Finite Group Databasebeta

← #213784

Subgroups — #213784

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35534C211yesmaximal, characteristic
2#17353C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#9102C212yesmaximal, complemented
5#9101C212yesmaximal, complemented
6#8960C212yesmaximal, complemented
7#8959C212yesmaximal, complemented
8#8732C212yesmaximal, complemented
9#8326C212yesmaximal, complemented
10C2×#540C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2982C2212yes—
13#2972C2212yes—
14#2566C411yescharacteristic
15#2565C411yescharacteristic
16#2548C411yescharacteristic
17#2541C411yescharacteristic
18#2504C411yescharacteristic
19#2501C411yescharacteristic
20#2471—22no—
21#2464—22no—
22#2456—22no—
23#2443—22no—
24#2431C411yescharacteristic
25#2431C411yescharacteristic
26#2328C2211yescharacteristic
27#2326C2211yescharacteristic
28C4×#127C2211yescharacteristic
29C4×#124C2211yescharacteristic
30#2298C2211yescharacteristic
31#1478C2214yes—
32#1478C2214yes—
33#1461C2212yes—
34#1460—22no—
35#1460C2212yes—
36#1459—22no—
37#1458—22no—
38#1458C2212yes—
39#1458C2212yes—
40C2×#324—22no—
41#1276C2214yes—
42#1274C2214yes—
43C22×#41C2×C411yescharacteristic
44#540—44no—
45C2×#127C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#125C2×C411yescharacteristic
49C2×#124C2×C411yescharacteristic
50C2×#124C2×C411yescharacteristic
51C2×#123—22no—
52C2×#123—21no—
53C2×#123—21no—
54C2×#121—22no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121C2×C411yescharacteristic
58C2×#121C2×C411yescharacteristic
59#520—44no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4—22no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117D412yes—
70C2×#117C2311yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4D412yes—
73C8×Q8—44no—
74C8×D4—44no—
75#431—24no—
76#431C2314yes—
77#431C2314yes—
78C2×C22⋊C8—24no—
79#399—22no—
80#398—22no—
81#397—22no—
82#396—22no—
83#394—22no—
84#392—22no—
85#392—22no—
86#392—22no—
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#390—22no—
90#390—22no—
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#390C2×C411yescharacteristic
94#390C2×C411yescharacteristic
95#390C2×C411yescharacteristic
96#390C2×C411yescharacteristic
97#388—22no—
98#388—21no—
99#388—21no—
100#388—22no—
101#387—21no—
102#387—21no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#382C2×C411yescharacteristic
114#382C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122D4⋊C8—44no—
123D4⋊C8—44no—
124#324—22no—
125#324—24no—
126#324—22no—
127#323—24no—
128#323—22no—
129#323—22no—
130C2×#41—24no—
131C2×#41—24no—
132C2×#41—42no—
133C22×Q8C22×C411yescharacteristic
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22⋊C412yes—
141C22×D4C22×C411yescharacteristic
142C22×D4C22×C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D412yes—
151C22×C8—44no—
152#127—24no—
153#127—24no—
154#127—42no—
155#125—42no—
156#124—24no—
157#124—24no—
158#123—44no—
159#123—44no—
160#123—84no—
161#121—44no—
162#121—44no—
163#121—42no—
164#121—42no—
165#121—84no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—44no—
172C4×D4—42no—
173C4×D4—44no—
174C4×D4—84no—
175#117—24no—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
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—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C2×D412yes—
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C22⋊C4C2×D411yescharacteristic
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—
228C4×C8—24no—
229C4×C8—24no—
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—21no—
240#95—21no—
241C24—21no—
242C24—21no—
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C412yes—
245#41—416no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8#13114yes—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4#13114yes—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—164no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—81no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—22no—
343C42—24no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—42no—
350C42—81no—
351C42H(5,𝔽2)14yes—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23#40814yes—
372C23C22×C22⋊C411yescharacteristic
373Q8—28no—
374D4—28no—
375D4—48no—
376D4—44no—
377D4—44no—
378D4—88no—
379D4—162no—
380D4—162no—
381D4—164no—
382D4—162no—
383D4—162no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—48no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—84no—
418C2×C4—162no—
419C2×C4—162no—
420C2×C4—162no—
421C2×C4—162no—
422C2×C4#40814yes—
423C2×C4#51812yes—
424C2×C4#53412yes—
425C2×C4#53412yes—
426C2×C4C2×#13112yes—
427C2×C4C2×#13112yes—
428C8—84no—
429C8—84no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—44no—
434C22—42no—
435C22—42no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—82no—
440C22—81no—
441C22—81no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#230811yescharacteristic
453C22C2×#40811yescharacteristic
454C22C2×#40811yescharacteristic
455C4—22no—
456C4—22no—
457C4—24no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4#297714yes—
470C2—21no—
471C2—21no—
472C2—44no—
473C2—81no—
474C2—161no—
475C2—161no—
476C2#2173911yescharacteristic
477C2#2178211yescharacteristic
478C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35534C2
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. #17353C2semidirect
3. #17351C2semidirect
4. #9102C2semidirect
5. #9101C2semidirect
6. #8960C2semidirect
7. #8959C2semidirect
8. #8732C2semidirect
9. #8326C2semidirect

Maximal subgroups

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

subgroupindex
#355342
#173532
#173512
#91022
#91012
#89602
#89592
#87322
#83262