Finite Group Databasebeta

← #200415

Subgroups — #200415

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic
2#35334C211yesmaximal, characteristic
3#20632C211yesmaximal, characteristic
4#17575C212yesmaximal, complemented
5#9383C218yesmaximal, complemented
6#8026C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9#2988—24no—
10#2982—21no—
11#2972—21no—
12#2972C2214yes—
13C8×#41—24no—
14C2×#431—22no—
15C2×#431C2212yes—
16C2×#410—21no—
17#2851—22no—
18#2628C412yescomplemented
19#2628C412yescomplemented
20#2581—21no—
21#2581C412yescomplemented
22#2580—21no—
23#2580C412yescomplemented
24#2564—22no—
25#2564C2214yes—
26#2316—21no—
27#2311—21no—
28#2285C2212yes—
29C2×#382C2211yescharacteristic
30#1799—22no—
31#1799C2214yes—
32#1740—21no—
33#1520C2218yes—
34#1518—24no—
35#1503—24no—
36#1294C2218yes—
37C22×#41D411yescharacteristic
38#533—22no—
39#533—22no—
40#533—22no—
41#533—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2312yes—
47C4×#41—22no—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4C2312yes—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117C2×C412yes—
62C2×#117D411yescharacteristic
63C22×C4⋊C4—21no—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C8×Q8—28no—
67C8×D4—28no—
68C8×D4—216no—
69#431—24no—
70#431—24no—
71#431—22no—
72#431—24no—
73#431—22no—
74#431D414yes—
75#431D414yes—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—22no—
78#410—44no—
79C2×#99—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8C2314yes—
82C2×C4×C8—24no—
83C2×C4×C8—22no—
84#400—22no—
85#400C2×C414yes—
86#399—22no—
87#399C2×C414yes—
88#395—22no—
89#395C2×C414yes—
90#393—22no—
91#393C2×C414yes—
92#391—22no—
93#390—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#387—24no—
99#387—24no—
100#386—24no—
101#385—24no—
102#385—24no—
103#384—22no—
104#382—21no—
105#382—21no—
106#382—22no—
107#382C2×C412yes—
108#382C2×C412yes—
109#380—22no—
110#380—22no—
111#380—24no—
112#380—22no—
113#380C2314yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C2×#95—21no—
117C2×#95C2312yes—
118#342—24no—
119#328—216no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—24no—
123C2×#41SD1614yes—
124C2×#41Q1614yes—
125C22×Q8—21no—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22×C412yes—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C2×D412yes—
130C23×C4—22no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—28no—
138C22×C8—24no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143#125—41no—
144#125—41no—
145#125—41no—
146#125—41no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167#117—22no—
168#117—22no—
169#117—24no—
170#117—24no—
171#117—42no—
172#117—42no—
173#117—44no—
174#117—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22×C412yes—
199C2×C22⋊C4C2×D412yes—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—42no—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C4⋊C8—28no—
218C4⋊C8—24no—
219C4⋊C8—24no—
220#99—48no—
221C22⋊C8—28no—
222C22⋊C8—24no—
223C22⋊C8C2×D418yes—
224C4×C8—28no—
225C4×C8—24no—
226C4×C8—24no—
227#95—22no—
228#95—22no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—22no—
234#95—24no—
235#95—42no—
236#95C22×C414yes—
237C24C2×C22⋊C412yes—
238C24C22×D411yescharacteristic
239#41—28no—
240#41—28no—
241#41—28no—
242#41—28no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—42no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8#10314yes—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4#10314yes—
255C2×D4C2×SD1614yes—
256C2×D4C2×Q1614yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4#10318yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4#12112yes—
284C22×C4#12112yes—
285C22×C4#12112yes—
286C22×C4#12112yes—
287C22×C4C2×SD1612yes—
288C22×C4C2×Q1612yes—
289C22×C4#13514yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4112yes—
294C2×C8—28no—
295C2×C8—84no—
296C2×C8—88no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—44no—
333C42—84no—
334C42C4×D412yes—
335C42C4×D412yes—
336C42C4×D412yes—
337C42C4×D412yes—
338C42#12011yescharacteristic
339C42#12012yes—
340C42#12011yescharacteristic
341C42#12311yescharacteristic
342C42#12311yescharacteristic
343C42#12312yes—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23C2×#10314yes—
351C23C22×C22⋊C411yescharacteristic
352C23C2×#12112yes—
353C23C22×SD1611yescharacteristic
354C23C22×Q1611yescharacteristic
355C23C2×#13512yes—
356Q8—28no—
357Q8—84no—
358Q8—84no—
359D4—28no—
360D4—28no—
361D4—28no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—164no—
395C2×C4#38611yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38612yes—
398C2×C4#38611yescharacteristic
399C2×C4#38611yescharacteristic
400C2×C4#38612yes—
401C2×C4C2×#10314yes—
402C2×C4C2×#10312yes—
403C2×C4C2×#10312yes—
404C2×C4#41612yes—
405C2×C4#41612yes—
406C2×C4#46514yes—
407C2×C4#46514yes—
408C2×C4#46714yes—
409C2×C4#46714yes—
410C2×C4C2×C4×D412yes—
411C2×C4C2×#12011yescharacteristic
412C2×C4C2×#12311yescharacteristic
413C8—48no—
414C22—21no—
415C22—22no—
416C22—21no—
417C22—22no—
418C22—24no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—44no—
423C22—42no—
424C22—42no—
425C22—82no—
426C22C2×#38611yescharacteristic
427C22C22×#10311yescharacteristic
428C22C2×#41611yescharacteristic
429C22C2×#46512yes—
430C22C2×#46712yes—
431C4—21no—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—24no—
438C4—42no—
439C4—42no—
440C4—84no—
441C4—82no—
442C4—82no—
443C4—164no—
444C4#189212yes—
445C4#189212yes—
446C4#189214yes—
447C2—21no—
448C2—22no—
449C2—21no—
450C2—22no—
451C2—44no—
452C2#2094711yescharacteristic
453C2C2×#189211yescharacteristic
454C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
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
4. #17575C2semidirect
5. #9383C2semidirect
6. #8026C2semidirect
18. #2628C4semidirect
19. #2628C4semidirect
21. #2581C4semidirect
23. #2580C4semidirect

Maximal subgroups

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

subgroupindex
#602762
#353342
#206322
#175752
#93832
#80262