Finite Group Databasebeta

← #474365

Subgroups — #474365

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14348C211yesmaximal, characteristic
2#14276C211yesmaximal, characteristic
3#11372C212yesmaximal
4#11358C212yesmaximal
5#11357C212yesmaximal
6#11293C212yesmaximal
7#11292C212yesmaximal
8#11267C212yesmaximal
9#10796C211yesmaximal, characteristic
10C2×#389C2212yes—
11C2×#384C2211yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#376C2211yescharacteristic
16C2×#340C2212yes—
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28#1442—22no—
29#1442—22no—
30#1441—22no—
31#1441—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C23×C8C2312yes—
36C23×C8C2312yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C42C2311yescharacteristic
46C22×C42C2311yescharacteristic
47C22×C42C2311yescharacteristic
48#389—42no—
49#389—42no—
50#384—24no—
51#384—24no—
52#382—24no—
53#382—24no—
54#376—24no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95D412yes—
66C2×#95D412yes—
67#340—24no—
68#340—24no—
69#340—42no—
70#340—42no—
71#340—42no—
72#340—42no—
73#340—42no—
74#340—42no—
75#336—24no—
76#336—24no—
77#336—41no—
78#336—41no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4C2×D412yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C23×C4C2411yescharacteristic
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4#4114yes—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42#4114yes—
147C2×C42#4114yes—
148#95—24no—
149#95—24no—
150#95—44no—
151#95—42no—
152#95—42no—
153#95—44no—
154#95—82no—
155#95—82no—
156#95—82no—
157#95—82no—
158#95—82no—
159#95—82no—
160C24C22×D411yescharacteristic
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4#12312yes—
211C22×C4#12312yes—
212C22×C4#12414yes—
213C22×C4C2×#4112yes—
214C22×C4C2×#4112yes—
215C22×C4C2×#4112yes—
216C22×C4H(5,𝔽2)12yes—
217C22×C4H(5,𝔽2)11yescharacteristic
218C22×C42−1+411yescharacteristic
219C22×C42−1+411yescharacteristic
220C22×C42−1+411yescharacteristic
221C22×C42−1+411yescharacteristic
222C22×C42−1+411yescharacteristic
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—88no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—88no—
257C42—24no—
258C42—24no—
259C42—24no—
260C42—24no—
261C42—42no—
262C42—42no—
263C23C2×#12311yescharacteristic
264C23#52511yescharacteristic
265C23C2×#12411yescharacteristic
266C23#53511yescharacteristic
267C23#53611yescharacteristic
268C23#53611yescharacteristic
269C23#53711yescharacteristic
270C23#57612yes—
271C23#57712yes—
272C23#57712yes—
273C23#57812yes—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#53912yes—
327C2×C4#53912yes—
328C2×C4#54012yes—
329C2×C4#54212yes—
330C2×C4#54212yes—
331C2×C4#54212yes—
332C2×C4#55212yes—
333C2×C4#55212yes—
334C2×C4#55212yes—
335C2×C4#55212yes—
336C2×C4#55612yes—
337C2×C4#55912yes—
338C2×C4#56311yescharacteristic
339C2×C4#56311yescharacteristic
340C2×C4#56411yescharacteristic
341C2×C4#56411yescharacteristic
342C8—82no—
343C8—82no—
344C22#248411yescharacteristic
345C22#250311yescharacteristic
346C22#259211yescharacteristic
347C22#262411yescharacteristic
348C22#263311yescharacteristic
349C22#264011yescharacteristic
350C22#282211yescharacteristic
351C22#309812yes—
352C22#309912yes—
353C22#312211yescharacteristic
354C22#312311yescharacteristic
355C22#312611yescharacteristic
356C22#312711yescharacteristic
357C22#313612yes—
358C22#313712yes—
359C22#322412yes—
360C22#322512yes—
361C22#324112yes—
362C22#324212yes—
363C22#324312yes—
364C22#324412yes—
365C22#325712yes—
366C22#325812yes—
367C4—44no—
368C4—42no—
369C4—42no—
370C4—84no—
371C4—82no—
372C4—162no—
373C4—162no—
374C4—162no—
375C4#276712yes—
376C4#276712yes—
377C4#277312yes—
378C4#277312yes—
379C2#1696111yescharacteristic
380C2#2762011yescharacteristic
381C2#2762111yescharacteristic
382C2#2797012yes—
383C2#2935811yescharacteristic
384C2#2935911yescharacteristic
385C2#2993512yes—
386C2#3008511yescharacteristic
387C2#3008711yescharacteristic
388C2#3021112yes—
389C2#3285412yes—
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
#143482
#142762
#113722
#113582
#113572
#112932
#112922
#112672
#107962