Finite Group Databasebeta

← #223162

Subgroups — #223162

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36181C211yesmaximal, characteristic
2#22748C212yesmaximal
3#9901C212yesmaximal, complemented
4#9128C212yesmaximal, complemented
5#9127C212yesmaximal, complemented
6#8728C211yesmaximal, characteristic, complemented
7#8723C211yesmaximal, characteristic, complemented
8#8594C211yesmaximal, characteristic, complemented
9C4×#333C211yesmaximal, characteristic, complemented
10#8038C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#552C412yes—
13C2×#431C2212yes—
14C22×C4⋊C8C2212yes—
15#2859C2211yescharacteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2565C412yes—
19#2506—22no—
20#2503C412yes—
21#2499C412yes—
22#2496—22no—
23#2486—22no—
24#2344—22no—
25C4×C4⋊Q8C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1923—22no—
30#1916—22no—
31#1869—22no—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35C4×C4⋊C8C2212yes—
36C8×C22⋊C4—22no—
37C2×#333C2212yes—
38C2×#333C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47C23×C8—22no—
48#552—48no—
49C2×C4⋊Q8C2311yescharacteristic
50C2×#126—22no—
51C2×#125—22no—
52C2×#125C2×C412yes—
53C2×#123—22no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×#117—22no—
60C2×#117C2×C412yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2×C412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#431—24no—
66#431—42no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79#400—24no—
80#398—24no—
81#397—24no—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2×C412yes—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#387—24no—
91#387—24no—
92#384—22no—
93#384C2×C412yes—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#382C2×C412yes—
98#382C2×C412yes—
99#381—24no—
100#380—22no—
101#380—22no—
102#380—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
107C4×C22⋊C4C2×C411yescharacteristic
108C4×C22⋊C4C2×C411yescharacteristic
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111C43C2311yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#333—24no—
118#333—42no—
119C22×Q8C22×C412yes—
120C22×D4C22⋊C411yescharacteristic
121C23×C4—22no—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—44no—
131C22×C8—44no—
132C22×C8—82no—
133C4⋊Q8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136#126—48no—
137#125—44no—
138#125—48no—
139#123—48no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C4×Q8—44no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146#117—48no—
147#117—44no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C22⋊C412yes—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C22⋊C411yescharacteristic
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2411yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—44no—
199C4⋊C8#4114yes—
200C22⋊C8—44no—
201C22⋊C8—44no—
202C22⋊C8—82no—
203C4×C8—44no—
204C4×C8—82no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212#95C22⋊C412yes—
213C24C2×C22⋊C411yescharacteristic
214C24C2×C22⋊C411yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×D4—22no—
220C2×D4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—44no—
244C22×C4—48no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4#13514yes—
253C22×C4#13514yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D412yes—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—88no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C22⋊C4—24no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4#10014yes—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—41no—
310C42—44no—
311C42—44no—
312C42C2×#4112yes—
313C42H(5,𝔽2)11yescharacteristic
314C422−1+411yescharacteristic
315C23—21no—
316C23—21no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23C2×#10012yes—
324C23#41614yes—
325C23C22×C22⋊C411yescharacteristic
326Q8—82no—
327Q8—82no—
328D4—44no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—88no—
369C2×C4C2×#10012yes—
370C2×C4C2×#10012yes—
371C2×C4#41614yes—
372C2×C4C4×#4111yescharacteristic
373C2×C4#52011yescharacteristic
374C2×C4#54612yes—
375C2×C4#54712yes—
376C2×C4C2×#13512yes—
377C2×C4C2×#13512yes—
378C2×C4#57712yes—
379C2×C4#57812yes—
380C8—44no—
381C8—82no—
382C8—82no—
383C8—82no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22—24no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—42no—
393C22#154314yes—
394C22#233911yescharacteristic
395C22C22×#10011yescharacteristic
396C22#288911yescharacteristic
397C22C2×#41612yes—
398C22#289812yes—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—24no—
404C4—41no—
405C4—42no—
406C4—41no—
407C4—42no—
408C4—82no—
409C4—82no—
410C4—82no—
411C4—82no—
412C4#329612yes—
413C4#330012yes—
414C4#333612yes—
415C4#333712yes—
416C2—24no—
417C2—81no—
418C2C2×#154311yescharacteristic
419C2#1772512yes—
420C2#1777911yescharacteristic
421C2#2247911yescharacteristic
422C2#2248612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9901C2semidirect
4. #9128C2semidirect
5. #9127C2semidirect
6. #8728C2semidirect
7. #8723C2semidirect
8. #8594C2semidirect
9. C4×#333C2semidirect
10. #8038C2semidirect
11. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#361812
#227482
#99012
#91282
#91272
#87282
#87232
#85942
C4×#3332
#80382
C2×#12962