Finite Group Databasebeta

← #197512

Subgroups — #197512

Order 512.

1303 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#17575C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9200C218yesmaximal
6#7981C212yesmaximal
7C2×#533C2211yescharacteristic
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2212yes—
11#2855—22no—
12#2627C2214yes—
13#2582—22no—
14#2581C414yescomplemented
15#2568C412yescomplemented
16#2567C412yescomplemented
17#2565—22no—
18#2351—24no—
19Q8×C4⋊C4—24no—
20#2317—21no—
21#2316—21no—
22#2316C2214yes—
23#2310—21no—
24#2285—22no—
25#2285C2211yescharacteristic
26#2285C2212yes—
27#1520C2218yes—
28#1518—24no—
29#1503—24no—
30#1461—22no—
31#1461—22no—
32C2×#324C2214yes—
33#1289C2218yes—
34C23×Q8D411yescharacteristic
35#533—22no—
36#533—22no—
37#533—22no—
38#533—22no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8C2312yes—
44#519—44no—
45#519—44no—
46C2×C4×Q8—28no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117—22no—
60C2×#117—24no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×C4×C8—22no—
72#400C2×C418yes—
73#398—28no—
74#395C2×C418yes—
75#391—22no—
76#391—22no—
77#391—24no—
78#391—24no—
79#389—28no—
80#389—24no—
81#389—24no—
82#389—216no—
83#387—28no—
84#385—24no—
85#384—24no—
86#384—22no—
87#384—21no—
88#384—22no—
89#382—24no—
90#382—22no—
91#382C2×C412yes—
92#382D414yes—
93#381—24no—
94#380—24no—
95#380—21no—
96#380—22no—
97#380C2312yes—
98#380C2314yes—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D414yes—
105#328—216no—
106#324—28no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8C22⋊C412yes—
113C22×Q8C22⋊C412yes—
114C22×Q8SD1614yes—
115C22×Q8Q1614yes—
116C22×Q8C22×C412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—24no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127#125—42no—
128#125—42no—
129C22⋊Q8—24no—
130C22⋊Q8—24no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—24no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—432no—
146C4×Q8—42no—
147C4×Q8—42no—
148#117—48no—
149#117—42no—
150#117—42no—
151#117—48no—
152#117—44no—
153#117—42no—
154#117—44no—
155#117—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C414yes—
178C2×C4⋊C4C2×D414yes—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—28no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42C22⋊C414yes—
191C2×C42C22⋊C414yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42#4112yes—
199C2×C42#4112yes—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C4×C8—42no—
205C4×C8—42no—
206#95—28no—
207#95—22no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—24no—
212#95C22×C414yes—
213#95C2×D418yes—
214C24C22×D411yescharacteristic
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—28no—
220C2×Q8—28no—
221C2×Q8—28no—
222C2×Q8—28no—
223C2×Q8—24no—
224C2×Q8—44no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—82no—
228C2×Q8#10318yes—
229C2×Q8#10318yes—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C414yes—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12012yes—
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12312yes—
255C22×C4C2×SD1614yes—
256C22×C4C2×SD1612yes—
257C22×C4C2×Q1614yes—
258C22×C4C2×Q1612yes—
259C22×C4AGL(1,ℤ/8)12yes—
260C22×C4#13712yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D412yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4112yes—
265C2×C8—44no—
266C2×C8—84no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—48no—
273C4⋊C4—44no—
274C4⋊C4—416no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—84no—
285C4⋊C4—82no—
286C4⋊C4—88no—
287C4⋊C4—88no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—42no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—416no—
299C42—44no—
300C42—44no—
301C42—48no—
302C42C4×D412yes—
303C42C4×D412yes—
304C42C4×D412yes—
305C42C4×D412yes—
306C42#12112yes—
307C42#12112yes—
308C42#12112yes—
309C42#12112yes—
310C23—41no—
311C23—41no—
312C23C22×C22⋊C411yescharacteristic
313C23C2×#12011yescharacteristic
314C23C2×#12311yescharacteristic
315C23C22×SD1611yescharacteristic
316C23C22×Q1611yescharacteristic
317C23C2×AGL(1,ℤ/8)11yescharacteristic
318C23C2×#13711yescharacteristic
319Q8—216no—
320Q8—216no—
321Q8—84no—
322Q8—82no—
323Q8—82no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4#38611yescharacteristic
355C2×C4#38611yescharacteristic
356C2×C4#38611yescharacteristic
357C2×C4#38612yes—
358C2×C4#38612yes—
359C2×C4#38611yescharacteristic
360C2×C4C2×#10318yes—
361C2×C4C2×#10312yes—
362C2×C4C2×#10312yes—
363C2×C4#41712yes—
364C2×C4#41712yes—
365C2×C4#45014yes—
366C2×C4#45114yes—
367C2×C4#48314yes—
368C2×C4#48414yes—
369C2×C4C2×C4×D412yes—
370C2×C4C2×#12112yes—
371C8—84no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22C2×#38611yescharacteristic
383C22C22×#10311yescharacteristic
384C22C2×#41711yescharacteristic
385C22C2×#45011yescharacteristic
386C22C2×#45111yescharacteristic
387C22C2×#48311yescharacteristic
388C22C2×#48411yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—22no—
394C4—28no—
395C4—44no—
396C4—88no—
397C4—84no—
398C4—82no—
399C4—82no—
400C4#187214yes—
401C4#187214yes—
402C4#191212yes—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2C2×#187211yescharacteristic
410C2C2×#191211yescharacteristic
411C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2581C4semidirect
15. #2568C4semidirect
16. #2567C4semidirect

Maximal subgroups

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

subgroupindex
#351792
#341822
#175752
C2×#14612
#92002
#79812