Finite Group Databasebeta

← #480134

Subgroups — #480134

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34199C211yesmaximal, characteristic
2#17350C212yesmaximal
3#11474C214yesmaximal, complemented
4#11472C214yesmaximal, complemented
5#11453C212yesmaximal, complemented
6C2×#1317C212yesmaximal, complemented
7C22×#123C411yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#117C411yescharacteristic
10#2972—22no—
11#2359—22no—
12#2359—22no—
13#2344—22no—
14#2342—22no—
15#2322C412yes—
16#2322C2212yes—
17#2315C2211yescharacteristic
18#2308C411yescharacteristic
19C4×#123C412yes—
20C4×#123C2212yes—
21C2×#381C2211yescharacteristic
22#2273C411yescharacteristic
23#1460C2214yes—
24#1459—22no—
25#1459C2214yes—
26#1458—22no—
27C2×#323—22no—
28C2×#323C2214yes—
29#1326C2218yescomplemented
30#1317C2214yescomplemented
31#1317C2214yescomplemented
32C23×D4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C2×#123—22no—
35C2×#123—24no—
36C2×#123C818yescomplemented
37C2×#123C2×C412yes—
38C2×#123C2×C412yes—
39#518—44no—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117—22no—
45C2×#117—21no—
46C2×#117—21no—
47C2×#117—44no—
48C2×#117D412yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C412yes—
51C22×C22⋊C4—21no—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4D412yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C42C2×C412yes—
59C8×D4—48no—
60#431—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63#389—22no—
64#389—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#386—22no—
74#386C2×C412yes—
75#386C2×C412yes—
76#385—22no—
77#385—22no—
78#385—22no—
79#385C2×C412yes—
80#385C2312yes—
81#381—21no—
82#381—22no—
83#381C2×C411yescharacteristic
84#381C2×C411yescharacteristic
85#381C2311yescharacteristic
86#381C2311yescharacteristic
87#380—22no—
88#379—21no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4C2×C412yes—
98C2×#95—22no—
99#324—24no—
100#323—24no—
101#323—24no—
102#323C2318yes—
103C25C22⋊C411yescharacteristic
104C25C22×C411yescharacteristic
105C22×D4—42no—
106C22×D4—42no—
107C22×D4C22⋊C412yes—
108C22×D4C2×C814yes—
109C22×D4M1614yes—
110C22×D4C22×C412yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C2×C814yescomplemented
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C22×C412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C22×C8—44no—
128C22×C8—44no—
129#123—216no—
130#123—24no—
131#123—24no—
132#123C2×C818yescomplemented
133#123C2×C818yescomplemented
134C4×D4—28no—
135C4×D4—44no—
136C4×D4—44no—
137C4×D4—44no—
138#117—44no—
139#117—44no—
140#117—48no—
141#117—44no—
142C2×C4⋊C4—28no—
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—44no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C2×C814yescomplemented
172C2×C22⋊C4C22×C412yes—
173C2×C22⋊C4C22×C411yescharacteristic
174C2×C22⋊C4C22×C411yescharacteristic
175C2×C22⋊C4C22×C411yescharacteristic
176C2×C22⋊C4C2×D412yes—
177C2×C22⋊C4C2×D412yes—
178C2×C22⋊C4C2411yescharacteristic
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—44no—
184C2×C42C22⋊C412yes—
185C2×C42C2×D412yes—
186C4⋊C8—44no—
187C22⋊C8—28no—
188C22⋊C8—44no—
189C22⋊C8—44no—
190C4×C8—44no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201C24—22no—
202C24—22no—
203C24—41no—
204C24—41no—
205C24—82no—
206C24—82no—
207C24C22⋊C814yes—
208C24C2×C22⋊C412yes—
209C24C2×C22⋊C411yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C24C22×C812yes—
212C24C2×M1612yes—
213C24C23×C411yescharacteristic
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—48no—
219C2×D4—48no—
220C2×D4C22⋊C818yes—
221C2×D4C22×C814yes—
222C2×D4C2×M1614yes—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—48no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—48no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4C22⋊C814yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C22×C812yes—
265C22×C4C22×C812yes—
266C22×C4C22×D412yes—
267C2×C8—48no—
268C2×C8—164no—
269C2×C8—164no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—22no—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4#9914yes—
297C22⋊C4C22×C814yes—
298C42—24no—
299C42—42no—
300C42—44no—
301C42—44no—
302C42—82no—
303C42—82no—
304C23—21no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—22no—
310C23—24no—
311C23—24no—
312C23—42no—
313C23—44no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—84no—
319C23—82no—
320C23—82no—
321C23—84no—
322C23—84no—
323C23—162no—
324C23—162no—
325C23C2×C22⋊C814yes—
326C23C2×C22⋊C812yes—
327C23C2×C22⋊C812yes—
328C23C2×#9912yes—
329C23#41312yes—
330C23C22×C22⋊C411yescharacteristic
331C23C23×C811yescharacteristic
332C23C22×M1611yescharacteristic
333D4—216no—
334D4—44no—
335D4—44no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—48no—
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—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—162no—
374C2×C4—162no—
375C2×C4C2×C22⋊C814yes—
376C2×C4C2×#9914yes—
377C8—88no—
378C22—22no—
379C22—24no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—24no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—82no—
397C22—84no—
398C22—84no—
399C22—164no—
400C22#145914yes—
401C22C22×C22⋊C811yescharacteristic
402C22C22×#9911yescharacteristic
403C22C2×#41311yescharacteristic
404C4—24no—
405C4—24no—
406C4—44no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—81no—
413C4—81no—
414C4—82no—
415C2—21no—
416C2—21no—
417C2—24no—
418C2—41no—
419C2—41no—
420C2—42no—
421C2—44no—
422C2C2×#145911yescharacteristic
423C2#3335012yes—
subgroupG/H
center, socleC22C22×#99
commutatorC23C23×C8
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11474C2semidirect
4. #11472C2semidirect
5. #11453C2semidirect
6. C2×#1317C2semidirect
29. #1326C22semidirect
30. #1317C22semidirect
31. #1317C22semidirect
36. C2×#123C8semidirect
120. C23×C4C2×C8semidirect
132. #123C2×C8semidirect
133. #123C2×C8semidirect
171. C2×C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
#341992
#173502
#114742
#114722
#114532
C2×#13172