Finite Group Databasebeta

← #480883

Subgroups — #480883

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36419C211yesmaximal, characteristic
2#17428C212yesmaximal, complemented
3#11567C212yesmaximal, complemented
4#11554C214yesmaximal, complemented
5#11532C211yesmaximal, characteristic, complemented
6#11507C212yesmaximal, complemented
7#11478C211yesmaximal, characteristic, complemented
8#11431C212yesmaximal, complemented
9C2×#556C2211yescharacteristic
10C2×#550C411yescharacteristic
11#2984—22no—
12#2858C411yescharacteristic
13#2850C412yes—
14#2568C411yescharacteristic
15#2559—24no—
16#2548C2211yescharacteristic
17#2544—22no—
18#2500C2212yes—
19#2432C412yes—
20#2367—22no—
21C4×C4⋊Q8C2211yescharacteristic
22#2297C2212yes—
23C42×D4C411yescharacteristic
24#1473—24no—
25#1469—22no—
26#1467C2212yes—
27#1464C2214yes—
28#1461C2212yes—
29#1458C2214yes—
30#1325C2212yes—
31#1325C2212yes—
32#1319C2212yes—
33#1319C2211yescharacteristic
34#1319C2211yescharacteristic
35#1318C2214yes—
36#1318C2214yes—
37C22×#41C2×C412yes—
38#556—41no—
39#556—42no—
40#556—41no—
41#550—42no—
42#550—41no—
43#550—41no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127—21no—
47C2×#127C2×C411yescharacteristic
48C2×#126D414yes—
49C2×#124—22no—
50C2×#124C2×C412yes—
51C2×#124D412yes—
52C2×C22⋊Q8C2×C412yes—
53C2×#121—22no—
54C2×#121C2×C412yes—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C411yescharacteristic
61C2×#117C2312yes—
62C22×C42C2×C412yes—
63#446—44no—
64#446—42no—
65C8×Q8—42no—
66#432—24no—
67#432—22no—
68#431—24no—
69#431—22no—
70#400—22no—
71#394—22no—
72#394—24no—
73#394—24no—
74#391—22no—
75#391C2×C411yescharacteristic
76#391C2×C412yes—
77#391C2311yescharacteristic
78#390—22no—
79#390—22no—
80#390C2×C412yes—
81#390C2×C412yes—
82#390C2312yes—
83#389—22no—
84#388—24no—
85#388—24no—
86#386—24no—
87#386—24no—
88#385—24no—
89#384C2311yescharacteristic
90#382C2×C412yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C22⋊C4—22no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4C2×C412yes—
97C43C2×C411yescharacteristic
98#324—22no—
99#324—22no—
100#324C2312yes—
101#324C2314yes—
102#324C2312yes—
103#323—24no—
104#323—24no—
105C2×#41—24no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41—42no—
109C2×#41—82no—
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C411yescharacteristic
112C22×Q8C22×C411yescharacteristic
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4C22⋊C411yescharacteristic
118C22×D4C22×C412yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C22×C412yes—
122C4⋊Q8—22no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126#127—41no—
127#127—41no—
128#127—42no—
129#127—42no—
130#126—44no—
131#126—44no—
132#124—42no—
133#124—42no—
134#124—48no—
135#124—82no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138#121—44no—
139#121—44no—
140#121—42no—
141#121—42no—
142C4×Q8—22no—
143C4×Q8—24no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×D4—22no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—41no—
155C4×D4—41no—
156#117—42no—
157#117—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2411yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C414yes—
177C2×C22⋊C4C2×D414yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—82no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C412yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D414yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C4⋊C8—42no—
191C4⋊C8—44no—
192C4⋊C8—42no—
193C22⋊C8—22no—
194C22⋊C8—24no—
195C22⋊C8—22no—
196#97—44no—
197#97—42no—
198C4×C8—44no—
199C4×C8—42no—
200#95—21no—
201#95—22no—
202#95—24no—
203#95—22no—
204#95—24no—
205#95—22no—
206C24—22no—
207C24C2×C22⋊C412yes—
208#41—84no—
209#41—84no—
210#41—84no—
211#41—84no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8#13112yes—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—42no—
228C2×D4—44no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—84no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C414yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D412yes—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—42no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—44no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4#9912yes—
279C4⋊C4#9911yescharacteristic
280C4⋊C4#9911yescharacteristic
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—84no—
291C22⋊C4—84no—
292C42—24no—
293C42—24no—
294C42—22no—
295C42—24no—
296C42—22no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—41no—
302C42—42no—
303C42—41no—
304C42—41no—
305C42—82no—
306C42—81no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—42no—
311C23—42no—
312C23—44no—
313C23—44no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23C22×C22⋊C411yescharacteristic
318Q8—42no—
319Q8—42no—
320Q8—44no—
321Q8—81no—
322Q8—81no—
323Q8—82no—
324D4—42no—
325D4—42no—
326D4—82no—
327D4—82no—
328D4—82no—
329D4—88no—
330D4—82no—
331D4—162no—
332D4—164no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—88no—
368C2×C4C2×#9911yescharacteristic
369C2×C4C2×#9912yes—
370C2×C4C2×#9912yes—
371C2×C4C2×#9911yescharacteristic
372C2×C4C2×#13111yescharacteristic
373C2×C4#56811yescharacteristic
374C8—82no—
375C8—84no—
376C8—82no—
377C22—21no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—82no—
384C22—84no—
385C22—84no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22#288011yescharacteristic
390C22C22×#9911yescharacteristic
391C22#288811yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—41no—
403C4—44no—
404C4—82no—
405C4—81no—
406C4#213212yes—
407C2—21no—
408C2—21no—
409C2—82no—
410C2—162no—
411C2#1741111yescharacteristic
412C2#3336211yescharacteristic
413C2C2×#213211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#556 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17428C2semidirect
3. #11567C2semidirect
4. #11554C2semidirect
5. #11532C2semidirect
6. #11507C2semidirect
7. #11478C2semidirect
8. #11431C2semidirect

Maximal subgroups

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

subgroupindex
#364192
#174282
#115672
#115542
#115322
#115072
#114782
#114312