Finite Group Databasebeta

← #213595

Subgroups — #213595

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2#21820C212yesmaximal
3#9086C214yesmaximal
4#9082C214yesmaximal
5#8701C212yesmaximal
6#8306C212yesmaximal
7C22×#125C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2212yes—
10C2×#431C2212yes—
11#2565—22no—
12#2565—22no—
13#2565—22no—
14#2565—22no—
15C2×#398C414yes—
16C2×#391C411yescharacteristic
17C2×#391C411yescharacteristic
18C2×#387C2214yes—
19C2×#384C411yescharacteristic
20C2×#382C411yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1985—22no—
23#1985—22no—
24#1984—22no—
25#1984—22no—
26#1817C2214yes—
27#1817C2214yes—
28C2×#324C2212yes—
29C2×#324C2214yes—
30C2×#324C2212yes—
31#1280C2212yes—
32#1280C2214yes—
33#1280C2212yes—
34C23×Q8C2×C411yescharacteristic
35C2×#125—22no—
36C2×#125—22no—
37C2×#125—41no—
38C2×#125—41no—
39C2×#125—42no—
40C2×C4×Q8—41no—
41C2×C4×Q8—42no—
42C2×C4×Q8—41no—
43C2×C4×Q8D412yes—
44C2×C4×Q8D412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8D412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59#432—42no—
60#432—42no—
61#431—42no—
62#431—42no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8C2314yes—
66C2×C22⋊C8C2314yes—
67C2×#97—22no—
68C2×C4×C8—22no—
69#398—24no—
70#398—24no—
71#398—24no—
72#398—24no—
73#391—22no—
74#391—21no—
75#391—21no—
76#391—22no—
77#391—21no—
78#391—21no—
79#387—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#384—22no—
84#384—21no—
85#384—21no—
86#382—22no—
87#382—21no—
88#382—21no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C2×#95C2×C414yes—
92C2×#95C2×C414yes—
93C2×#95C2×C414yes—
94C2×#95C2312yes—
95#336—24no—
96#336—24no—
97#336—24no—
98#336—24no—
99#324—42no—
100#324—44no—
101#324—42no—
102C22×Q8—24no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—42no—
106C22×Q8C22⋊C412yes—
107C22×Q8C22⋊C411yescharacteristic
108C22×Q8C22⋊C411yescharacteristic
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C2411yescharacteristic
118C22×C8—24no—
119C22×C8—24no—
120#125—44no—
121#125—44no—
122#125—44no—
123#125—44no—
124C4×Q8—42no—
125C4×Q8—42no—
126C4×Q8—42no—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C412yes—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C4⋊C8—44no—
185C4⋊C8—44no—
186C22⋊C8—24no—
187C22⋊C8—24no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190#97—44no—
191C4×C8—44no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199C24C23×C411yescharacteristic
200C2×Q8—24no—
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—48no—
209C2×Q8—48no—
210C2×Q8#10114yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#13114yes—
254C22×C4C22×D412yes—
255C22×C4H(5,𝔽2)11yescharacteristic
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C4H(5,𝔽2)12yes—
258C2×C8—48no—
259C2×C8—48no—
260C2×C8—44no—
261C2×C8—44no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—48no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23C22×C22⋊C411yescharacteristic
299C23#51811yescharacteristic
300C23#51811yescharacteristic
301C23C2×#13112yes—
302C23#56812yes—
303Q8—44no—
304Q8—48no—
305Q8—44no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4C2×#10112yes—
355C2×C4C2×#10114yes—
356C2×C4#41312yes—
357C2×C4#53412yes—
358C2×C4#53412yes—
359C8—84no—
360C8—84no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22#230811yescharacteristic
370C22#288012yes—
371C22#288412yes—
372C22C22×#10111yescharacteristic
373C22C2×#41311yescharacteristic
374C22#297911yescharacteristic
375C22#298011yescharacteristic
376C22#298112yes—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—24no—
382C4—44no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—82no—
389C4—82no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2#1742312yes—
394C2#1743012yes—
395C2#2174511yescharacteristic
396C2#2175111yescharacteristic
397C2#2175311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25652
#218202
#90862
#90822
#87012
#83062