/8)12yes—
147C24#13712yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—48no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4—88no—
186C22×C4—84no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4#10214yes—
194C22×C4#10314yes—
195C22×C4#11711yescharacteristic
196C22×C4#11711yescharacteristic
197C22×C4C4×D412yes—
198C22×C4C4×Q812yes—
199C22×C4C22⋊Q812yes—
200C22×C4C22⋊Q812yes—
201C22×C4C22⋊Q812yes—
202C22×C4#12312yes—
203C22×C4#12312yes—
204C22×C4#12312yes—
205C22×C4#12411yescharacteristic
206C22×C4#12512yes—
207C22×C4#12512yes—
208C22×C4#12612yes—
209C22×C4#12612yes—
210C22×C4#12711yescharacteristic
211C2×C8—44no—
212C2×C8—84no—
213C2×C8—84no—
214C2×C8—84no—
215C4⋊C4—88no—
216C4⋊C4—88no—
217C4⋊C4—84no—
218C4⋊C4—82no—
219C4⋊C4—84no—
220C4⋊C4—82no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—82no—
232C42—82no—
233C23—22no—
234C23—24no—
235C23—22no—
236C23—24no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—42no—
243C23—44no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23—44no—
249C23#38111yescharacteristic
250C23#39011yescharacteristic
251C23#39111yescharacteristic
252C23#39712yes—
253C23#39812yes—
254C23C2×#10211yescharacteristic
255C23C2×#10311yescharacteristic
256C23#41811yescharacteristic
257C23#41911yescharacteristic
258C23#45912yes—
259C23#46012yes—
260C23#46112yes—
261C23#46212yes—
262C23#48012yes—
263C23#48112yes—
264C23#48312yes—
265C23#48412yes—
266C23#48611yescharacteristic
267C23#48711yescharacteristic
268C23#48912yes—
269C23#49811yescharacteristic
270C23#50011yescharacteristic
271C23#50112yes—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—88no—
305C2×C4—88no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—84no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—164no—
321C2×C4—164no—
322C2×C4#38712yes—
323C2×C4#38712yes—
324C2×C4#39912yes—
325C2×C4#40012yes—
326C8—84no—
327C22—22no—
328C22—22no—
329C22—24no—
330C22—24no—
331C22—22no—
332C22—22no—
333C22—24no—
334C22—24no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—42no—
353C22#144511yescharacteristic
354C22#182311yescharacteristic
355C22#182411yescharacteristic
356C22#182712yes—
357C22#182812yes—
358C22#198111yescharacteristic
359C22#198211yescharacteristic
360C22#199011yescharacteristic
361C22#199311yescharacteristic
362C22#207112yes—
363C22#207212yes—
364C22#208312yes—
365C22#208412yes—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—22no—
370C4—44no—
371C4—44no—
372C4—84no—
373C4—161no—
374C4—161no—
375C4—164no—
376C2—21no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—22no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#876012yes—
385C2#876112yes—
386C2#1134711yescharacteristic
387C2#1135111yescharacteristic
388C2#1135711yescharacteristic
subgroupG/H
center, socleC23#381
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
16. C4⋊#95C4semidirect
17. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#139652
#108462
C2×#12942
C2×#12892
#79072