Finite Group Databasebeta

← #417049

Subgroups — #417049

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1992C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic
6#14283C211yesmaximal, characteristic
7#11373C212yesmaximal
8#11372C212yesmaximal
9#11372C212yesmaximal
10#11371C212yesmaximal
11#10694C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#384C2212yes—
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2212yes—
20C2×#376C2211yescharacteristic
21#1992—24no—
22#1991—24no—
23#1836—24no—
24#1834—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1444—22no—
38#1444—22no—
39#1441—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125Q814yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#97—22no—
68C2×#97—22no—
69C2×#97D414yes—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8D414yes—
73#384—22no—
74#384—22no—
75#384—22no—
76#384—22no—
77#384—24no—
78#384—24no—
79#382—22no—
80#382—22no—
81#382—24no—
82#382—24no—
83#376—22no—
84#376—22no—
85C2×#95—24no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#336—24no—
103#336—24no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C2×D414yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C2×D412yes—
129C22×C8C2×D412yes—
130C22×C8C2×D414yes—
131#125—28no—
132#125—28no—
133#125—44no—
134#125—44no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—84no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4C2×Q812yes—
181C2×C4⋊C4C2×Q812yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×Q812yes—
189C2×C42#4114yes—
190C2×C42#4114yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194#97—28no—
195C4×C8—28no—
196#95—24no—
197#95—24no—
198#95—48no—
199#95—42no—
200#95—42no—
201#95—84no—
202#95—84no—
203C24C22×D411yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C22⋊Q812yes—
242C22×C4C22⋊Q812yes—
243C22×C4#12312yes—
244C22×C4#12312yes—
245C22×C4#12418yes—
246C22×C4#12712yes—
247C22×C4#12712yes—
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×Q811yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4112yes—
255C22×C4C2×#4112yes—
256C22×C42−1+412yes—
257C22×C42−1+412yes—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—24no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—24no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—816no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—24no—
307C42C22⋊Q814yes—
308C42C22⋊Q814yes—
309C42#12314yes—
310C42#12314yes—
311C23C2×C22⋊Q811yescharacteristic
312C23C2×#12311yescharacteristic
313C23C2×#12412yes—
314C23C2×#12711yescharacteristic
315C23#53612yes—
316C23#57612yes—
317C23#57712yes—
318C23#57712yes—
319C23#57812yes—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—88no—
362C2×C4#39914yes—
363C2×C4#39914yes—
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4C2×C22⋊Q811yescharacteristic
366C2×C4C2×#12311yescharacteristic
367C2×C4C2×#12311yescharacteristic
368C2×C4#53211yescharacteristic
369C2×C4#53211yescharacteristic
370C2×C4#53212yes—
371C2×C4#53611yescharacteristic
372C2×C4#53611yescharacteristic
373C2×C4#53612yes—
374C2×C4#55212yes—
375C2×C4#55212yes—
376C2×C4#55212yes—
377C2×C4#55212yes—
378C2×C4#55612yes—
379C2×C4#55612yes—
380C2×C4#55712yes—
381C2×C4#55712yes—
382C8—42no—
383C8—42no—
384C8—44no—
385C22C2×#39911yescharacteristic
386C22#250312yes—
387C22#256812yes—
388C22#265811yescharacteristic
389C22#285811yescharacteristic
390C22#308312yes—
391C22#308412yes—
392C22#309812yes—
393C22#309912yes—
394C22#313614yes—
395C22#313714yes—
396C22#315411yescharacteristic
397C22#315411yescharacteristic
398C22#315511yescharacteristic
399C22#315611yescharacteristic
400C22#324114yes—
401C22#324214yes—
402C4—24no—
403C4—24no—
404C4—48no—
405C4—82no—
406C4—82no—
407C4—162no—
408C4—161no—
409C4—162no—
410C4—161no—
411C4#249911yescharacteristic
412C4#249911yescharacteristic
413C4#249912yes—
414C4#256511yescharacteristic
415C4#256511yescharacteristic
416C4#256512yes—
417C2#1524011yescharacteristic
418C2#2373711yescharacteristic
419C2#2373811yescharacteristic
420C2#2797014yes—
421C2#2922814yes—
422C2#3028711yescharacteristic
423C2#3028811yescharacteristic
424C2#3322811yescharacteristic
425C2#3322911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19922
C2×#19912
C2×#18362
C2×#18342
#143542
#142832
#113732
#113722
#113712
#106942