Finite Group Databasebeta

← #162675

Subgroups — #162675

Order 512.

966 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×#2859C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20364C211yesmaximal, characteristic
4#7700C214yesmaximal, complemented
5#7683C214yesmaximal, complemented
6#7610C214yesmaximal
7C22×#125—21no—
8C2×C4×M16—21no—
9#2859—21no—
10#2859—21no—
11#2859—21no—
12#2859—21no—
13#2859C412yes—
14#2859C412yes—
15#2859C2212yes—
16#2859C2212yes—
17C2×#389—21no—
18C2×#389—21no—
19C2×#389C412yes—
20C2×#389C2212yes—
21C2×#384C411yescharacteristic
22C2×#384C411yescharacteristic
23C2×#382—21no—
24C2×C43C2211yescharacteristic
25#1839C2212yes—
26#1831—21no—
27#1817C2212yes—
28#1797—21no—
29#1767C2212yes—
30#1749—21no—
31#1745—21no—
32#1745—21no—
33#1745C2212yes—
34#1744—21no—
35#1744C2212yes—
36#1728—21no—
37#1728C2212yes—
38#1271C22116yes—
39C22×M16—22no—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125—48no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C4×M16—28no—
63C2×#97—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389C2×C414yes—
79#389C2×C414yes—
80#389C2×C414yes—
81#389C2314yes—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384C2×C412yes—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2×C412yes—
90#382—22no—
91#382—22no—
92#382—22no—
93#382—22no—
94C2×#95—22no—
95C2×#95—22no—
96C43C2×C411yescharacteristic
97C43C2×C411yescharacteristic
98C43D411yescharacteristic
99C43D411yescharacteristic
100C43D411yescharacteristic
101C43D411yescharacteristic
102C43C2311yescharacteristic
103C43C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336C2314yes—
111#336C2314yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—28no—
122C2×M16—44no—
123C2×M16—44no—
124C22×C8—24no—
125C22×C8—24no—
126#125—48no—
127#125—48no—
128#125—48no—
129#125—48no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2411yescharacteristic
189C4⋊C8—44no—
190C22⋊C8—44no—
191#97—28no—
192#97—44no—
193C4×C8—28no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198C24C2×C22⋊C411yescharacteristic
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4#10414yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4AGL(1,ℤ/8)14yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242M16—416no—
243C2×C8—28no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C4⋊C4—28no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42#10114yes—
292C42#13714yes—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23#41012yes—
298C23#41812yes—
299C23C2×#10412yes—
300C23C22×C22⋊C411yescharacteristic
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—88no—
345C2×C4—88no—
346C2×C4C2×#10112yes—
347C2×C4C2×#10112yes—
348C2×C4C2×#10112yes—
349C2×C4#41812yes—
350C2×C4#41912yes—
351C2×C4#41912yes—
352C2×C4C2×#10412yes—
353C2×C4C2×#10412yes—
354C2×C4#42112yes—
355C2×C4#56812yes—
356C2×C4C2×AGL(1,ℤ/8)12yes—
357C2×C4C2×#13712yes—
358C8—48no—
359C8—84no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22#150514yes—
367C22#287911yescharacteristic
368C22C2×#41011yescharacteristic
369C22C22×#10111yescharacteristic
370C22C2×#41811yescharacteristic
371C22C2×#41911yescharacteristic
372C22C22×#10411yescharacteristic
373C22C2×#42111yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—84no—
388C4—84no—
389C4#152814yes—
390C2—24no—
391C2#1739411yescharacteristic
392C2#1749011yescharacteristic
393C2#1750111yescharacteristic
394C2C2×#150511yescharacteristic
395C2#1756211yescharacteristic
396C2C2×#152811yescharacteristic
397C2#1774211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7700C2semidirect
5. #7683C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
#207782
#203642
#77002
#76832
#76102