Finite Group Databasebeta

← #120046

Subgroups — #120046

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21955C211yesmaximal, characteristic
2#14298C211yesmaximal, characteristic
3C4×#343C211yesmaximal, characteristic
4#8033C212yesmaximal
5C2×#1288C212yesmaximal
6C2×#432C2212yes—
7C2×C4×M16C2212yes—
8C2×#383C2211yescharacteristic
9C2×#382C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1940—24no—
12#1797—21no—
13#1797—21no—
14C2×#343—21no—
15C2×#343—21no—
16#1444—21no—
17#1444C411yescharacteristic
18#1444C411yescharacteristic
19#1442—21no—
20#1442C411yescharacteristic
21#1442C411yescharacteristic
22#1436—21no—
23#1436—21no—
24#1295—24no—
25#1288—24no—
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36#432—22no—
37#432—22no—
38#432—22no—
39#432—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C22⋊C8—22no—
44C4×M16—22no—
45C4×M16—22no—
46C4×M16—22no—
47C4×M16Q814yes—
48C2×#97—22no—
49C2×#97D414yes—
50C2×C4×C8—22no—
51#383—21no—
52#383—21no—
53#383D412yes—
54#383Q812yes—
55#382—22no—
56#382—22no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95C2×C411yescharacteristic
70C2×#95C2×C411yescharacteristic
71C2×#95C2×C411yescharacteristic
72C2×#95C2×C411yescharacteristic
73#343—24no—
74#343—24no—
75#336—22no—
76#336—22no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C2×M16—24no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—42no—
93C2×M16—42no—
94C22×C8—24no—
95C22×C8—42no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—81no—
107C2×C4⋊C4—81no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4—81no—
110C2×C4⋊C4—81no—
111C2×C4⋊C4—81no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—81no—
117C2×C42—81no—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D412yes—
120C2×C42C2×Q812yes—
121C2×C42C2×Q811yescharacteristic
122C2×C42#4112yes—
123C2×C42#4111yescharacteristic
124C2×C42#4112yes—
125C2×C42#4111yescharacteristic
126C4⋊C8—42no—
127C4⋊C8—42no—
128C4⋊C8—82no—
129C4⋊C8—82no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132#97—24no—
133#97—24no—
134#97—24no—
135#97—42no—
136C4×C8—24no—
137C4×C8—42no—
138#95—21no—
139#95—21no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—42no—
147#95—42no—
148#95—41no—
149#95—42no—
150#95—41no—
151#95—81no—
152#95—81no—
153#95—81no—
154#95—81no—
155#95C4⋊C412yes—
156#95C4⋊C412yes—
157C24C2×C22⋊C411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4#10412yes—
213C22×C4#10412yes—
214C22×C4C2×C4⋊C411yescharacteristic
215C22×C4#11711yescharacteristic
216C22×C4C4×D412yes—
217C22×C4C4×Q812yes—
218C22×C4#12011yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12112yes—
221C22×C4#12111yescharacteristic
222C22×C4C22⋊Q812yes—
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12411yescharacteristic
227C22×C4C4⋊Q811yescharacteristic
228M16—28no—
229M16—44no—
230M16—44no—
231M16—44no—
232C2×C8—44no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—84no—
243C4⋊C4—84no—
244C42—42no—
245C42—42no—
246C42—81no—
247C42—81no—
248C42#12112yes—
249C42C22⋊Q812yes—
250C42C22⋊Q812yes—
251C42#12312yes—
252C42#12512yes—
253C42#12512yes—
254C42#12612yes—
255C42C4⋊Q812yes—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#38011yescharacteristic
261C23#38611yescharacteristic
262C23#39111yescharacteristic
263C23C2×#10411yescharacteristic
264C23C2×#10411yescharacteristic
265C23#42111yescharacteristic
266C23#42111yescharacteristic
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4#38712yes—
337C2×C4#38912yes—
338C2×C4#39411yescharacteristic
339C2×C4#39511yescharacteristic
340C2×C4#39612yes—
341C2×C4#39811yescharacteristic
342C2×C4#39812yes—
343C2×C4#40011yescharacteristic
344C8—48no—
345C8—84no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22#144111yescharacteristic
359C22#180611yescharacteristic
360C22#180611yescharacteristic
361C22#187111yescharacteristic
362C22#187111yescharacteristic
363C22#191111yescharacteristic
364C22#191211yescharacteristic
365C22#191411yescharacteristic
366C22#191511yescharacteristic
367C22#199411yescharacteristic
368C22#199511yescharacteristic
369C4—22no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—82no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4—161no—
384C4—161no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2—21no—
389C2#918711yescharacteristic
390C2#918811yescharacteristic
391C2#918911yescharacteristic
392C2#919011yescharacteristic
393C2#1120611yescharacteristic
394C2#1120911yescharacteristic
395C2#1122411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219552
#142982
C4×#3432
#80332
C2×#12882