Finite Group Databasebeta

← #213787

Subgroups — #213787

Order 512.

1009 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#35536C211yesmaximal, characteristic
2#17350C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#9101C212yesmaximal, complemented
5#9099C212yesmaximal, complemented
6#8959C212yesmaximal, complemented
7#8957C212yesmaximal, complemented
8#8732C212yesmaximal, complemented
9#8325C212yesmaximal, complemented
10C2×#542C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2972C2212yes—
13#2972C2212yes—
14#2563C411yescharacteristic
15#2544C411yescharacteristic
16#2544C411yescharacteristic
17#2542C411yescharacteristic
18#2500C411yescharacteristic
19#2499C411yescharacteristic
20#2472—22no—
21#2467—22no—
22#2463—22no—
23#2435C411yescharacteristic
24#2433C411yescharacteristic
25#2418—22no—
26#2328C2211yescharacteristic
27#2326C2211yescharacteristic
28C4×#127C2211yescharacteristic
29C4×#124C2211yescharacteristic
30#2298C2211yescharacteristic
31#1479C2214yescomplemented
32#1478C2214yescomplemented
33#1460C2212yes—
34#1460C2212yes—
35#1459—22no—
36#1459—22no—
37#1458C2212yes—
38#1458C2212yes—
39C2×#323—22no—
40C2×#323—22no—
41#1276C2214yescomplemented
42#1274C2214yescomplemented
43C23×D4C2×C411yescharacteristic
44#542—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#126—21no—
49C2×#124—21no—
50C2×#124C2×C411yescharacteristic
51C2×#124C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#123—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×#121C2×C411yescharacteristic
58C2×#120—22no—
59#518—44no—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C22⋊C4D412yes—
72C22×C22⋊C4D412yes—
73C8×D4—44no—
74C8×D4—44no—
75#431C2314yes—
76#431C2314yes—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79#399—22no—
80#397—22no—
81#394—22no—
82#394—22no—
83#394—22no—
84#394—22no—
85#393—22no—
86#392—22no—
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#390—22no—
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#390C2×C411yescharacteristic
94#390C2×C411yescharacteristic
95#390C2×C411yescharacteristic
96#388—22no—
97#388—21no—
98#388—22no—
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#385—22no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113#380—22no—
114#379—22no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122Q8⋊C8—44no—
123D4⋊C8—44no—
124#324—22no—
125#324—22no—
126#323—22no—
127#323—22no—
128#323—24no—
129#323—24no—
130C25C22⋊C412yes—
131C22×Q8—21no—
132C22×Q8C22×C411yescharacteristic
133C22×D4—24no—
134C22×D4—24no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4C22⋊C412yes—
140C22×D4C22×C411yescharacteristic
141C22×D4C22×C411yescharacteristic
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C22×C8—44no—
150C22×C8—44no—
151C4⋊Q8—42no—
152#127—24no—
153#127—24no—
154#126—44no—
155#124—24no—
156#124—24no—
157#124—42no—
158#124—44no—
159#123—44no—
160C22⋊Q8—42no—
161C22⋊Q8—44no—
162C22⋊Q8—84no—
163#121—42no—
164#120—84no—
165C4×Q8—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—84no—
174#117—24no—
175#117—42no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C22⋊C4C2×D412yes—
212C2×C22⋊C4C2×D412yes—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C22⋊C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—24no—
224C4⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227C22⋊C8—44no—
228C22⋊C8—44no—
229C4×C8—24no—
230C4×C8—24no—
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—24no—
242C24—24no—
243C24—21no—
244C24—42no—
245C24—82no—
246C24—82no—
247C24C2×C22⋊C411yescharacteristic
248C24C2×C22⋊C412yes—
249C24C2×C22⋊C412yes—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×Q8—82no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—416no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4#13114yes—
275C2×D4#13114yes—
276C22×C4—21no—
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—21no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—164no—
307C2×C8—164no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—81no—
320C4⋊C4—82no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—82no—
338C22⋊C4—81no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—81no—
352C42H(5,𝔽2)14yes—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—48no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—162no—
373C23—162no—
374C23—164no—
375C23—164no—
376C23#40814yes—
377C23#40814yes—
378C23C22×C22⋊C411yescharacteristic
379Q8—44no—
380Q8—162no—
381D4—28no—
382D4—28no—
383D4—48no—
384D4—44no—
385D4—88no—
386D4—164no—
387D4—162no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
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—84no—
422C2×C4—84no—
423C2×C4—162no—
424C2×C4—162no—
425C2×C4—162no—
426C2×C4#51812yes—
427C2×C4#53412yes—
428C2×C4#53412yes—
429C2×C4C2×#13112yes—
430C2×C4C2×#13112yes—
431C8—84no—
432C8—84no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—24no—
437C22—44no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—82no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—82no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—168no—
456C22#230811yescharacteristic
457C22C2×#40811yescharacteristic
458C22C2×#40811yescharacteristic
459C4—22no—
460C4—22no—
461C4—42no—
462C4—42no—
463C4—41no—
464C4—42no—
465C4—41no—
466C4—41no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4#297714yes—
474C2—21no—
475C2—21no—
476C2—44no—
477C2—44no—
478C2—81no—
479C2—161no—
480C2#2173911yescharacteristic
481C2#2178211yescharacteristic
482C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2433C4
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. #17350C2semidirect
3. #17350C2semidirect
4. #9101C2semidirect
5. #9099C2semidirect
6. #8959C2semidirect
7. #8957C2semidirect
8. #8732C2semidirect
9. #8325C2semidirect
31. #1479C22semidirect
32. #1478C22semidirect
41. #1276C22semidirect
42. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#355362
#173502
#91012
#90992
#89592
#89572
#87322
#83252