Finite Group Databasebeta

← #480034

Subgroups — #480034

Order 512.

1082 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C2×#1458C212yesmaximal
3#11450C212yesmaximal, complemented
4#11449C212yesmaximal, complemented
5#11448C214yesmaximal, complemented
6#11447C214yesmaximal, complemented
7C22×#117C411yescharacteristic
8C22×#117C411yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#431—22no—
11#2317C411yescharacteristic
12#2317C411yescharacteristic
13#2317C412yes—
14#2317C2211yescharacteristic
15#2317C2211yescharacteristic
16#2317C2212yes—
17#2316—22no—
18#2315—22no—
19#2296—22no—
20#2296—22no—
21#2285C412yes—
22#2285C2212yes—
23#1458—22no—
24#1458—22no—
25#1458C2214yes—
26#1458C2214yes—
27C2×#324C2214yes—
28C2×#323—22no—
29#1325C2218yescomplemented
30#1324C2218yescomplemented
31C24×C4C2×C411yescharacteristic
32C2×#117—21no—
33C2×#117—21no—
34C2×#117—24no—
35C2×#117—21no—
36C2×#117—21no—
37C2×#117—42no—
38C2×#117—42no—
39C2×#117—42no—
40C2×#117—42no—
41C2×#117C818yescomplemented
42C2×#117C2×C412yes—
43C2×#117C2×C412yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C412yes—
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D412yes—
50C2×#117D412yes—
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4D412yes—
61C22×C42—21no—
62C22×C42C2×C412yes—
63C22×C42D412yes—
64#431—24no—
65#431—24no—
66#431—42no—
67#431—42no—
68C2×C4⋊C8—42no—
69C2×C22⋊C8—24no—
70C2×C4×C8—42no—
71#389—24no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#385—24no—
78#385—24no—
79#385—22no—
80#385—22no—
81#385—22no—
82#385—22no—
83#384C2×C412yes—
84#384C2×C412yes—
85#384C2×C412yes—
86#384C2312yes—
87#382C2×C412yes—
88#382C2×C412yes—
89#382C2×C412yes—
90#382C2312yes—
91#380—24no—
92#380—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95#324—24no—
96#324C2318yes—
97#323—24no—
98#323—24no—
99C25C22⋊C411yescharacteristic
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C2×C814yes—
111C23×C4M1614yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D412yes—
116C22×C8—44no—
117#117—28no—
118#117—28no—
119#117—24no—
120#117—22no—
121#117—22no—
122#117—42no—
123#117—41no—
124#117—41no—
125#117—42no—
126#117—42no—
127#117—42no—
128#117—42no—
129#117—42no—
130#117—41no—
131#117—41no—
132#117—44no—
133#117—42no—
134#117—42no—
135#117—44no—
136#117—42no—
137#117—42no—
138#117C2×C818yes—
139#117C2×C814yescomplemented
140#117C2×C814yescomplemented
141C2×C4⋊C4—28no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2411yescharacteristic
163C2×C22⋊C4—28no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C2×D412yes—
175C2×C22⋊C4C2×D412yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C2×C818yescomplemented
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C4⋊C8—44no—
194C4⋊C8—44no—
195C22⋊C8—28no—
196C22⋊C8—44no—
197C4×C8—44no—
198C4×C8—44no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203C24—22no—
204C24—22no—
205C24—81no—
206C24—81no—
207C24C22⋊C814yes—
208C24C2×C22⋊C411yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—28no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4C22⋊C814yes—
245C22×C4C22⋊C814yes—
246C22×C4C22⋊C814yes—
247C22×C4#9912yes—
248C22×C4#9912yes—
249C22×C4#10114yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C22×C812yes—
257C22×C4C22×C812yes—
258C22×C4C22×C812yes—
259C22×C4C2×M1612yes—
260C22×C4C2×M1612yes—
261C22×C4C2×M1612yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D412yes—
264C2×C8—48no—
265C2×C8—88no—
266C4⋊C4—28no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C22⋊C4—28no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C42—22no—
293C42—22no—
294C42—28no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—44no—
304C42—42no—
305C42—42no—
306C42C22×C814yes—
307C42C22×C814yes—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—28no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—82no—
320C23—82no—
321C23—84no—
322C23—84no—
323C23C2×C22⋊C812yes—
324C23C2×C22⋊C812yes—
325C23C2×C22⋊C812yes—
326C23C2×#9912yes—
327C23C2×#10112yes—
328C23C22×C22⋊C411yescharacteristic
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—88no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4#32418yes—
370C2×C4#32418yes—
371C2×C4C2×C22⋊C812yes—
372C2×C4C2×C22⋊C812yes—
373C2×C4C2×C22⋊C812yes—
374C2×C4C2×#9911yescharacteristic
375C2×C4C2×#9912yes—
376C2×C4C2×#9911yescharacteristic
377C2×C4C2×#10112yes—
378C2×C4C2×#10112yes—
379C2×C4C23×C811yescharacteristic
380C2×C4C22×M1611yescharacteristic
381C8—88no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—88no—
399C22C2×#32414yes—
400C22C22×C22⋊C811yescharacteristic
401C22C22×#9911yescharacteristic
402C22C22×#10111yescharacteristic
403C4—21no—
404C4—21no—
405C4—22no—
406C4—44no—
407C4—44no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—82no—
413C4—82no—
414C4C2×#32412yes—
415C4C2×#32414yes—
416C4C2×#32412yes—
417C2—21no—
418C2—21no—
419C2—24no—
420C2—42no—
421C2—42no—
422C2C22×#32411yescharacteristic
423C2#3335811yescharacteristic
424C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11450C2semidirect
4. #11449C2semidirect
5. #11448C2semidirect
6. #11447C2semidirect
29. #1325C22semidirect
30. #1324C22semidirect
41. C2×#117C8semidirect
139. #117C2×C8semidirect
140. #117C2×C8semidirect
188. C2×C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#14582
#114502
#114492
#114482
#114472