Finite Group Databasebeta

← #197521

Subgroups — #197521

Order 512.

1333 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic, complemented
3#17574C212yesmaximal
4C2×#1461C211yesmaximal, characteristic, complemented
5#9206C218yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×#536—21no—
8C2×#532C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2212yes—
11#2854—22no—
12#2618C2214yes—
13#2548C412yescomplemented
14#2546C412yescomplemented
15#2545—22no—
16#2538—22no—
17#2536C414yescomplemented
18#2356—24no—
19#2355—24no—
20#2313C2214yes—
21#2311—21no—
22#2310—21no—
23#2285C2211yescharacteristic
24#2285C2212yes—
25#2283—22no—
26#2274—21no—
27#1519C2218yes—
28#1517—24no—
29#1502—24no—
30#1461—22no—
31#1461—22no—
32C2×#324C2214yes—
33#1289C2218yes—
34C22×#41—21no—
35C23×Q8D411yescharacteristic
36#536—48no—
37#532—22no—
38#532—22no—
39#532—22no—
40#532—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127C2×C411yescharacteristic
44C2×#124—22no—
45C2×#124—21no—
46C2×#123—24no—
47C2×C22⋊Q8—24no—
48C2×#121—21no—
49C2×#121C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×#117—22no—
61C2×#117—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8C2314yes—
72C2×C4×C8—22no—
73#398C2×C418yes—
74#396—28no—
75#394C2×C418yes—
76#391—28no—
77#391—22no—
78#390—22no—
79#390—24no—
80#389—24no—
81#388—28no—
82#387—216no—
83#386—24no—
84#384—22no—
85#384—21no—
86#383—24no—
87#382—24no—
88#382C2×C412yes—
89#382D414yes—
90#381—24no—
91#380—21no—
92#380—22no—
93#380—24no—
94#380C2312yes—
95#380C2314yes—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—28no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D414yes—
102C4×C22⋊C4—24no—
103#376—24no—
104#327—216no—
105#324—28no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41—44no—
109C2×#41—44no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8C22⋊C412yes—
116C22×Q8C22⋊C412yes—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4C22×C412yes—
121C23×C4—22no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—24no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—48no—
131#127—22no—
132#127—22no—
133#124—48no—
134#124—42no—
135#124—42no—
136#123—416no—
137C22⋊Q8—416no—
138#121—24no—
139#121—24no—
140#121—41no—
141#121—41no—
142#121—41no—
143#121—41no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—42no—
155C4×D4—42no—
156#117—44no—
157#117—44no—
158#117—44no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—28no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—28no—
189C2×C42C22⋊C414yes—
190C2×C42C22⋊C414yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42#4112yes—
198C2×C42#4112yes—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C4×C8—42no—
204C4×C8—42no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—22no—
209#95—28no—
210#95—24no—
211#95C22×C414yes—
212#95C2×D418yes—
213C24—22no—
214C24C22×D411yescharacteristic
215#41—84no—
216#41—84no—
217#41—88no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—24no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—48no—
230C2×D4—24no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—82no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—81no—
256C22×C4—82no—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4#12011yescharacteristic
265C22×C4#12012yes—
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12312yes—
269C22×C4#13514yes—
270C22×C4AGL(1,ℤ/8)14yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4112yes—
275C2×C8—44no—
276C2×C8—84no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—416no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—84no—
290C4⋊C4—82no—
291C22⋊C4—416no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—88no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—44no—
304C42—48no—
305C42—88no—
306C42C4×D412yes—
307C42C4×D412yes—
308C42C4×D412yes—
309C42C4×D412yes—
310C42#12112yes—
311C42#12112yes—
312C42#12112yes—
313C42#12112yes—
314C23—41no—
315C23—41no—
316C23—44no—
317C23—44no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C22×C22⋊C411yescharacteristic
322C23C2×#12011yescharacteristic
323C23C2×#12311yescharacteristic
324C23C2×#13512yes—
325C23C2×AGL(1,ℤ/8)12yes—
326Q8—48no—
327Q8—48no—
328Q8—84no—
329D4—84no—
330D4—82no—
331D4—84no—
332D4—84no—
333D4—84no—
334D4—82no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—48no—
354C2×C4—44no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—88no—
364C2×C4#38611yescharacteristic
365C2×C4#38611yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38612yes—
368C2×C4#38612yes—
369C2×C4#38611yescharacteristic
370C2×C4#41612yes—
371C2×C4#41612yes—
372C2×C4#41812yes—
373C2×C4#41812yes—
374C2×C4#44914yes—
375C2×C4#44914yes—
376C2×C4#48218yes—
377C2×C4C2×C4×D412yes—
378C2×C4C2×#12112yes—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—42no—
387C22—42no—
388C22—84no—
389C22—84no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22C2×#38611yescharacteristic
394C22C2×#41611yescharacteristic
395C22C2×#41811yescharacteristic
396C22C2×#44912yes—
397C22C2×#48212yes—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—22no—
403C4—44no—
404C4—44no—
405C4—84no—
406C4—88no—
407C4—82no—
408C4#187514yes—
409C4#187514yes—
410C4#191212yes—
411C2—21no—
412C2—22no—
413C2—21no—
414C2—22no—
415C2—162no—
416C2C2×#187511yescharacteristic
417C2C2×#191211yescharacteristic
418C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34187C2semidirect
4. C2×#1461C2semidirect
5. #9206C2semidirect
6. #7981C2semidirect
13. #2548C4semidirect
14. #2546C4semidirect
17. #2536C4semidirect

Maximal subgroups

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

subgroupindex
#351782
#341872
#175742
C2×#14612
#92062
#79812