Finite Group Databasebeta

← #449449

Subgroups — #449449

Order 512.

858 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#15151C211yesmaximal, characteristic
2#13955C211yesmaximal, characteristic
3#11316C212yesmaximal
4#11315C212yesmaximal
5#11267C212yesmaximal
6#11208C212yesmaximal
7#10885C212yesmaximal
8#10875C211yesmaximal, characteristic
9#10821C212yesmaximal
10C22×#97C2211yescharacteristic
11C2×#398—22no—
12C2×#389C2212yes—
13C2×#384C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28#1445—22no—
29#1442—22no—
30C4⋊#95C2212yes—
31C4⋊#95C2211yescharacteristic
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34#1434—22no—
35C23×C8C2312yes—
36C23×C8C2312yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4C2312yes—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46C22×C42C2311yescharacteristic
47C22×C42C2311yescharacteristic
48C2×#97—24no—
49C2×#97—41no—
50#398—48no—
51#389—24no—
52#389—24no—
53#384—41no—
54#384—41no—
55#382—41no—
56#382—41no—
57C4×C4⋊C4—24no—
58C4×C4⋊C4—41no—
59C4×C4⋊C4—41no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—21no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—21no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95D412yes—
72#340—24no—
73#340—24no—
74#340—24no—
75#340—24no—
76#340—24no—
77#340—24no—
78#340—42no—
79#340—42no—
80#336—24no—
81#336—24no—
82#336—41no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4C2×D412yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2411yescharacteristic
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—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8#4114yes—
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—24no—
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—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4#4114yes—
138C2×C4⋊C4#4114yes—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—82no—
152C2×C42#4114yes—
153#97—44no—
154#95—24no—
155#95—42no—
156#95—44no—
157#95—42no—
158#95—42no—
159#95—44no—
160#95—44no—
161#95—44no—
162#95—44no—
163#95—44no—
164#95—82no—
165#95—82no—
166C24C22×D411yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
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—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4#12412yes—
217C22×C4#12412yes—
218C22×C4AGL(1,ℤ/8)14yes—
219C22×C4C2×#4112yes—
220C22×C4C2×#4112yes—
221C22×C4C2×#4112yes—
222C22×C4C2×#4112yes—
223C22×C4H(5,𝔽2)11yescharacteristic
224C22×C4H(5,𝔽2)11yescharacteristic
225C22×C42−1+412yes—
226C22×C42−1+412yes—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—48no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—88no—
259C4⋊C4—88no—
260C42—24no—
261C42—24no—
262C42—42no—
263C42—42no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—41no—
268C42—84no—
269C23#52511yescharacteristic
270C23C2×#12411yescharacteristic
271C23#53411yescharacteristic
272C23#54712yes—
273C23#54812yes—
274C23C2×AGL(1,ℤ/8)12yes—
275C23#57512yes—
276C23#57712yes—
277C23#57812yes—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
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—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#52912yes—
331C2×C4#52912yes—
332C2×C4#54112yes—
333C2×C4#54112yes—
334C2×C4#54212yes—
335C2×C4#54212yes—
336C2×C4#55111yescharacteristic
337C2×C4#55212yes—
338C2×C4#55212yes—
339C2×C4#55212yes—
340C2×C4#55212yes—
341C2×C4#55312yes—
342C2×C4#55312yes—
343C2×C4#55511yescharacteristic
344C2×C4#56311yescharacteristic
345C2×C4#56311yescharacteristic
346C8—44no—
347C8—82no—
348C22#247512yes—
349C22#248912yes—
350C22#259111yescharacteristic
351C22#271711yescharacteristic
352C22#273111yescharacteristic
353C22#311512yes—
354C22#312311yescharacteristic
355C22#312711yescharacteristic
356C22#313511yescharacteristic
357C22#313712yes—
358C22#313811yescharacteristic
359C22#319412yes—
360C22#320112yes—
361C22#334312yes—
362C22#334512yes—
363C22#335112yes—
364C22#335212yes—
365C22#336712yes—
366C22#337112yes—
367C22#337312yes—
368C22#337712yes—
369C4—44no—
370C4—42no—
371C4—42no—
372C4—82no—
373C4—84no—
374C4—84no—
375C4—162no—
376C4—162no—
377C4#251512yes—
378C4#252912yes—
379C4#276612yes—
380C4#276612yes—
381C2#1594511yescharacteristic
382C2#2738212yes—
383C2#2738612yes—
384C2#2773412yes—
385C2#2774412yes—
386C2#2933711yescharacteristic
387C2#2934111yescharacteristic
388C2#3227512yes—
389C2#3263912yes—
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
#151512
#139552
#113162
#113152
#112672
#112082
#108852
#108752
#108212