Finite Group Databasebeta

← #223940

Subgroups — #223940

Order 512.

1259 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#22750C212yesmaximal, complemented
3#9903C212yesmaximal, complemented
4#9142C212yesmaximal, complemented
5#9133C212yesmaximal, complemented
6#8725C211yesmaximal, characteristic, complemented
7#8723C211yesmaximal, characteristic, complemented
8#8592C211yesmaximal, characteristic, complemented
9#8231C211yesmaximal, characteristic, complemented
10#8040C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#547—22no—
13C2×#547C412yes—
14C2×#432C2212yes—
15C22×C4⋊C8C2212yes—
16#2868C2211yescharacteristic
17#2856C2211yescharacteristic
18#2856C2211yescharacteristic
19#2542—22no—
20#2542C412yes—
21#2501—22no—
22#2501—22no—
23#2501C412yes—
24#2501C412yes—
25C4×#127C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1869—22no—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1767C2212yes—
36#1752—22no—
37C2×#331C2212yes—
38C2×#331C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47C22×#41—22no—
48C23×C8—22no—
49#547—48no—
50#547—48no—
51#547—48no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C412yes—
54C2×#127C2311yescharacteristic
55C2×#123—24no—
56C2×#123—24no—
57C2×#123—22no—
58C2×#123—22no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121C2×C412yes—
64C2×#121C2×C412yes—
65C2×C4×D4—22no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D412yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#432—24no—
76#432—42no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—22no—
88C2×#97—22no—
89#399—24no—
90#399—24no—
91#399—24no—
92#390—22no—
93#390—22no—
94#390—22no—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2×C412yes—
98#384—22no—
99#384—22no—
100#384—22no—
101#384C2×C412yes—
102#384C2×C412yes—
103#384C2×C412yes—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4D412yes—
109C4×C22⋊C4C2×C411yescharacteristic
110C4×C22⋊C4C2×C411yescharacteristic
111C4×C22⋊C4D412yes—
112C4×C22⋊C4D412yes—
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#331—24no—
120#331—42no—
121C2×#41—416no—
122C2×#41—48no—
123C22×Q8—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4C22⋊C411yescharacteristic
128C22×D4C22×C412yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—44no—
141C22×C8—44no—
142C22×C8—82no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146#127—22no—
147#127—21no—
148#127—21no—
149#123—48no—
150#123—48no—
151#123—416no—
152#123—416no—
153C22⋊Q8—48no—
154C22⋊Q8—416no—
155#121—44no—
156#121—44no—
157#121—44no—
158#121—44no—
159#121—44no—
160#121—44no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—44no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C22⋊C411yescharacteristic
194C2×C22⋊C4C22⋊C411yescharacteristic
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—44no—
220C4⋊C8#4114yescomplemented
221C22⋊C8—44no—
222C22⋊C8—44no—
223C22⋊C8—82no—
224#97—44no—
225#97—82no—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229C24—22no—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232#41—832no—
233C2×Q8—48no—
234C2×Q8—44no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—48no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—48no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4#10014yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—48no—
277C22×C4—44no—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4AGL(1,ℤ/8)14yes—
286C22×C4AGL(1,ℤ/8)14yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D412yes—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—88no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—48no—
310C4⋊C4—44no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—48no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—44no—
324C22⋊C4—48no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—41no—
343C42C2×#4112yes—
344C42H(5,𝔽2)11yescharacteristic
345C422−1+411yescharacteristic
346C23—21no—
347C23—21no—
348C23—22no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—22no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—44no—
358C23C2×#10012yes—
359C23C2×#10012yes—
360C23#41814yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—88no—
363D4—28no—
364D4—82no—
365D4—82no—
366D4—88no—
367D4—88no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—82no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—88no—
410C2×C4—88no—
411C2×C4—88no—
412C2×C4C2×#10012yes—
413C2×C4#41814yes—
414C2×C4C4×#4111yescharacteristic
415C2×C4#52011yescharacteristic
416C2×C4#54612yes—
417C2×C4#54712yes—
418C2×C4C2×AGL(1,ℤ/8)12yes—
419C2×C4C2×AGL(1,ℤ/8)12yes—
420C2×C4#57712yes—
421C2×C4#57812yes—
422C8—44no—
423C8—82no—
424C8—82no—
425C8—82no—
426C22—24no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—82no—
436C22—84no—
437C22—82no—
438C22—82no—
439C22—84no—
440C22#155514yes—
441C22#233911yescharacteristic
442C22C22×#10011yescharacteristic
443C22#288911yescharacteristic
444C22C2×#41812yes—
445C22#289812yes—
446C4—22no—
447C4—21no—
448C4—21no—
449C4—24no—
450C4—42no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—42no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4#330612yes—
459C4#330812yes—
460C4#334312yes—
461C4#334512yes—
462C2—24no—
463C2—42no—
464C2—162no—
465C2C2×#155511yescharacteristic
466C2#1775912yes—
467C2#1777911yescharacteristic
468C2#2247911yescharacteristic
469C2#2249512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36179C2
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
2. #22750C2semidirect
3. #9903C2semidirect
4. #9142C2semidirect
5. #9133C2semidirect
6. #8725C2semidirect
7. #8723C2semidirect
8. #8592C2semidirect
9. #8231C2semidirect
10. #8040C2semidirect
11. C2×#1296C2semidirect
220. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#361792
#227502
#99032
#91422
#91332
#87252
#87232
#85922
#82312
#80402
C2×#12962