Finite Group Databasebeta

← #479292

Subgroups — #479292

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15133C212yesmaximal
2#11377C211yesmaximal, characteristic
3#11371C212yesmaximal
4#11346C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11316C212yesmaximal
7#11256C211yesmaximal, characteristic
8#11219C212yesmaximal
9#11207C212yesmaximal
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C2×#398—22no—
13C2×#389C2212yes—
14C2×#389C2212yes—
15C2×#384C2212yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#336C2211yescharacteristic
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29#1441—22no—
30#1441—22no—
31#1441—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C23×C8C2312yes—
36C23×C8C2311yescharacteristic
37C23×C8C2311yescharacteristic
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C42C2311yescharacteristic
49C22×C42C2312yes—
50C2×#107—41no—
51C2×#107—41no—
52C2×#107—42no—
53C2×#107—41no—
54C2×#107Q814yes—
55#398—48no—
56#389—24no—
57#389—42no—
58#389—42no—
59#389—42no—
60#384—24no—
61#384—42no—
62C4×C4⋊C4—41no—
63C4×C4⋊C4—41no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95D412yes—
73#340—24no—
74#340—22no—
75#340—24no—
76#340—41no—
77#340—42no—
78#340—42no—
79#340—42no—
80#340—42no—
81#336—41no—
82#336—41no—
83#336—42no—
84#336—42no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2411yescharacteristic
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×Q812yes—
110C22×C8#4112yes—
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—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4Q1614yes—
150C2×C4⋊C4C2×Q814yes—
151C2×C4⋊C4#4114yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162#107—44no—
163#107—44no—
164#95—42no—
165#95—44no—
166#95—44no—
167#95—44no—
168#95—44no—
169#95—44no—
170#95—82no—
171#95—82no—
172#95—82no—
173C24C22×D411yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
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—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×Q1612yes—
224C22×C4C2×Q1614yes—
225C22×C4AGL(1,ℤ/8)14yes—
226C22×C4C22×Q811yescharacteristic
227C22×C4C2×#4111yescharacteristic
228C22×C4C2×#4112yes—
229C22×C4H(5,𝔽2)11yescharacteristic
230C22×C4H(5,𝔽2)12yes—
231C22×C4H(5,𝔽2)11yescharacteristic
232C22×C4H(5,𝔽2)11yescharacteristic
233C22×C42−1+412yes—
234C22×C42−1+411yescharacteristic
235C22×C42−1+412yes—
236C2×C8—22no—
237C2×C8—24no—
238C2×C8—22no—
239C2×C8—24no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—88no—
272C4⋊C4—88no—
273C4⋊C4—88no—
274C4⋊C4—88no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—41no—
280C42—42no—
281C42—41no—
282C23#53411yescharacteristic
283C23#53612yes—
284C23#54612yes—
285C23D4×Q811yescharacteristic
286C23#55011yescharacteristic
287C23C22×Q1611yescharacteristic
288C23C2×AGL(1,ℤ/8)12yes—
289C23#57511yescharacteristic
290C23#57611yescharacteristic
291C23#57712yes—
292C23#57811yescharacteristic
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#53911yescharacteristic
346C2×C4#53912yes—
347C2×C4#53912yes—
348C2×C4#54012yes—
349C2×C4#54112yes—
350C2×C4#54311yescharacteristic
351C2×C4#54412yes—
352C2×C4#55212yes—
353C2×C4#55411yescharacteristic
354C2×C4#55511yescharacteristic
355C2×C4#55712yes—
356C2×C4#56011yescharacteristic
357C2×C4#56211yescharacteristic
358C2×C4#56212yes—
359C2×C4#56312yes—
360C2×C4#56311yescharacteristic
361C2×C4#56312yes—
362C2×C4#56311yescharacteristic
363C8—42no—
364C8—42no—
365C8—82no—
366C22#269412yes—
367C22#269911yescharacteristic
368C22#271312yes—
369C22#273211yescharacteristic
370C22#281711yescharacteristic
371C22#318911yescharacteristic
372C22#319211yescharacteristic
373C22#320311yescharacteristic
374C22#320411yescharacteristic
375C22#323012yes—
376C22#323512yes—
377C22#324112yes—
378C22#324212yes—
379C22#329912yes—
380C22#330812yes—
381C22#331812yes—
382C22#332312yes—
383C22Q8×Q1611yescharacteristic
384C22#338412yes—
385C22#339011yescharacteristic
386C22#339511yescharacteristic
387C22#339712yes—
388C22#340311yescharacteristic
389C4—42no—
390C4—44no—
391C4—44no—
392C4—81no—
393C4—84no—
394C4—84no—
395C4—162no—
396C4—162no—
397C4#276911yescharacteristic
398C4#279112yes—
399C4#280911yescharacteristic
400C4#283411yescharacteristic
401C4#283612yes—
402C4#284111yescharacteristic
403C2#1719711yescharacteristic
404C2#3157612yes—
405C2#3159412yes—
406C2#3172211yescharacteristic
407C2#3173611yescharacteristic
408C2#3213612yes—
409C2#3216012yes—
410C2#3265011yescharacteristic
411C2#3266611yescharacteristic
412C2#3279011yescharacteristic
413C2#3279511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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
#151332
#113772
#113712
#113462
#113282
#113162
#112562
#112192
#112072