Finite Group Databasebeta

← #214699

Subgroups — #214699

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35535C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#9106C212yesmaximal, complemented
5#9099C212yesmaximal, complemented
6#8962C212yesmaximal, complemented
7#8959C212yesmaximal, complemented
8#8732C212yesmaximal, complemented
9#8325C212yesmaximal, complemented
10C2×#542C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2983C2212yes—
13#2972C2212yes—
14#2567C411yescharacteristic
15#2545C411yescharacteristic
16#2544C411yescharacteristic
17#2542C411yescharacteristic
18#2502C411yescharacteristic
19#2501C411yescharacteristic
20#2479—22no—
21#2461—22no—
22#2455—22no—
23#2449—22no—
24#2432C411yescharacteristic
25#2431C411yescharacteristic
26#2326C2211yescharacteristic
27#2326C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×#127C2211yescharacteristic
30#2298C2211yescharacteristic
31#1479C2214yes—
32#1478C2214yes—
33#1466—22no—
34#1464—22no—
35#1461C2212yes—
36#1460C2212yes—
37#1459—22no—
38#1458C2212yes—
39#1458C2212yes—
40C2×#324—22no—
41#1276C2214yes—
42#1274C2214yes—
43C22×#41C2×C411yescharacteristic
44#542—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#126—21no—
49C2×#124—21no—
50C2×#124C2×C411yescharacteristic
51C2×#124C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#123—22no—
54C2×#123—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×#121C2×C411yescharacteristic
59#520—44no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4—22no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D412yes—
66C2×C4×D4D412yes—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117D412yes—
70C2×#117C2311yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4D412yes—
73#446—44no—
74C8×D4—44no—
75#432—24no—
76#431C2314yes—
77#431C2314yes—
78C2×C22⋊C8—24no—
79#399—22no—
80#399—22no—
81#398—22no—
82#396—22no—
83#395—22no—
84#394—22no—
85#394—22no—
86#392—22no—
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#390—22no—
92#390C2×C411yescharacteristic
93#390C2×C411yescharacteristic
94#390C2×C411yescharacteristic
95#390C2×C411yescharacteristic
96#389—22no—
97#389—22no—
98#388—22no—
99#388—21no—
100#388—21no—
101#388—22no—
102#388—22no—
103#387—21no—
104#387—21no—
105#386—22no—
106#386—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114#384C2×C411yescharacteristic
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122Q8⋊C8—44no—
123D4⋊C8—44no—
124#324—22no—
125#324—24no—
126#324—22no—
127#323—22no—
128#323—22no—
129#323—24no—
130C2×#41—24no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8—21no—
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×D4—22no—
137C22×D4—21no—
138C22×D4C22⋊C412yes—
139C22×D4C22×C411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C23×C4—21no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C22×C8—44no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150#127—24no—
151#126—44no—
152#124—24no—
153#124—42no—
154#124—42no—
155#124—44no—
156#123—44no—
157#123—84no—
158C22⋊Q8—42no—
159C22⋊Q8—44no—
160C22⋊Q8—84no—
161#121—42no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—42no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—44no—
168C4×D4—42no—
169C4×D4—44no—
170C4×D4—84no—
171#117—24no—
172#117—44no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C2×D412yes—
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22⋊C412yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2411yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—24no—
224C4⋊C8—44no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227C22⋊C8—44no—
228#97—44no—
229C4×C8—24no—
230C4×C8—24no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—21no—
241C24—21no—
242C24C2×C22⋊C411yescharacteristic
243C24C2×C22⋊C412yes—
244#41—416no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—82no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4#13114yes—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—164no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—24no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—22no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42H(5,𝔽2)14yes—
347C23—22no—
348C23—22no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23#40814yes—
362C23C22×C22⋊C411yescharacteristic
363Q8—44no—
364Q8—44no—
365Q8—162no—
366D4—28no—
367D4—48no—
368D4—44no—
369D4—88no—
370D4—162no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—84no—
409C2×C4—162no—
410C2×C4—162no—
411C2×C4—162no—
412C2×C4#51812yes—
413C2×C4#53412yes—
414C2×C4#53412yes—
415C2×C4C2×#13112yes—
416C2×C4#56812yes—
417C8—84no—
418C8—84no—
419C22—21no—
420C22—21no—
421C22—24no—
422C22—44no—
423C22—42no—
424C22—42no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—82no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22—82no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#230811yescharacteristic
437C22C2×#40811yescharacteristic
438C22#287911yescharacteristic
439C4—22no—
440C4—22no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—42no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4#297614yes—
455C2—21no—
456C2—21no—
457C2—44no—
458C2—81no—
459C2—161no—
460C2#2174111yescharacteristic
461C2#2178211yescharacteristic
462C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2431C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17372C2semidirect
3. #17351C2semidirect
4. #9106C2semidirect
5. #9099C2semidirect
6. #8962C2semidirect
7. #8959C2semidirect
8. #8732C2semidirect
9. #8325C2semidirect

Maximal subgroups

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

subgroupindex
#355352
#173722
#173512
#91062
#90992
#89622
#89592
#87322
#83252