Finite Group Databasebeta

← #201252

Subgroups — #201252

Order 512.

1065 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#60248C211yesmaximal, characteristic
2#35248C211yesmaximal, characteristic
3#17734C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9456C214yesmaximal, complemented
6#9452C214yesmaximal, complemented
7#7951C212yesmaximal
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—21no—
12#2983C2214yes—
13#2982—21no—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17C2×#432C2212yes—
18#2855—22no—
19#2634—21no—
20#2634—21no—
21#2581—21no—
22#2581—21no—
23#2568C2212yes—
24#2546C2212yes—
25#2503C412yes—
26#2484—22no—
27#2434C412yes—
28#2327C414yes—
29#2283C2211yescharacteristic
30#1559—24no—
31#1558C2214yes—
32#1556C2214yes—
33#1542—24no—
34#1467—21no—
35#1467—21no—
36#1467C2214yes—
37C2×#324—22no—
38#1285C2218yes—
39C23×Q8C2×C411yescharacteristic
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8C2×C414yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2312yes—
62C2×#117—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67#446—28no—
68#446—216no—
69C8×Q8—28no—
70#432—24no—
71#432—24no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#432C2314yes—
77#432C2314yes—
78#431—24no—
79#431—24no—
80#431—22no—
81#431D414yes—
82#423—24no—
83#423D414yes—
84C2×C4⋊C8—22no—
85#407—22no—
86C2×C22⋊C8—22no—
87C4×M16—24no—
88C4×M16—22no—
89C2×#97—22no—
90#400—22no—
91#400—22no—
92#400—22no—
93#400—22no—
94#398—22no—
95#398—22no—
96#395—22no—
97#395—22no—
98#391—22no—
99#391—22no—
100#391C2×C414yes—
101#391C2×C414yes—
102#389—22no—
103#389—22no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#385—24no—
109#383—21no—
110#382—21no—
111#382C2×C414yes—
112#381—24no—
113#381—21no—
114#380—21no—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4C2×C412yes—
117#376C2×C412yes—
118#333—28no—
119#332—28no—
120#324—24no—
121#324—24no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—44no—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22×C412yes—
129C22×Q8C22×C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×M16—24no—
136C22×C8—42no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—48no—
142#125—42no—
143#125—42no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—42no—
165C4×Q8—42no—
166#117—42no—
167#117—42no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22×C414yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213C4⋊C8C2×D418yes—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218#97—28no—
219#97—24no—
220#97—24no—
221#97—42no—
222C4×C8—28no—
223C4×C8—24no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—22no—
234C24C2×C22⋊C411yescharacteristic
235C2×Q8—24no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—41no—
248C2×Q8—41no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C4×D412yes—
270C22×C4C4×D412yes—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4#12311yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—48no—
281C2×C8—28no—
282C2×C8—44no—
283C2×C8—84no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—48no—
302C22⋊C4—44no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—44no—
314C42—44no—
315C42C4×D412yes—
316C42C4×D412yes—
317C42#12012yes—
318C42#12112yes—
319C42#12112yes—
320C42#12312yes—
321C42C22×D412yes—
322C42C2×#4112yes—
323C23—22no—
324C23—22no—
325C23—41no—
326C23#38611yescharacteristic
327C23#38611yescharacteristic
328C23C22×C22⋊C411yescharacteristic
329Q8—28no—
330Q8—48no—
331Q8—44no—
332Q8—84no—
333Q8—82no—
334Q8—82no—
335C2×C4—22no—
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—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4#38611yescharacteristic
370C2×C4#38611yescharacteristic
371C2×C4#38612yes—
372C2×C4#38612yes—
373C2×C4#41314yes—
374C2×C4C2×C4×D412yes—
375C2×C4C2×#12011yescharacteristic
376C2×C4C2×#12112yes—
377C2×C4C2×#12311yescharacteristic
378C8—48no—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—44no—
388C22#190912yes—
389C22C2×#38611yescharacteristic
390C22C2×#41312yes—
391C4—22no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—22no—
398C4—24no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—84no—
403C4—82no—
404C4—82no—
405C4#302012yes—
406C2—22no—
407C2—22no—
408C2—41no—
409C2#2095411yescharacteristic
410C2#2098911yescharacteristic
411C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9456C2semidirect
6. #9452C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#352482
#177342
C2×#14672
#94562
#94522
#79512