Finite Group Databasebeta

← #215827

Subgroups — #215827

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9875C212yesmaximal, complemented
5#9871C212yesmaximal, complemented
6#9867C212yesmaximal, complemented
7#9866C212yesmaximal, complemented
8#8710C212yesmaximal
9#8706C212yesmaximal
10#2571C412yes—
11#2570—22no—
12#2569—22no—
13#2552C412yes—
14#2442C412yes—
15#2442C412yes—
16#2402—21no—
17#2402—21no—
18C4×#125C2212yes—
19C4×#125C2212yes—
20#2296—21no—
21#2285—21no—
22#2285C2211yescharacteristic
23C4×#117C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1923—22no—
26#1916—22no—
27#1840—21no—
28#1831C2212yes—
29#1818—21no—
30#1797C2212yes—
31#1767C2211yescharacteristic
32C4×C4⋊C8C2211yescharacteristic
33C4×C4⋊C8C2212yes—
34#1750—21no—
35C4×C22⋊C8C2212yes—
36#1746—21no—
37#1744C2211yescharacteristic
38C42×C8C2211yescharacteristic
39#1279C2212yes—
40#1279C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47C2×#125C2×C412yes—
48C2×#125C2×C412yes—
49C2×#117—21no—
50C2×#117D411yescharacteristic
51C2×#117D411yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×#97—21no—
61C2×C4×C8—22no—
62C2×C4×C8—21no—
63C2×C4×C8C2312yes—
64C2×C4×C8C2312yes—
65#400—22no—
66#400—22no—
67#400—22no—
68#400—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#398—22no—
73#389—22no—
74#389—22no—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#385—22no—
84#385—22no—
85#384—22no—
86#384C2×C412yes—
87#382—22no—
88#382—21no—
89#382C2×C412yes—
90#382C2×C412yes—
91#380—21no—
92#380—22no—
93#380—22no—
94#380C2×C411yescharacteristic
95#380C2×C411yescharacteristic
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4C2312yes—
108C4×C4⋊C4C2312yes—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111#376D411yescharacteristic
112#376D411yescharacteristic
113C2×#95—21no—
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—21no—
119#336—21no—
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124C23×C4—22no—
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132#125—24no—
133#125—24no—
134#125—42no—
135#125—42no—
136#117—24no—
137#117—41no—
138#117—41no—
139#117—42no—
140#117—42no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C412yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—44no—
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2411yescharacteristic
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—44no—
188C22⋊C8—44no—
189C22⋊C8—44no—
190#97—42no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—42no—
196C4×C8—44no—
197#95—21no—
198#95—21no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—21no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95C22⋊C412yes—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24C2×C22⋊C411yescharacteristic
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4#13514yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4#13114yes—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—21no—
281C42—21no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—42no—
295C42—42no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—42no—
301C422−1+412yes—
302C422−1+412yes—
303C23—21no—
304C23—22no—
305C23—21no—
306C23—22no—
307C23#41612yes—
308C23C22×C22⋊C411yescharacteristic
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—88no—
347C2×C4#41612yes—
348C2×C4#51911yescharacteristic
349C2×C4#51911yescharacteristic
350C2×C4#53612yes—
351C2×C4#53612yes—
352C2×C4C2×#13112yes—
353C2×C4#56812yes—
354C2×C4C2×#13512yes—
355C2×C4#57511yescharacteristic
356C2×C4#57511yescharacteristic
357C8—44no—
358C8—82no—
359C8—82no—
360C8—82no—
361C22—22no—
362C22—24no—
363C22—41no—
364C22—41no—
365C22—42no—
366C22#231011yescharacteristic
367C22#288012yes—
368C22C2×#41611yescharacteristic
369C22#289611yescharacteristic
370C22#290112yes—
371C4—22no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—42no—
376C4—42no—
377C4—44no—
378C4—42no—
379C4—42no—
380C4—82no—
381C4—82no—
382C4—82no—
383C4#298512yes—
384C4#298611yescharacteristic
385C4#298711yescharacteristic
386C4#323212yes—
387C4#323212yes—
388C2—42no—
389C2#1750812yes—
390C2#1751612yes—
391C2#2182411yescharacteristic
392C2#2183611yescharacteristic
393C2#2184611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9875C2semidirect
5. #9871C2semidirect
6. #9867C2semidirect
7. #9866C2semidirect

Maximal subgroups

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

subgroupindex
#349482
#214302
#207862
#98752
#98712
#98672
#98662
#87102
#87062