Finite Group Databasebeta

← #496317

Subgroups — #496317

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35621C211yesmaximal, characteristic
2#20725C212yesmaximal, complemented
3#13225C214yesmaximal, complemented
4#13224C214yesmaximal, complemented
5#12698C212yesmaximal, complemented
6#12695C212yesmaximal
7C2×#536C411yescharacteristic, complemented
8C2×#536C2211yescharacteristic
9C2×#532C2211yescharacteristic
10C2×#520C2211yescharacteristic
11#2547C2212yes—
12#2546C412yescomplemented
13#2545C412yescomplemented
14#2544C2212yes—
15#2425—22no—
16#2425—22no—
17#2417—22no—
18#2415—22no—
19#2313—22no—
20#2311C411yescharacteristic, complemented
21#2296C411yescharacteristic, complemented
22C2×#391C411yescharacteristic, complemented
23C2×#354C2214yes—
24C2×#354C2218yes—
25C2×#353C2214yes—
26#1820—22no—
27#1799—22no—
28#1799C2214yes—
29#1782C2214yes—
30C4×#99—22no—
31C4×#99C2214yes—
32C22×#41C2×C411yescharacteristic
33C22×#41C2311yescharacteristic
34C23×Q8C2×C411yescharacteristic
35#536—44no—
36#536—44no—
37#532—44no—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C4⋊Q8C2×C412yes—
40C2×C4⋊Q8C2×C412yes—
41C2×#127C2311yescharacteristic
42C2×#124C2×C411yescharacteristic
43C2×#124C2×C412yes—
44C2×#124C2311yescharacteristic
45C2×#124C2312yes—
46C2×#123—22no—
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×#121—22no—
52#520—44no—
53C2×C4×Q8C2×C412yes—
54C2×C4×D4—22no—
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4C2312yes—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117D412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4D412yes—
63C22×C42C2×C411yescharacteristic
64C2×#99—24no—
65C2×#99C2318yes—
66#398—22no—
67#398—22no—
68#397—22no—
69#397—22no—
70#396—22no—
71#396—22no—
72#394—22no—
73#394—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#390C2×C412yes—
79#389—21no—
80#389—21no—
81#389—22no—
82#389—22no—
83#388—22no—
84#388—22no—
85#387—48no—
86#386—21no—
87#386—21no—
88#386—22no—
89#386—22no—
90#385—22no—
91#381—24no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—24no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C412yes—
98C4×C22⋊C4—24no—
99C2×#95—21no—
100C2×#95—21no—
101#354—48no—
102#354—44no—
103#353—44no—
104#342—24no—
105#342—24no—
106C2×#41—24no—
107C2×#41—42no—
108C2×#41—42no—
109C2×#41—42no—
110C2×#41—42no—
111C2×#41—42no—
112C22×Q8—21no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8C22×C412yes—
117C22×Q8C22×C411yescharacteristic
118C22×Q8C22×C411yescharacteristic
119C22×D4—21no—
120C22×D4—22no—
121C22×D4C22×C411yescharacteristic
122C22×D4C2×D412yes—
123C22×D4C2411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C4⋊Q8—42no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136#127—24no—
137#124—24no—
138#124—28no—
139#124—42no—
140#124—44no—
141#123—48no—
142#123—48no—
143C22⋊Q8—44no—
144C22⋊Q8—48no—
145C22⋊Q8—44no—
146#121—48no—
147C4×Q8—44no—
148C4×D4—44no—
149C4×D4—44no—
150C4×D4—44no—
151C4×D4—48no—
152#117—24no—
153#117—42no—
154#117—44no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—21no—
186C2×C42—42no—
187C2×C42—44no—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C412yes—
194#99—28no—
195#99—28no—
196#99—48no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—44no—
206#95—44no—
207#95—48no—
208C24—21no—
209C24C2×C22⋊C411yescharacteristic
210C24C22×D412yes—
211#41—416no—
212#41—88no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—41no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—88no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—42no—
239C2×D4—82no—
240C2×D4H(5,𝔽2)14yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—81no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C23×C411yescharacteristic
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—82no—
290C4⋊C4—164no—
291C4⋊C4#9914yes—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—48no—
303C22⋊C4—48no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—24no—
307C42—22no—
308C42—22no—
309C42—24no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—164no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23#41012yes—
328C23C22×C22⋊C411yescharacteristic
329C23#53514yes—
330Q8—44no—
331Q8—84no—
332Q8—82no—
333Q8—84no—
334D4—44no—
335D4—48no—
336D4—82no—
337D4—88no—
338D4—84no—
339C2×C4—28no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—48no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—88no—
376C2×C4—88no—
377C2×C4—88no—
378C2×C4—161no—
379C2×C4—161no—
380C2×C4C2×#9914yes—
381C2×C4C2×#9912yes—
382C2×C4#51812yes—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—24no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—42no—
393C22—41no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22#230911yescharacteristic
401C22C22×#9911yescharacteristic
402C22C2×#41011yescharacteristic
403C4—22no—
404C4—22no—
405C4—24no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—82no—
415C4—168no—
416C2—21no—
417C2—21no—
418C2—44no—
419C2—42no—
420C2—161no—
421C2#2179311yescharacteristic
422C2#3336011yescharacteristic
423C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20725C2semidirect
3. #13225C2semidirect
4. #13224C2semidirect
5. #12698C2semidirect
7. C2×#536C4semidirect
12. #2546C4semidirect
13. #2545C4semidirect
20. #2311C4semidirect
21. #2296C4semidirect
22. C2×#391C4semidirect

Maximal subgroups

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

subgroupindex
#356212
#207252
#132252
#132242
#126982
#126952