Finite Group Databasebeta

← #224936

Subgroups — #224936

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#22823C212yesmaximal, complemented
3#9707C212yesmaximal, complemented
4#9706C212yesmaximal
5#9699C212yesmaximal, complemented
6#8594C211yesmaximal, characteristic, complemented
7#8592C211yesmaximal, characteristic, complemented
8#8585C211yesmaximal, characteristic, complemented
9#8232C211yesmaximal, characteristic, complemented
10#8231C211yesmaximal, characteristic, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#559C412yes—
13#2867C2211yescharacteristic
14#2858C2211yescharacteristic
15#2858C2211yescharacteristic
16#2548—22no—
17#2548C412yes—
18#2544C412yes—
19#2544C412yes—
20#2445—22no—
21#2445—22no—
22#2366—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#1985—22no—
28#1917—22no—
29#1916—22no—
30#1842C2212yes—
31#1831C2212yes—
32#1831C2212yes—
33#1771—22no—
34#1767C2212yes—
35#1767C2212yes—
36C4×C4⋊C8C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39C2×#331C2212yes—
40C2×#331C2212yes—
41C2×#331C2212yes—
42C2×#331C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#559—48no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×#127—22no—
50C2×#127—22no—
51C2×#127C2×C412yes—
52C2×#127C2×C412yes—
53C2×#127C2311yescharacteristic
54C2×#127C2311yescharacteristic
55C2×#124—22no—
56C2×#124C2×C412yes—
57C2×#124C2×C412yes—
58C2×#124C2×C412yes—
59C2×#124C2×C412yes—
60C2×#123—22no—
61C2×#123—22no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D412yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C412yes—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—22no—
79#394—24no—
80#394—24no—
81#394—24no—
82#391C2×C412yes—
83#390—24no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390—22no—
90#390C2×C412yes—
91#390C2×C412yes—
92#389—22no—
93#386—24no—
94#386—24no—
95#384C2311yescharacteristic
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C43C2311yescharacteristic
104C43C2311yescharacteristic
105C43C2311yescharacteristic
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#333—24no—
114#333—42no—
115#331—24no—
116#331—24no—
117#331—42no—
118#331—42no—
119C22×Q8C22⋊C411yescharacteristic
120C22×Q8C22×C412yes—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C22×C412yes—
127C22×D4C22×C412yes—
128C23×C4—22no—
129C23×C4—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137#127—22no—
138#127—22no—
139#127—21no—
140#127—21no—
141#127—21no—
142#127—21no—
143#127—44no—
144#127—44no—
145#127—44no—
146#127—82no—
147#127—82no—
148#127—82no—
149#124—44no—
150#124—44no—
151#124—48no—
152#124—82no—
153#124—82no—
154#123—48no—
155#123—48no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—44no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—48no—
163C4×D4—82no—
164C4×D4—82no—
165C4×D4—82no—
166C4×D4—82no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C42—22no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2411yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—44no—
209C4⋊C8—82no—
210C4⋊C8#4114yes—
211#97—44no—
212#97—82no—
213C4×C8—82no—
214#95—24no—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218C24—22no—
219C24—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—48no—
235C2×D4—44no—
236C2×D4—48no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4—84no—
251C2×D4—84no—
252C2×D4—84no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D412yes—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4#10014yes—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—84no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—44no—
323C42C2×#4112yes—
324C42H(5,𝔽2)11yescharacteristic
325C422−1+411yescharacteristic
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23C22×C22⋊C411yescharacteristic
337Q8—44no—
338Q8—82no—
339Q8—82no—
340D4—82no—
341D4—82no—
342D4—82no—
343D4—82no—
344D4—88no—
345D4—82no—
346D4—82no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—82no—
352D4—82no—
353D4—168no—
354D4—164no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—164no—
381C2×C4—164no—
382C2×C4C2×#10012yes—
383C2×C4C2×#10012yes—
384C2×C4C2×#10012yes—
385C2×C4C4×#4111yescharacteristic
386C2×C4#52011yescharacteristic
387C2×C4#54812yes—
388C2×C4#55012yes—
389C2×C4#57512yes—
390C2×C4#57512yes—
391C2×C4#57612yes—
392C2×C4#57712yes—
393C8—44no—
394C8—82no—
395C8—82no—
396C8—82no—
397C22—82no—
398C22—82no—
399C22—84no—
400C22—84no—
401C22—82no—
402C22—82no—
403C22—84no—
404C22—84no—
405C22—82no—
406C22—82no—
407C22#234111yescharacteristic
408C22C22×#10011yescharacteristic
409C22#288811yescharacteristic
410C22#289612yes—
411C22#289712yes—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—41no—
423C4—82no—
424C4—82no—
425C4#337012yes—
426C4#337112yes—
427C4#340112yes—
428C4#340312yes—
429C2—162no—
430C2—162no—
431C2#1773811yescharacteristic
432C2#1776612yes—
433C2#1777011yescharacteristic
434C2#2253611yescharacteristic
435C2#2254912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2867C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22823C2semidirect
3. #9707C2semidirect
5. #9699C2semidirect
6. #8594C2semidirect
7. #8592C2semidirect
8. #8585C2semidirect
9. #8232C2semidirect
10. #8231C2semidirect
11. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#228232
#97072
#97062
#96992
#85942
#85922
#85852
#82322
#82312
C4×#3312