Finite Group Databasebeta

← #493989

Subgroups — #493989

Order 512.

1041 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#20730C212yesmaximal, complemented
3#12860C212yesmaximal, complemented
4#12804C212yesmaximal, complemented
5#12801C212yesmaximal, complemented
6#12702C212yesmaximal
7#12632C212yesmaximal, complemented
8C4×#356C212yesmaximal, complemented
9C2×#536C2211yescharacteristic
10C2×#520C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2921—22no—
13#2362—22no—
14#2360—22no—
15#2359—22no—
16#2359—22no—
17#2327C411yescharacteristic
18#2327C2211yescharacteristic
19#2326C411yescharacteristic
20#2326C2211yescharacteristic
21#2317C411yescharacteristic
22C4×C4⋊Q8C411yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26#2297C411yescharacteristic
27C2×#391C2211yescharacteristic
28C4×#117C411yescharacteristic
29C2×#356C2214yes—
30C2×#355C2214yes—
31#1822—22no—
32#1801—22no—
33#1801C2214yes—
34#1784C2214yes—
35C4×#101—22no—
36C4×#101C2214yes—
37#1394C2214yes—
38#1394C2214yes—
39C22×#41C2×C411yescharacteristic
40C23×Q8C2311yescharacteristic
41#536—24no—
42#536—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2311yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49#520—44no—
50#519—44no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C2×#117D412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72#433—48no—
73#423—24no—
74C2×#101—24no—
75C2×#101C2318yes—
76#407—24no—
77C4×M16—24no—
78#391—22no—
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#391C2×C411yescharacteristic
82#391C2×C411yescharacteristic
83#391C2311yescharacteristic
84#391C2311yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#389—22no—
88#389—22no—
89#389—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#384C2×C411yescharacteristic
103#384C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#382C2×C411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115#376C2×C411yescharacteristic
116#376C2×C411yescharacteristic
117C2×#95—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120#356—28no—
121#355—28no—
122#343—24no—
123#343—24no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—42no—
127C2×#41—42no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C22×C411yescharacteristic
134C22×Q8C2×D412yes—
135C22×Q8C2411yescharacteristic
136C22×D4—21no—
137C22×D4C22×C411yescharacteristic
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D412yes—
145C2×M16—28no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8#4114yes—
150#124—22no—
151#124—22no—
152#124—22no—
153#124#4114yes—
154#123—216no—
155C22⋊Q8—216no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—44no—
161C4×Q8—44no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—44no—
167C4×D4—44no—
168#117—24no—
169#117—28no—
170#117—22no—
171#117—22no—
172#117—42no—
173#117—44no—
174#117—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C411yescharacteristic
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C411yescharacteristic
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—42no—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22⋊C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C4⋊C8—48no—
225#101—28no—
226#101—28no—
227#101—48no—
228C22⋊C8—48no—
229#97—44no—
230C4×C8—44no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24—21no—
244C24C2×C22⋊C411yescharacteristic
245#41—48no—
246#41—48no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8#11712yes—
262C2×Q8C2×#4112yes—
263C2×Q8C2×#4112yes—
264C2×D4—24no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4#11712yes—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D412yes—
307M16—48no—
308M16—48no—
309C2×C8—48no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C22⋊C4—24no—
329C22⋊C4—28no—
330C22⋊C4—24no—
331C22⋊C4—22no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—82no—
337C22⋊C4#9914yes—
338C42—21no—
339C42—21no—
340C42—24no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—41no—
348C42—41no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—81no—
355C42—81no—
356C42#11712yes—
357C42#11712yes—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23C2×#9912yes—
368C23C22×C22⋊C411yescharacteristic
369Q8—42no—
370Q8—42no—
371Q8—48no—
372Q8—42no—
373Q8—42no—
374Q8—42no—
375Q8—42no—
376D4—42no—
377D4—42no—
378D4—48no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
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—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—84no—
422C2×C4—162no—
423C2×C4—162no—
424C2×C4—162no—
425C2×C4C2×#9912yes—
426C2×C4C2×#9912yes—
427C2×C4#41012yes—
428C2×C4C2×#11711yescharacteristic
429C2×C4C2×#11711yescharacteristic
430C2×C4#52514yes—
431C8—88no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—42no—
438C22—42no—
439C22—44no—
440C22—42no—
441C22—82no—
442C22—81no—
443C22—81no—
444C22#228011yescharacteristic
445C22C22×#9911yescharacteristic
446C22C2×#41011yescharacteristic
447C4—22no—
448C4—22no—
449C4—24no—
450C4—22no—
451C4—21no—
452C4—21no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—41no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—81no—
463C4—81no—
464C4—81no—
465C2—21no—
466C2—21no—
467C2—42no—
468C2—82no—
469C2#2072211yescharacteristic
470C2#3337011yescharacteristic
471C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20730C2semidirect
3. #12860C2semidirect
4. #12804C2semidirect
5. #12801C2semidirect
7. #12632C2semidirect
8. C4×#356C2semidirect

Maximal subgroups

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

subgroupindex
#342762
#207302
#128602
#128042
#128012
#127022
#126322
C4×#3562