Finite Group Databasebeta

← #495411

Subgroups — #495411

Order 512.

994 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#35639C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#20393C211yesmaximal, characteristic
4#13297C214yesmaximal, complemented
5#13295C214yesmaximal, complemented
6#12575C212yesmaximal
7#12573C212yesmaximal
8C22×#125C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2905—21no—
11C2×#413C2212yes—
12C2×#410C2212yes—
13#2579C412yescomplemented
14#2579C2212yes—
15#2487C412yescomplemented
16#2487C2212yes—
17#2454—22no—
18#2453—22no—
19#2442—21no—
20#2442—21no—
21#2441—21no—
22#2441—21no—
23#2317C411yescharacteristic, complemented
24#2316C411yescharacteristic, complemented
25#2311C2211yescharacteristic
26#2297C2211yescharacteristic
27#2296C411yescharacteristic, complemented
28#2274—21no—
29#1801—22no—
30#1800C2214yes—
31C4×#99—22no—
32C4×#99C2214yes—
33C2×#343—21no—
34#1740—21no—
35#1390C2214yes—
36#1390C2214yes—
37#1390C2218yes—
38C22×#41C2311yescharacteristic
39C22×M16—22no—
40C2×#125—21no—
41C2×#125—22no—
42C2×#125—21no—
43C2×#125—44no—
44#520—44no—
45C2×C4×Q8D412yes—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D412yes—
48C2×C4×D4C2312yes—
49C2×#117—22no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117D411yescharacteristic
54C2×#117D411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C42C2×C411yescharacteristic
62#424—44no—
63#423—24no—
64#413—28no—
65C2×#101—22no—
66C2×#100C2314yes—
67#410—24no—
68#410Q818yes—
69C2×#99—22no—
70C2×#99C2314yes—
71#400—22no—
72#400—22no—
73#399—22no—
74#399—22no—
75#398—22no—
76#398—22no—
77#396—22no—
78#396—22no—
79#391—21no—
80#391—21no—
81#390—21no—
82#390—21no—
83#389—21no—
84#389—22no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389—22no—
90#389—22no—
91#387—22no—
92#387—22no—
93#386C2×C412yes—
94#386C2×C411yescharacteristic
95#386C2×C411yescharacteristic
96#386C2312yes—
97#386C2311yescharacteristic
98#386C2311yescharacteristic
99#385—22no—
100#385—22no—
101#385C2×C412yes—
102#385C2×C412yes—
103#385C2×C412yes—
104#385C2×C412yes—
105#384—21no—
106#382—21no—
107#380—22no—
108#380—21no—
109#380—21no—
110#380—22no—
111#380C2×C412yes—
112#380C2×C412yes—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—44no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—24no—
118#343—24no—
119#342—24no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41C2×Q814yes—
123C2×#41#4114yes—
124C22×Q8C2×D411yescharacteristic
125C22×D4C22×C411yescharacteristic
126C22×D4C2×D411yescharacteristic
127C22×D4C2411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C2×M16—24no—
138C2×M16—42no—
139C2×M16—44no—
140C2×M16—44no—
141C22×C8—42no—
142#125—48no—
143#125—44no—
144#125—44no—
145C4×Q8—44no—
146C4×D4—44no—
147C4×D4—44no—
148C4×D4—44no—
149#117—24no—
150#117—28no—
151#117—42no—
152#117—42no—
153#117—44no—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22×C412yes—
193C2×C22⋊C4C22×C411yescharacteristic
194C2×C22⋊C4C22×C411yescharacteristic
195C2×C22⋊C4C22×C412yes—
196C2×C22⋊C4C2×D412yes—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—44no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C2×D412yes—
207C4⋊C8—48no—
208#101—44no—
209#100—28no—
210#99—28no—
211#99—24no—
212#99C2×Q818yes—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—44no—
226C24C2×C22⋊C411yescharacteristic
227C24C23×C411yescharacteristic
228C24C22×D411yescharacteristic
229#41—44no—
230#41—44no—
231#41—44no—
232#41—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8#12412yes—
236C2×Q8C4⋊Q812yes—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4C4×Q814yes—
244C2×D4C22⋊Q814yes—
245C2×D4C22×Q812yes—
246C2×D4C2×#4112yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—82no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C4×Q814yes—
280C22×C4C22⋊Q814yes—
281C22×C4#12412yes—
282C22×C4C4⋊Q812yes—
283C22×C4C22×D411yescharacteristic
284M16—44no—
285M16—44no—
286M16—84no—
287C2×C8—44no—
288C2×C8—84no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—48no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4#9914yes—
311C22⋊C4—24no—
312C22⋊C4—28no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C42—24no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—48no—
331C42—42no—
332C42—42no—
333C42—164no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#39112yes—
347C23C22×C22⋊C411yescharacteristic
348C23C2×C4×Q812yes—
349C23C2×C22⋊Q812yes—
350Q8—42no—
351Q8—42no—
352D4—44no—
353D4—44no—
354D4—42no—
355D4—42no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—162no—
402C2×C4#39112yes—
403C2×C4#39112yes—
404C2×C4#39112yes—
405C2×C4C2×#9914yes—
406C2×C4C2×#9912yes—
407C2×C4#41012yes—
408C2×C4C2×#12411yescharacteristic
409C2×C4C2×C4⋊Q811yescharacteristic
410C8—84no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—81no—
428C22C2×#39111yescharacteristic
429C22C22×#9911yescharacteristic
430C22C2×#41011yescharacteristic
431C4—24no—
432C4—21no—
433C4—21no—
434C4—22no—
435C4—22no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—82no—
440C4—81no—
441C4—81no—
442C4—88no—
443C4—81no—
444C4—81no—
445C4—82no—
446C4#199614yes—
447C2—21no—
448C2—21no—
449C2—42no—
450C2—42no—
451C2—42no—
452C2C2×#199611yescharacteristic
453C2#3335011yescharacteristic
454C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13297C2semidirect
5. #13295C2semidirect
8. C22×#125C4semidirect
13. #2579C4semidirect
15. #2487C4semidirect
23. #2317C4semidirect
24. #2316C4semidirect
27. #2296C4semidirect

Maximal subgroups

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

subgroupindex
#356392
#206402
#203932
#132972
#132952
#125752
#125732