Finite Group Databasebeta

← #474409

Subgroups — #474409

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2100C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1941C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#15242C211yesmaximal, characteristic
6#15197C211yesmaximal, characteristic
7#11379C212yesmaximal
8#11378C212yesmaximal
9#11372C212yesmaximal
10#11325C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2212yes—
21#2100—24no—
22#2083—24no—
23#1941—24no—
24#1834—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1445—22no—
36#1442—22no—
37#1441—22no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#107—24no—
61C2×#107—24no—
62C2×#107—42no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—24no—
68C2×C4×C8—24no—
69C2×C4×C8D414yes—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#384—22no—
75#384—22no—
76#384—22no—
77#384—22no—
78#382—24no—
79#382—42no—
80C2×#95—24no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—24no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95D412yes—
87C2×#95D412yes—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—22no—
94#340—22no—
95#336—24no—
96#336—42no—
97#336D414yes—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2×D414yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8C2×D414yes—
122C22×C8C2×D412yes—
123C22×C8C2×D412yes—
124C22×C8#4114yes—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4#4112yes—
166C2×C4⋊C4#4112yes—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177#107—48no—
178#106—28no—
179#106—44no—
180C4×C8—28no—
181#95—24no—
182#95—24no—
183#95—48no—
184#95—42no—
185#95—42no—
186#95—84no—
187#95—84no—
188#95—84no—
189C24C22×D411yescharacteristic
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4#12712yes—
229C22×C4#12712yes—
230C22×C4#13514yes—
231C22×C4#13714yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C2×#4112yes—
235C22×C4C2×#4111yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C22×C4H(5,𝔽2)11yescharacteristic
238C22×C42−1+412yes—
239C22×C42−1+412yes—
240C22×C42−1+412yes—
241C22×C42−1+411yescharacteristic
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8#12314yes—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C42—22no—
287C42—22no—
288C42—24no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C23C2×#12711yescharacteristic
294C23#53612yes—
295C23#54611yescharacteristic
296C23#54712yes—
297C23#54811yescharacteristic
298C23C2×#13512yes—
299C23C2×#13712yes—
300C23#57612yes—
301C23#57712yes—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—88no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—84no—
341C2×C4—82no—
342C2×C4#49514yes—
343C2×C4#49714yes—
344C2×C4C2×#12311yescharacteristic
345C2×C4#52511yescharacteristic
346C2×C4#53611yescharacteristic
347C2×C4#53611yescharacteristic
348C2×C4#53611yescharacteristic
349C2×C4#53612yes—
350C2×C4#53711yescharacteristic
351C2×C4#54112yes—
352C2×C4#54112yes—
353C2×C4#54112yes—
354C2×C4#54212yes—
355C2×C4#54611yescharacteristic
356C2×C4#54712yes—
357C2×C4#54811yescharacteristic
358C2×C4#55212yes—
359C2×C4#55212yes—
360C2×C4#56312yes—
361C2×C4#56412yes—
362C8—24no—
363C8—42no—
364C8—44no—
365C22#250111yescharacteristic
366C22#254011yescharacteristic
367C22#265811yescharacteristic
368C22#269812yes—
369C22#271912yes—
370C22C2×#49511yescharacteristic
371C22C2×#49711yescharacteristic
372C22#315411yescharacteristic
373C22#315511yescharacteristic
374C22#323114yes—
375C22#324114yes—
376C22#330512yes—
377C22#333112yes—
378C22#333714yes—
379C22#334614yes—
380C22#336112yes—
381C22#336912yes—
382C4—24no—
383C4—24no—
384C4—82no—
385C4—84no—
386C4—82no—
387C4—161no—
388C4—161no—
389C4—162no—
390C4—162no—
391C4#248411yescharacteristic
392C4#262511yescharacteristic
393C4#263411yescharacteristic
394C4#269812yes—
395C4#271912yes—
396C4#282211yescharacteristic
397C2#1696411yescharacteristic
398C2#2792611yescharacteristic
399C2#2793811yescharacteristic
400C2#2839911yescharacteristic
401C2#2841911yescharacteristic
402C2#3028111yescharacteristic
403C2#3028711yescharacteristic
404C2#3168714yes—
405C2#3231114yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#21002
C2×#20832
C2×#19412
C2×#18342
#152422
#151972
#113792
#113782
#113722
#113252
#107972