Finite Group Databasebeta

← #481483

Subgroups — #481483

Order 512.

1237 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36020C211yesmaximal, characteristic
2#20358C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11700C212yesmaximal, complemented
5#11698C214yesmaximal, complemented
6#11697C212yesmaximal, complemented
7C2×#1346C214yesmaximal, complemented
8C2×#568—21no—
9C2×#547—21no—
10C2×#547C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2972—21no—
13#2621C2211yescharacteristic
14#2618C2212yes—
15#2546C411yescharacteristic
16#2543C412yes—
17#2536C411yescharacteristic
18#2486C412yes—
19#2429—21no—
20#2425—22no—
21#2417—21no—
22#2413—22no—
23#2392C2211yescharacteristic
24C2×#398C2212yes—
25#2349C411yescharacteristic
26#2339—21no—
27#1742—22no—
28#1742C2212yes—
29C2×#343—21no—
30C2×#343C2214yes—
31#1461—21no—
32#1459—22no—
33#1458C2212yes—
34C2×#324C2214yes—
35#1346C2214yes—
36#1346C2218yes—
37#1345C2214yes—
38C22×#41—21no—
39C22×#41D411yescharacteristic
40#568—28no—
41#568—44no—
42C2×#131—44no—
43C22×M16—22no—
44C22×M16—24no—
45#547—44no—
46#547—44no—
47#547—44no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#123—22no—
51C2×#123—22no—
52C2×#123—22no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×#121—22no—
57C2×#121C2×C412yes—
58#520—44no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D412yes—
67C2×C4×D4D411yescharacteristic
68C2×#117D412yes—
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C8×D4—44no—
75#431—22no—
76C2×C22⋊C8—24no—
77#398—21no—
78#398—22no—
79#398—22no—
80#398C2×C412yes—
81#398C2×C412yes—
82#398C2×C412yes—
83#398C2312yes—
84#398C2311yescharacteristic
85#397—21no—
86#397C2×C412yes—
87#397C2×C411yescharacteristic
88#396—22no—
89#396—22no—
90#396—21no—
91#394—21no—
92#394C2×C412yes—
93#394C2311yescharacteristic
94#392C2×C411yescharacteristic
95#391—21no—
96#391—21no—
97#389—21no—
98#389—22no—
99#389—22no—
100#388—22no—
101#388—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#385—22no—
106#382C2×C411yescharacteristic
107#381—22no—
108#380—22no—
109#380—21no—
110#380C2×C412yes—
111C4×C4⋊C4C2×C411yescharacteristic
112C4×C22⋊C4—21no—
113C2×#95—22no—
114C2×#95C2×C412yes—
115#343—24no—
116#343—22no—
117#343C2314yes—
118#324—24no—
119#324C2314yes—
120#323—22no—
121C2×#41—24no—
122C2×#41—24no—
123C2×#41—44no—
124C2×#41—44no—
125C2×#41—44no—
126C2×#41D814yes—
127C2×#41SD1614yes—
128C22×Q8—21no—
129C22×Q8C22⋊C411yescharacteristic
130C22×D4—22no—
131C22×D4—21no—
132C22×D4C22⋊C412yes—
133C22×D4C22⋊C411yescharacteristic
134C22×D4C22×C411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4C22⋊C412yes—
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143#131—432no—
144C2×M16—216no—
145C2×M16—28no—
146C2×M16—24no—
147C2×M16—44no—
148C2×M16—42no—
149C2×M16—48no—
150C2×M16—44no—
151C22×C8—44no—
152C22×C8—42no—
153C22×C8—44no—
154C4⋊Q8—42no—
155C4⋊Q8—44no—
156#123—48no—
157#123—48no—
158#123—44no—
159#123—44no—
160C22⋊Q8—44no—
161C22⋊Q8—44no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164#121—44no—
165#121—48no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—42no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—44no—
174C4×D4—42no—
175C4×D4—42no—
176#117—42no—
177#117—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22×C411yescharacteristic
215C2×C22⋊C4C2×D411yescharacteristic
216C2×C42—21no—
217C2×C42C22⋊C412yes—
218C2×C42C22⋊C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C4⋊C8—42no—
222C22⋊C8—24no—
223C22⋊C8—44no—
224C4×C8—42no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95C22×C411yescharacteristic
235#95C22×C412yes—
236#95C22×C411yescharacteristic
237C24—21no—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C412yes—
240#41—216no—
241#41—216no—
242#41—816no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8#10214yes—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—81no—
258C2×D4—81no—
259C2×D4#9914yes—
260C2×D4#10218yes—
261C2×D4#10214yes—
262C2×D4#10414yes—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×D814yes—
294C22×C4C2×D812yes—
295C22×C4C2×SD1614yes—
296C22×C4C2×SD1612yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300M16—28no—
301M16—416no—
302M16—48no—
303M16—44no—
304C2×C8—44no—
305C2×C8—416no—
306C2×C8—48no—
307C2×C8—44no—
308C2×C8—164no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—82no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C22⋊C4—81no—
345C42—22no—
346C42—22no—
347C42—42no—
348C42—42no—
349C42—42no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C2×#9914yes—
364C23C2×#10212yes—
365C23C2×#10214yes—
366C23C2×#10414yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—28no—
369Q8—84no—
370D4—216no—
371D4—84no—
372D4—82no—
373D4—88no—
374D4#327116yes—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—88no—
413C2×C4—88no—
414C2×C4—84no—
415C2×C4—162no—
416C2×C4C2×#9912yes—
417C2×C4C2×#10214yes—
418C2×C4C2×#10212yes—
419C2×C4C2×#10412yes—
420C2×C4C22×D811yescharacteristic
421C2×C4C22×SD1611yescharacteristic
422C8—48no—
423C8—84no—
424C22—21no—
425C22—24no—
426C22—21no—
427C22—24no—
428C22—22no—
429C22—44no—
430C22—44no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22C2×#32718yes—
443C22C22×#9911yescharacteristic
444C22C22×#10211yescharacteristic
445C22C22×#10411yescharacteristic
446C4—21no—
447C4—21no—
448C4—24no—
449C4—42no—
450C4—41no—
451C4—41no—
452C4—42no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—81no—
457C4—82no—
458C4—81no—
459C4—82no—
460C4C2×#32714yes—
461C2—21no—
462C2—21no—
463C2—28no—
464C2—42no—
465C2—161no—
466C2C22×#32711yescharacteristic
467C2#3335011yescharacteristic
468C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#36020C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20358C2semidirect
3. #17351C2semidirect
4. #11700C2semidirect
5. #11698C2semidirect
6. #11697C2semidirect
7. C2×#1346C2semidirect

Maximal subgroups

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

subgroupindex
#360202
#203582
#173512
#117002
#116982
#116972
C2×#13462