Finite Group Databasebeta

← #497651

Subgroups — #497651

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39173C211yesmaximal, characteristic
2#22513C212yesmaximal
3#13224C214yesmaximal
4#13224C212yesmaximal
5#12800C212yesmaximal
6#12798C211yesmaximal, characteristic
7#12627C212yesmaximal
8C4×#354C211yesmaximal, characteristic
9C2×#557C411yescharacteristic, complemented
10C2×#554C2211yescharacteristic
11#2862C411yescharacteristic, complemented
12#2690—24no—
13#2642C412yescomplemented
14#2567C411yescharacteristic, complemented
15#2565C412yescomplemented
16#2547C2212yes—
17#2547C2211yescharacteristic
18#2442—22no—
19#2421—22no—
20#2345—22no—
21#2326C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic, complemented
24C2×#354C2212yes—
25C2×#354C2214yes—
26C2×#354C2214yes—
27C2×#354C2212yes—
28#1996—22no—
29#1996—24no—
30#1799C2214yes—
31#1799C2212yes—
32C4×#99C2212yes—
33C4×#99C2214yes—
34#1393C2211yescharacteristic
35#1393C2212yes—
36#1393C2211yescharacteristic
37#557—42no—
38#557—42no—
39#554—42no—
40#554—42no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—22no—
45C2×#125—21no—
46C2×#125C2×C412yes—
47C2×#124C2312yes—
48C2×#124C2311yescharacteristic
49C2×C22⋊Q8D412yes—
50C2×C22⋊Q8D414yes—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×D4C2312yes—
56C2×C4×D4C2311yescharacteristic
57C22×C4⋊C4D412yes—
58C2×#99C2312yes—
59C2×#99C2314yes—
60C2×#99C2312yes—
61#400—24no—
62#400—24no—
63#398—24no—
64#397—24no—
65#397—22no—
66#395—22no—
67#391—22no—
68#391—21no—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2×C411yescharacteristic
74#390C2311yescharacteristic
75#389—22no—
76#389—21no—
77#389—42no—
78#388—24no—
79#388—22no—
80#387—24no—
81#387—22no—
82#387—44no—
83#385—24no—
84#384C2×C412yes—
85#382—22no—
86#382C2×C411yescharacteristic
87#380—22no—
88#380—24no—
89#380—24no—
90#380—22no—
91C4×C4⋊C4—42no—
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—28no—
101#354—44no—
102#354—42no—
103#342—24no—
104#342—22no—
105#342—22no—
106#342—24no—
107C22×Q8C22⋊C411yescharacteristic
108C22×Q8C22⋊C412yes—
109C22×Q8C22×C412yes—
110C22×Q8C22×C411yescharacteristic
111C22×D4C2411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D414yes—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—44no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119#125—44no—
120#125—48no—
121#125—42no—
122#125—42no—
123#124—24no—
124#124—22no—
125#124—22no—
126#124—21no—
127#124—22no—
128#124—21no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—44no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×D4—44no—
141C4×D4—41no—
142C4×D4—41no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C414yes—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4C2×D412yes—
172C2×C22⋊C4C2×D414yes—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—42no—
178C2×C42—44no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C412yes—
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184#99—22no—
185#99—22no—
186#99—24no—
187#99—22no—
188#99—24no—
189#99#4114yes—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—42no—
201#95—44no—
202#95—44no—
203#95—42no—
204C24C22×D412yes—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—22no—
208C2×Q8—21no—
209C2×Q8—22no—
210C2×Q8—21no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—82no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4C2×#4112yes—
221C2×D4H(5,𝔽2)11yescharacteristic
222C2×D42−1+411yescharacteristic
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—84no—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C23×C411yescharacteristic
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—44no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—82no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C4⋊C4—164no—
266C4⋊C4#9912yes—
267C4⋊C4#9912yes—
268C22⋊C4—24no—
269C22⋊C4—44no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—82no—
281C42—24no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—22no—
286C42—21no—
287C42—22no—
288C42—41no—
289C42—42no—
290C42—44no—
291C42—41no—
292C42—41no—
293C42—81no—
294C42—84no—
295C42—164no—
296C23—22no—
297C23—22no—
298C23—42no—
299C23—42no—
300C23C22×C22⋊C411yescharacteristic
301C23#54812yes—
302C23#55012yes—
303Q8—42no—
304Q8—42no—
305Q8—82no—
306Q8—81no—
307Q8—82no—
308Q8—81no—
309Q8—162no—
310D4—42no—
311D4—41no—
312D4—41no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4C2×#9911yescharacteristic
356C2×C4C2×#9914yes—
357C2×C4C2×#9911yescharacteristic
358C2×C4C4×#4111yescharacteristic
359C2×C4#52011yescharacteristic
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22#234111yescharacteristic
370C22C22×#9911yescharacteristic
371C22#288411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—24no—
376C4—42no—
377C4—42no—
378C4—44no—
379C4—41no—
380C4—81no—
381C4—82no—
382C4—81no—
383C4—84no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—162no—
388C4—164no—
389C4#212912yes—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—42no—
394C2#2253411yescharacteristic
395C2C2×#212911yescharacteristic
396C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, prime radical, cosocleC22×D4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#557C4semidirect
11. #2862C4semidirect
13. #2642C4semidirect
14. #2567C4semidirect
15. #2565C4semidirect
23. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#391732
#225132
#132242
#128002
#127982
#126272
C4×#3542