Finite Group Databasebeta

← #497650

Subgroups — #497650

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39171C211yesmaximal, characteristic
2Q8×#99C212yesmaximal, complemented
3#13225C214yesmaximal, complemented
4#13223C212yesmaximal
5#12799C212yesmaximal, complemented
6#12798C211yesmaximal, characteristic, complemented
7#12628C212yesmaximal, complemented
8C4×#354C211yesmaximal, characteristic, complemented
9C2×#559C411yescharacteristic, complemented
10C2×D4×Q8C2211yescharacteristic
11#2858C411yescharacteristic, complemented
12#2696—24no—
13#2640C412yescomplemented
14#2568C411yescharacteristic, complemented
15#2563C412yescomplemented
16#2544C2212yes—
17#2541C2211yescharacteristic
18#2436—22no—
19#2427—22no—
20Q8×C22⋊C4—22no—
21#2328C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic, complemented
24C2×#354C2212yes—
25C2×#354C2212yes—
26C2×#353C2214yes—
27C2×#353C2214yes—
28#1996—22no—
29#1996—24no—
30#1799C2212yes—
31#1799C2214yes—
32C4×#99C2212yes—
33C4×#99C2214yes—
34#1393C2211yescharacteristic
35#1393C2211yescharacteristic
36#1393C2212yes—
37C23×Q8D412yes—
38#559—42no—
39#559—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#127C2312yes—
46C2×#124—21no—
47C2×#124—22no—
48C2×#124C2×C412yes—
49C2×#124C2311yescharacteristic
50C2×#123—22no—
51C2×C22⋊Q8D412yes—
52C2×C22⋊Q8D414yes—
53C2×#121—22no—
54C2×C4×Q8—42no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2312yes—
60C2×C4×D4C2311yescharacteristic
61C2×#99C2314yes—
62C2×#99C2312yes—
63C2×#99C2312yes—
64#400—22no—
65#397—24no—
66#394—24no—
67#393—24no—
68#393—24no—
69#392—22no—
70#391—22no—
71#391—21no—
72#391—42no—
73#391—44no—
74#391C2×C412yes—
75#391C2×C411yescharacteristic
76#390—22no—
77#390C2×C412yes—
78#390C2×C412yes—
79#390C2×C412yes—
80#390C2311yescharacteristic
81#389—21no—
82#388—22no—
83#387—22no—
84#386—24no—
85#386—24no—
86#386—24no—
87#382C2×C411yescharacteristic
88#382C2×C412yes—
89#380—24no—
90#380—22no—
91#380—24no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#354—24no—
100#354—42no—
101#353—28no—
102#353—44no—
103#342—24no—
104#342—22no—
105#342—24no—
106#342—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—24no—
110C22×Q8—84no—
111C22×Q8C22⋊C412yes—
112C22×Q8C22⋊C414yes—
113C22×Q8C22⋊C411yescharacteristic
114C22×Q8C22×C411yescharacteristic
115C22×D4—24no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4C22×C412yes—
119C22×D4C2411yescharacteristic
120C23×C4—22no—
121C23×C4C2×D412yes—
122C23×C4C2×D414yes—
123C4⋊Q8—44no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126#127—24no—
127#127—22no—
128#127—22no—
129#127—41no—
130#127—41no—
131#124—22no—
132#124—21no—
133#124—21no—
134#124—44no—
135#124—42no—
136#124—42no—
137#124—84no—
138#123—48no—
139C22⋊Q8—42no—
140C22⋊Q8—42no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143#121—84no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—88no—
149C4×D4—44no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4C2×D414yes—
179C2×C22⋊C4C2×D412yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—44no—
184C2×C42—42no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190#99—22no—
191#99—24no—
192#99—22no—
193#99—22no—
194#99—24no—
195#99#4114yes—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—21no—
202#95—24no—
203#95—24no—
204#95—42no—
205#95—44no—
206#95—42no—
207#95—44no—
208C24—22no—
209C24C22×D412yes—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—48no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—88no—
224C2×Q8#9912yes—
225C2×Q8#9912yes—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—44no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—84no—
239C2×D4—82no—
240C2×D4—84no—
241C2×D4C2×#4112yes—
242C2×D4H(5,𝔽2)11yescharacteristic
243C2×D42−1+411yescharacteristic
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C414yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C23×C411yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—88no—
279C4⋊C4—84no—
280C22⋊C4—24no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—82no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C22⋊C4—84no—
297C42—24no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—22no—
302C42—21no—
303C42—22no—
304C42—41no—
305C42—42no—
306C42—41no—
307C42—41no—
308C42—81no—
309C42—88no—
310C42—84no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23—44no—
316C23—44no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23C22×C22⋊C411yescharacteristic
321C23#54812yes—
322C23#55012yes—
323Q8—24no—
324Q8—42no—
325Q8—48no—
326Q8—44no—
327Q8—82no—
328Q8—84no—
329Q8—81no—
330Q8—81no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—82no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—162no—
339D4—164no—
340D4—164no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—48no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—164no—
381C2×C4C2×#9911yescharacteristic
382C2×C4C2×#9914yes—
383C2×C4C2×#9911yescharacteristic
384C2×C4C4×#4111yescharacteristic
385C2×C4#52011yescharacteristic
386C22—21no—
387C22—21no—
388C22—22no—
389C22—22no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—82no—
395C22—84no—
396C22—84no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22#234111yescharacteristic
401C22C22×#9911yescharacteristic
402C22#288411yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—24no—
407C4—44no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—81no—
412C4—81no—
413C4—82no—
414C4—84no—
415C4—81no—
416C4—81no—
417C4—164no—
418C4—162no—
419C4#212712yes—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—42no—
424C2—162no—
425C2#2253411yescharacteristic
426C2C2×#212711yescharacteristic
427C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×D4×Q8 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#99C2semidirect
3. #13225C2semidirect
5. #12799C2semidirect
6. #12798C2semidirect
7. #12628C2semidirect
8. C4×#354C2semidirect
9. C2×#559C4semidirect
11. #2858C4semidirect
13. #2640C4semidirect
14. #2568C4semidirect
15. #2563C4semidirect
23. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#391712
Q8×#992
#132252
#132232
#127992
#127982
#126282
C4×#3542