Finite Group Databasebeta

← #182012

Subgroups — #182012

Order 512.

1172 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
1C2×#2298C211yesmaximal, characteristic
2C4×#432C212yesmaximal, complemented
3C2×#1464C212yesmaximal
4#8155C214yesmaximal
5#8154C214yesmaximal, complemented
6#7910C212yesmaximal, complemented
7C22×#117C2211yescharacteristic
8C2×#432C2214yes—
9#2298—22no—
10#2298—22no—
11#2298—22no—
12#2298—22no—
13C2×#388C412yes—
14C2×#388C2212yes—
15C2×#387C412yes—
16C2×#387C2212yes—
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20C2×C4×C22⋊C4C411yescharacteristic
21C2×#376C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#1767—22no—
24C4×C4⋊C8—22no—
25#1749C2214yes—
26C4×C22⋊C8C2214yes—
27#1745—22no—
28C4×#97—22no—
29#1464C4116yescomplemented
30C2×#324C2214yes—
31C2×#323C2214yes—
32#1280C2218yes—
33C24×C4C2×C411yescharacteristic
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—24no—
37C2×#117C2×C418yes—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2311yescharacteristic
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2311yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C412yes—
52C22×C42C2312yes—
53C22×C42C2311yescharacteristic
54#432—28no—
55#432C2×C4116yes—
56C2×C4⋊C8—24no—
57C2×C22⋊C8C2318yes—
58C2×#97—24no—
59C2×#97—24no—
60C2×C4×C8—24no—
61#388—22no—
62#388—22no—
63#388—22no—
64#388—22no—
65#388—22no—
66#388—22no—
67#388—22no—
68#388—22no—
69#387—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—22no—
90#376—22no—
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2×C412yes—
94C2×#95C2×C412yes—
95C2×#95C2×C412yes—
96C2×#95C2312yes—
97C43D412yes—
98C43D412yes—
99C43D412yes—
100C43D412yes—
101#336—24no—
102#336—24no—
103#324C2×C4116yes—
104#323C2×C4116yes—
105C25C22×C411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4C4214yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C412yes—
117C23×C4C22×C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—28no—
123#117—216no—
124#117—216no—
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—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4C4214yes—
136C2×C4⋊C4C22×C414yes—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4C4214yes—
148C2×C22⋊C4C22×C414yes—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42C4214yes—
163C2×C42C22⋊C412yes—
164C2×C42C22⋊C412yes—
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C412yes—
173C2×C42C22×C414yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42#4114yes—
181C4⋊C8—216no—
182C22⋊C8—28no—
183C22⋊C8C22×C4116yes—
184#97—216no—
185#97—48no—
186C4×C8—48no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—42no—
204C24—42no—
205C24C2×C4212yes—
206C24C23×C411yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—48no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C2×C4212yes—
244C22×C4C2×C4212yes—
245C22×C4C2×C4212yes—
246C22×C4C2×C4212yes—
247C22×C4C2×C4212yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4#11712yes—
256C22×C4#11712yes—
257C22×C4C23×C412yes—
258C22×C4C22×D412yes—
259C22×C4C2×#4112yes—
260C2×C8—216no—
261C2×C8—48no—
262C2×C8—48no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C22⋊C4—28no—
270C22⋊C4—28no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C42—24no—
276C42—24no—
277C42—24no—
278C42—24no—
279C42—24no—
280C42—24no—
281C42—41no—
282C42—41no—
283C42—42no—
284C42—42no—
285C42—41no—
286C42—41no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42C4×D418yes—
296C42C4×D418yes—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—84no—
311C23—84no—
312C23C22×C4211yescharacteristic
313C23C22×C22⋊C411yescharacteristic
314C23C2×#11711yescharacteristic
315C23#56812yes—
316C23#56812yes—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
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—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4C4×C22⋊C412yes—
367C2×C4C4×C22⋊C412yes—
368C2×C4C4×C22⋊C414yes—
369C2×C4#41014yes—
370C2×C4#41314yes—
371C2×C4C2×C4×D414yes—
372C8—416no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—24no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—84no—
383C22—84no—
384C22—82no—
385C22—82no—
386C22C2×C4×C22⋊C411yescharacteristic
387C22#287312yes—
388C22#287911yescharacteristic
389C22#287911yescharacteristic
390C22C2×#41012yes—
391C22C2×#41312yes—
392C22#292112yes—
393C4—22no—
394C4—22no—
395C4—24no—
396C4—24no—
397C4—24no—
398C4—24no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—82no—
405C4—82no—
406C2—22no—
407C2—22no—
408C2—82no—
409C2#1739512yes—
410C2#1739912yes—
411C2#2037711yescharacteristic
412C2C4×#41011yescharacteristic
413C2C4×#41311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
5. #8154C2semidirect
6. #7910C2semidirect
29. #1464C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22982
C4×#4322
C2×#14642
#81552
#81542
#79102