Finite Group Databasebeta

← #481503

Subgroups — #481503

Order 512.

1067 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36031C211yesmaximal, characteristic
2#20358C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#11723C212yesmaximal, complemented
5#11722C214yesmaximal, complemented
6#11721C212yesmaximal, complemented
7C2×#1353C214yesmaximal, complemented
8C2×#568—21no—
9C2×#554C411yescharacteristic
10C2×#520C2211yescharacteristic
11#2972—21no—
12#2628C2211yescharacteristic
13#2627C2212yes—
14#2581C411yescharacteristic
15#2568C411yescharacteristic
16#2564C412yes—
17#2487C412yes—
18#2453—21no—
19#2450—22no—
20#2442—21no—
21#2438—22no—
22#2396C2211yescharacteristic
23C2×#400C2212yes—
24#2349—21no—
25#2349C411yescharacteristic
26#2339—21no—
27#1742—22no—
28#1742C2212yes—
29C2×#343—21no—
30C2×#343C2214yes—
31#1461—21no—
32#1459—22no—
33#1458C2212yes—
34C2×#324C2214yes—
35#1354C2214yes—
36#1353C2214yes—
37#1353C2218yes—
38C22×#41D411yescharacteristic
39#568—28no—
40#568—44no—
41C2×#131—44no—
42C22×M16—22no—
43C22×M16—24no—
44#554—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2×C412yes—
49#520—44no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D412yes—
58C2×C4×D4D411yescharacteristic
59C2×#117D412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C8×D4—44no—
65#431—22no—
66C2×C22⋊C8—24no—
67#400—22no—
68#400—22no—
69#400C2×C412yes—
70#400C2×C412yes—
71#400C2×C412yes—
72#400C2312yes—
73#400C2311yescharacteristic
74#399C2×C412yes—
75#399C2×C411yescharacteristic
76#398—22no—
77#398—22no—
78#398—21no—
79#396—21no—
80#395—21no—
81#395C2×C412yes—
82#395C2311yescharacteristic
83#393—21no—
84#393C2×C411yescharacteristic
85#391—21no—
86#391—21no—
87#391—22no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—21no—
92#387—22no—
93#387—22no—
94#386—22no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#382—21no—
101#382C2×C411yescharacteristic
102#381—22no—
103#380—22no—
104#380—21no—
105#380—22no—
106#380C2×C412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95C2×C412yes—
113#343—24no—
114#343—22no—
115#343C2314yes—
116#324—24no—
117#324C2314yes—
118#323—22no—
119C2×#41—24no—
120C2×#41—24no—
121C2×#41SD1614yes—
122C2×#41Q1614yes—
123C22×Q8—21no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C22×D4C22⋊C412yes—
127C22×D4C22⋊C411yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C22⋊C412yes—
131C23×C4C22×C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135#131—432no—
136C2×M16—216no—
137C2×M16—28no—
138C2×M16—24no—
139C2×M16—44no—
140C2×M16—42no—
141C2×M16—44no—
142C2×M16—48no—
143C22×C8—44no—
144C22×C8—42no—
145C22×C8—44no—
146C4⋊Q8—42no—
147#125—44no—
148C22⋊Q8—48no—
149C22⋊Q8—44no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—44no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—44no—
159#117—42no—
160#117—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2411yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22×C411yescharacteristic
192C2×C22⋊C4C2×D411yescharacteristic
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C4⋊C8—42no—
200C22⋊C8—24no—
201C22⋊C8—44no—
202C4×C8—42no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—21no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95C22×C411yescharacteristic
219#95C22×C412yes—
220#95C22×C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C24C2×C22⋊C412yes—
223#41—216no—
224#41—216no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—81no—
230C2×Q8—82no—
231C2×Q8#10314yes—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4#9914yes—
235C2×D4#10318yes—
236C2×D4#10314yes—
237C2×D4#10414yes—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×SD1614yes—
267C22×C4C2×SD1612yes—
268C22×C4C2×Q1614yes—
269C22×C4C2×Q1612yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273M16—28no—
274M16—416no—
275M16—48no—
276M16—44no—
277C2×C8—44no—
278C2×C8—416no—
279C2×C8—48no—
280C2×C8—44no—
281C2×C8—164no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—81no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—81no—
309C22⋊C4—82no—
310C22⋊C4—81no—
311C42—22no—
312C42—22no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C23—21no—
318C23—21no—
319C23—42no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23C2×#9914yes—
324C23C2×#10312yes—
325C23C2×#10314yes—
326C23C2×#10414yes—
327C23C22×C22⋊C411yescharacteristic
328Q8—28no—
329Q8—82no—
330Q8—162no—
331D4—216no—
332D4#328116yes—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—162no—
374C2×C4—164no—
375C2×C4—164no—
376C2×C4C2×#9912yes—
377C2×C4C2×#10314yes—
378C2×C4C2×#10312yes—
379C2×C4C2×#10412yes—
380C2×C4C22×SD1611yescharacteristic
381C2×C4C22×Q1611yescharacteristic
382C8—48no—
383C8—84no—
384C22—21no—
385C22—24no—
386C22—21no—
387C22—24no—
388C22—22no—
389C22—44no—
390C22—44no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—81no—
395C22C2×#32818yes—
396C22C22×#9911yescharacteristic
397C22C22×#10311yescharacteristic
398C22C22×#10411yescharacteristic
399C4—21no—
400C4—21no—
401C4—24no—
402C4—41no—
403C4—41no—
404C4—42no—
405C4—42no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—82no—
414C4C2×#32814yes—
415C2—21no—
416C2—21no—
417C2—28no—
418C2—42no—
419C2C22×#32811yescharacteristic
420C2#3335011yescharacteristic
421C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11723C2semidirect
5. #11722C2semidirect
6. #11721C2semidirect
7. C2×#1353C2semidirect

Maximal subgroups

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

subgroupindex
#360312
#203582
#173512
#117232
#117222
#117212
C2×#13532