Finite Group Databasebeta

← #480884

Subgroups — #480884

Order 512.

981 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38951C211yesmaximal, characteristic
2#17416C212yesmaximal, complemented
3#11564C212yesmaximal, complemented
4#11555C214yesmaximal, complemented
5#11529C211yesmaximal, characteristic, complemented
6#11508C212yesmaximal, complemented
7#11475C211yesmaximal, characteristic, complemented
8#11432C212yesmaximal, complemented
9C2×#547C411yescharacteristic
10C2×#547C2211yescharacteristic
11#2880—22no—
12#2862C411yescharacteristic
13#2706—24no—
14#2620C412yes—
15#2547C2211yescharacteristic
16#2542C411yescharacteristic
17#2502C2212yes—
18#2431C412yes—
19#2417—22no—
20#2407—22no—
21#2311C2212yes—
22C4×#127C2211yescharacteristic
23#2293C411yescharacteristic
24#1472—22no—
25#1468—24no—
26#1465C2212yes—
27#1463C2214yes—
28#1459C2212yes—
29C2×#324C2214yes—
30#1325C2212yes—
31#1325C2212yes—
32#1319C2214yes—
33#1319C2214yes—
34#1318C2212yes—
35#1318C2211yescharacteristic
36#1318C2211yescharacteristic
37C22×#41C2×C412yes—
38C22×#41D412yes—
39C2×#131—42no—
40#547—42no—
41#547—42no—
42#547—42no—
43#547—42no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#124C2×C412yes—
48C2×#123—22no—
49C2×#123D414yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8D412yes—
53C2×#121—21no—
54C2×#121C2×C412yes—
55C2×C4×D4—22no—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C2×#117C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62#408—42no—
63#408—44no—
64#407—24no—
65#407—22no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—22no—
68#399—22no—
69#398—24no—
70#397—22no—
71#396—24no—
72#394—24no—
73#392—24no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2312yes—
78#390—21no—
79#390—22no—
80#390C2×C411yescharacteristic
81#390C2×C412yes—
82#390C2311yescharacteristic
83#389—21no—
84#388—22no—
85#386—24no—
86#386—24no—
87#385—24no—
88#384C2×C412yes—
89#384C2311yescharacteristic
90#382—22no—
91#380—22no—
92#380—24no—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C22⋊C4C2×C412yes—
95C43C2×C411yescharacteristic
96#324—24no—
97#324—24no—
98#324C2312yes—
99#324C2314yes—
100#324C2312yes—
101#323—22no—
102#323—22no—
103C2×#41—22no—
104C2×#41—24no—
105C2×#41—22no—
106C2×#41—24no—
107C2×#41—42no—
108C2×#41—44no—
109C2×#41—84no—
110C2×#41—82no—
111C22×Q8C22⋊C412yes—
112C22×Q8C22×C412yes—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C22⋊C414yes—
116C22×D4C22⋊C411yescharacteristic
117C22×D4C22×C411yescharacteristic
118C22×D4C22×C411yescharacteristic
119C23×C4—21no—
120C23×C4—22no—
121C23×C4C22×C412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D414yes—
124#131—88no—
125C2×M16—44no—
126C2×M16—42no—
127C22×C8—44no—
128C22×C8—42no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133#127—22no—
134#127—41no—
135#124—82no—
136#123—44no—
137#123—44no—
138#123—48no—
139C22⋊Q8—42no—
140C22⋊Q8—42no—
141C22⋊Q8—44no—
142C22⋊Q8—84no—
143#121—44no—
144#121—42no—
145#121—42no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—84no—
153#117—82no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2411yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22⋊C414yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—24no—
190C22⋊C8—44no—
191C22⋊C8—42no—
192C22⋊C8—44no—
193C22⋊C8—42no—
194#95—22no—
195#95—21no—
196#95—24no—
197#95—24no—
198#95—24no—
199C24—21no—
200C24C2×C22⋊C412yes—
201C24C2×C22⋊C412yes—
202#41—48no—
203#41—84no—
204#41—88no—
205#41—88no—
206#41—88no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—44no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—84no—
229C2×D4—82no—
230C2×D4—82no—
231C2×D4#9912yes—
232C2×D4#9912yes—
233C2×D4#13112yescomplemented
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—82no—
254C22×C4—84no—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D412yes—
261M16—88no—
262M16—84no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—88no—
267C2×C8—84no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—84no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—82no—
290C22⋊C4—84no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—84no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—41no—
299C42—42no—
300C42—42no—
301C42—42no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23C2×#9914yes—
317C23C22×C22⋊C411yescharacteristic
318Q8—44no—
319Q8—82no—
320Q8—82no—
321Q8—82no—
322D4—24no—
323D4—42no—
324D4—42no—
325D4—42no—
326D4—48no—
327D4—81no—
328D4—81no—
329D4—84no—
330D4—88no—
331D4—88no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—48no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—84no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4C2×#9911yescharacteristic
368C2×C4C2×#9911yescharacteristic
369C2×C4C2×#13111yescharacteristic
370C2×C4#56811yescharacteristic
371C8—82no—
372C8—84no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—81no—
390C22—81no—
391C22#288011yescharacteristic
392C22C22×#9911yescharacteristic
393C22#288811yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—44no—
401C4—41no—
402C4—81no—
403C4—82no—
404C4—82no—
405C4—81no—
406C4#212712yes—
407C2—21no—
408C2—21no—
409C2—44no—
410C2—82no—
411C2—82no—
412C2#1741111yescharacteristic
413C2C2×#212711yescharacteristic
414C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#38951C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#547 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17416C2semidirect
3. #11564C2semidirect
4. #11555C2semidirect
5. #11529C2semidirect
6. #11508C2semidirect
7. #11475C2semidirect
8. #11432C2semidirect
233. C2×D4#131semidirect

Maximal subgroups

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

subgroupindex
#389512
#174162
#115642
#115552
#115292
#115082
#114752
#114322