Finite Group Databasebeta

← #214701

Subgroups — #214701

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35622C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#9104C212yesmaximal, complemented
5#9099C212yesmaximal, complemented
6#8962C212yesmaximal, complemented
7#8957C212yesmaximal, complemented
8#8733C212yesmaximal, complemented
9#8327C212yesmaximal, complemented
10C2×#541C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2983C2212yes—
13#2972C2212yes—
14#2566C411yescharacteristic
15#2565C411yescharacteristic
16#2547C411yescharacteristic
17#2546C411yescharacteristic
18#2503C411yescharacteristic
19#2502C411yescharacteristic
20#2475—22no—
21#2473—22no—
22#2468—22no—
23#2466—22no—
24#2432C411yescharacteristic
25#2432C411yescharacteristic
26#2327C2211yescharacteristic
27#2326C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×#124C2211yescharacteristic
30#2298C2211yescharacteristic
31#1479C2214yes—
32#1479C2214yes—
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#541—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#126—21no—
49C2×#126—21no—
50C2×#125—21no—
51C2×#125—21no—
52C2×#125C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#124C2×C411yescharacteristic
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8C2×C411yescharacteristic
57#520—44no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×C4×D4C2311yescharacteristic
66C2×#117D412yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D412yes—
70#446—44no—
71C8×D4—44no—
72#432—24no—
73#431C2314yes—
74#431C2314yes—
75C2×C22⋊C8—24no—
76#400—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#397—22no—
81#396—22no—
82#394—22no—
83#394—22no—
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#389—22no—
93#389—22no—
94#388—22no—
95#388—21no—
96#388—21no—
97#388—22no—
98#387—22no—
99#387—21no—
100#387—21no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382C2×C411yescharacteristic
111#382C2×C411yescharacteristic
112#381—22no—
113#380—22no—
114C4×C4⋊C4—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—
123Q8⋊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×Q8C22×C411yescharacteristic
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×D4C22⋊C412yes—
137C22×D4C22×C411yescharacteristic
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D412yes—
143C22×C8—44no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#126—44no—
148#126—44no—
149#125—42no—
150#125—44no—
151#125—44no—
152#124—24no—
153#124—24no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—44no—
162C4×D4—42no—
163C4×D4—44no—
164#117—24no—
165#117—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C22⋊C4C2×D411yescharacteristic
202C2×C22⋊C4C2×D411yescharacteristic
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C22⋊C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—44no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—44no—
221#97—44no—
222C4×C8—24no—
223C4×C8—24no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C412yes—
240#41—416no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4#13114yes—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—164no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C4⋊C4—81no—
310C4⋊C4—82no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—24no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—81no—
322C22⋊C4—82no—
323C42—22no—
324C42—24no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—81no—
334C42H(5,𝔽2)14yes—
335C23—22no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23#40814yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—44no—
347Q8—44no—
348Q8—44no—
349D4—28no—
350D4—48no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—48no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—84no—
392C2×C4—162no—
393C2×C4—162no—
394C2×C4#51812yes—
395C2×C4#53412yes—
396C2×C4#53412yes—
397C2×C4C2×#13112yes—
398C2×C4#56812yes—
399C8—84no—
400C8—84no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—44no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—82no—
411C22—81no—
412C22—81no—
413C22#230811yescharacteristic
414C22C2×#40811yescharacteristic
415C22#287911yescharacteristic
416C4—22no—
417C4—22no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—41no—
422C4—42no—
423C4—41no—
424C4—42no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4#297614yes—
433C2—21no—
434C2—21no—
435C2—44no—
436C2—81no—
437C2#2174111yescharacteristic
438C2#2178311yescharacteristic
439C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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
4. #9104C2semidirect
5. #9099C2semidirect
6. #8962C2semidirect
7. #8957C2semidirect
8. #8733C2semidirect
9. #8327C2semidirect

Maximal subgroups

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

subgroupindex
#356222
#173722
#173512
#91042
#90992
#89622
#89572
#87332
#83272