Finite Group Databasebeta

← #481882

Subgroups — #481882

Order 512.

863 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36031C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#17370C211yesmaximal, characteristic
4#11817C214yesmaximal, complemented
5#11814C214yesmaximal, complemented
6#11723C212yesmaximal, complemented
7#11721C212yesmaximal, complemented
8C2×#554C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2975—21no—
11#2879—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#1822—22no—
28#1821C2214yes—
29#1742C2212yes—
30C2×#343—21no—
31#1465—22no—
32#1463C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1354C2214yes—
36#1353C2214yes—
37#1353C2218yes—
38C22×#41D411yescharacteristic
39C22×M16—22no—
40#554—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8C2×C412yes—
45#520—44no—
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×#117D412yes—
56C2×#117C2311yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60#436—44no—
61#431—22no—
62#408—44no—
63#407—24no—
64#407—24no—
65#400—22no—
66#400—22no—
67#400C2×C412yes—
68#400C2×C412yes—
69#400C2×C412yes—
70#400C2312yes—
71#400C2311yescharacteristic
72#399C2×C412yes—
73#399C2×C411yescharacteristic
74#398—22no—
75#398—22no—
76#398—21no—
77#396—21no—
78#395—21no—
79#395C2×C412yes—
80#395C2311yescharacteristic
81#393—21no—
82#393C2×C411yescharacteristic
83#391—21no—
84#391—21no—
85#391—22no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389—21no—
90#387—22no—
91#387—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#382—21no—
99#382C2×C411yescharacteristic
100#381—22no—
101#380—22no—
102#380—21no—
103#380—22no—
104#380C2×C412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C22⋊C4—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95C2×C412yes—
111#343—24no—
112#343—22no—
113#343C2314yes—
114#324—24no—
115#324C2314yes—
116#323—22no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8—21no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C411yescharacteristic
123C22×D4C22⋊C412yes—
124C22×D4C22⋊C411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C22⋊C412yes—
128C23×C4C22×C412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C2×M16—24no—
133C2×M16—42no—
134C2×M16—44no—
135C2×M16—44no—
136C2×M16—44no—
137C22×C8—42no—
138C4⋊Q8—42no—
139#125—44no—
140C22⋊Q8—48no—
141C22⋊Q8—44no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—44no—
148C4×D4—44no—
149C4×D4—44no—
150C4×D4—44no—
151#117—42no—
152#117—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2411yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22×C411yescharacteristic
184C2×C22⋊C4C2×D411yescharacteristic
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C4⋊C8—42no—
192C22⋊C8—24no—
193C22⋊C8—48no—
194C22⋊C8—44no—
195C4×C8—42no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—21no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95C22×C411yescharacteristic
212#95C22×C412yes—
213#95C22×C411yescharacteristic
214C24C2×C22⋊C412yes—
215C24C2×C22⋊C411yescharacteristic
216#41—48no—
217#41—48no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—81no—
224C2×Q8—82no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4#9914yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#13714yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264M16—44no—
265M16—44no—
266M16—84no—
267M16—164no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—84no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—81no—
298C22⋊C4—82no—
299C22⋊C4—81no—
300C42—22no—
301C42—22no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23C2×#9914yes—
315C23#41912yes—
316C23C22×C22⋊C411yescharacteristic
317Q8—44no—
318Q8—82no—
319Q8—162no—
320D4—28no—
321D4—48no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—48no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—162no—
363C2×C4—164no—
364C2×C4—164no—
365C2×C4C2×#9912yes—
366C2×C4#41912yes—
367C2×C4#42112yes—
368C2×C4C2×#13712yes—
369C8—84no—
370C8—84no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22—22no—
376C22—24no—
377C22—42no—
378C22—44no—
379C22—44no—
380C22—41no—
381C22—41no—
382C22—81no—
383C22C22×#9911yescharacteristic
384C22C2×#41911yescharacteristic
385C22C2×#42111yescharacteristic
386C4—22no—
387C4—41no—
388C4—41no—
389C4—42no—
390C4—42no—
391C4—41no—
392C4—41no—
393C4—42no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—82no—
400C4#151114yes—
401C2—21no—
402C2—21no—
403C2—44no—
404C2—42no—
405C2C2×#151111yescharacteristic
406C2#3335011yescharacteristic
407C2#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. #11817C2semidirect
5. #11814C2semidirect
6. #11723C2semidirect
7. #11721C2semidirect

Maximal subgroups

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

subgroupindex
#360312
#206952
#173702
#118172
#118142
#117232
#117212