Finite Group Databasebeta

← #493022

Subgroups — #493022

Order 512.

1231 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2C2×#2133C212yesmaximal
3#20399C212yesmaximal, complemented
4#12632C212yesmaximal, complemented
5#12631C212yesmaximal, complemented
6C4×#356C212yesmaximal, complemented
7C4×#355C212yesmaximal, complemented
8C2×#1394C212yesmaximal, complemented
9C2×#536C2211yescharacteristic
10C2×#520C411yescharacteristic
11C2×#519C2211yescharacteristic
12C22×#101C2214yes—
13#2873—22no—
14#2362—22no—
15#2360—22no—
16#2359—22no—
17#2359—22no—
18#2327C411yescharacteristic
19#2327C2211yescharacteristic
20#2326C411yescharacteristic
21#2326C2211yescharacteristic
22#2317C411yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124C411yescharacteristic
26C4×#124C2211yescharacteristic
27#2297C411yescharacteristic
28C2×#391C2211yescharacteristic
29C4×#117C411yescharacteristic
30#2133C4116yescomplemented
31C2×#356C2214yes—
32C2×#355C2214yes—
33C4×#101—22no—
34C4×#101—22no—
35C4×#101C2214yes—
36C4×#101C2214yes—
37#1742—22no—
38#1394C2214yes—
39#1394C2214yes—
40C22×#41C2×C411yescharacteristic
41C23×Q8C2311yescharacteristic
42C22×M16—24no—
43#536—24no—
44#536C2×C418yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#124C2×C411yescharacteristic
48C2×#124C2311yescharacteristic
49C2×#123—22no—
50C2×C22⋊Q8—22no—
51#520—44no—
52#519—44no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×#117—21no—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117D412yes—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C2×#101—24no—
75C2×#101—28no—
76C2×#101C2×C4116yescomplemented
77C2×#101C2318yes—
78#405—48no—
79C4×M16—24no—
80C4×M16—24no—
81#391—22no—
82#391C2×C411yescharacteristic
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2311yescharacteristic
87#391C2311yescharacteristic
88#390C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#389—22no—
91#389—22no—
92#389—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#384C2×C411yescharacteristic
106#384C2×C411yescharacteristic
107#382C2×C411yescharacteristic
108#382C2×C411yescharacteristic
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118#376C2×C411yescharacteristic
119#376C2×C411yescharacteristic
120C2×#95—22no—
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123#356C2×C4116yescomplemented
124#355C2×C4116yescomplemented
125#343—24no—
126#343—24no—
127C2×#41—22no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41C4214yes—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8C22×C411yescharacteristic
134C22×Q8C22×C414yes—
135C22×Q8C22×C411yescharacteristic
136C22×Q8C2×D412yes—
137C22×Q8#4114yes—
138C22×Q8C2411yescharacteristic
139C22×D4—21no—
140C22×D4C22×C411yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D412yes—
148C2×M16—216no—
149C2×M16—28no—
150C2×M16—48no—
151C2×M16—44no—
152C22×C8—44no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8C4214yes—
156C4⋊Q8C22×C414yes—
157#124—22no—
158#124—22no—
159#124C4214yes—
160#124C22×C414yes—
161#123—216no—
162C22⋊Q8—216no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—44no—
168C4×Q8—44no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—44no—
174C4×D4—44no—
175#117—22no—
176#117—24no—
177#117—28no—
178#117—42no—
179#117—44no—
180#117—44no—
181#117C4214yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊C411yescharacteristic
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C411yescharacteristic
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—42no—
220C2×C42C22⋊C412yes—
221C2×C42C22⋊C412yes—
222C2×C42C22⋊C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231#101—216no—
232#101—28no—
233#101C22×C4116yes—
234#97—44no—
235#97—44no—
236C4×C8—44no—
237C4×C8—44no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250C24—21no—
251C24C2×C22⋊C411yescharacteristic
252#41—48no—
253#41—48no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—48no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8C2×C4212yes—
267C2×Q8C2×C4212yes—
268C2×Q8#11712yes—
269C2×Q8C4×D418yes—
270C2×Q8C23×C412yes—
271C2×Q8C2×#4112yes—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4C2×C4212yes—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4C2×C4212yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4#11712yes—
312C22×C4C4×D418yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D412yes—
315M16—216no—
316M16—48no—
317M16—416no—
318C2×C8—48no—
319C2×C8—416no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—22no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—44no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—82no—
347C22⋊C4#9914yes—
348C42—21no—
349C42—21no—
350C42—24no—
351C42—24no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—81no—
365C42—81no—
366C42C2×C4212yes—
367C42C2×C4212yes—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23C4×C22⋊C412yes—
376C23C2×#9912yes—
377C23#41012yes—
378C23C22×C22⋊C411yescharacteristic
379Q8—42no—
380Q8—42no—
381Q8—48no—
382Q8—42no—
383Q8—42no—
384Q8—42no—
385Q8—42no—
386D4—42no—
387D4—42no—
388D4—48no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—48no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—84no—
432C2×C4—162no—
433C2×C4—162no—
434C2×C4—162no—
435C2×C4C4×C22⋊C414yes—
436C2×C4C4×C22⋊C412yes—
437C2×C4C2×#9912yes—
438C2×C4C2×#9912yes—
439C2×C4C22×C4211yescharacteristic
440C2×C4C2×#11711yescharacteristic
441C2×C4C2×C4×D414yes—
442C8—416no—
443C22—21no—
444C22—22no—
445C22—21no—
446C22—22no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—42no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22C4×#9914yes—
455C22C2×C4×C22⋊C411yescharacteristic
456C22C22×#9911yescharacteristic
457C22C2×#41011yescharacteristic
458C4—22no—
459C4—22no—
460C4—24no—
461C4—22no—
462C4—21no—
463C4—24no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—81no—
476C2—21no—
477C2—21no—
478C2—24no—
479C2—82no—
480C2C2×C4×#9911yescharacteristic
481C2#3337011yescharacteristic
482C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C22×C42
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C2×C4
LCSG ▷ C2×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
3. #20399C2semidirect
4. #12632C2semidirect
5. #12631C2semidirect
6. C4×#356C2semidirect
7. C4×#355C2semidirect
8. C2×#1394C2semidirect
30. #2133C4semidirect
76. C2×#101C2×C4semidirect
123. #356C2×C4semidirect
124. #355C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342762
C2×#21332
#203992
#126322
#126312
C4×#3562
C4×#3552
C2×#13942