Finite Group Databasebeta

← #214696

Subgroups — #214696

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35678C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#9106C212yesmaximal, complemented
5#9102C212yesmaximal, complemented
6#8964C212yesmaximal, complemented
7#8960C212yesmaximal, complemented
8#8732C212yesmaximal, complemented
9#8326C212yesmaximal
10C2×#540C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13#2982C2212yes—
14#2565C411yescharacteristic
15#2565C411yescharacteristic
16#2548C411yescharacteristic
17#2548C411yescharacteristic
18#2500C411yescharacteristic
19#2500C411yescharacteristic
20#2476—22no—
21#2476—22no—
22#2458—22no—
23#2458—22no—
24#2436C411yescharacteristic
25#2436C411yescharacteristic
26#2326C2211yescharacteristic
27#2326C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C4×#124C2211yescharacteristic
30#2298C2211yescharacteristic
31#1478C2214yes—
32#1478C2214yes—
33#1467—22no—
34#1464—22no—
35#1461—22no—
36#1461C2212yes—
37#1461C2212yes—
38#1458—22no—
39#1458C2212yes—
40#1458C2212yes—
41#1276C2214yes—
42#1274C2214yes—
43C23×Q8C2×C411yescharacteristic
44#540—44no—
45C2×#127C2×C411yescharacteristic
46C2×#125C2×C411yescharacteristic
47C2×#124C2×C411yescharacteristic
48C2×#124C2×C411yescharacteristic
49C2×#124C2×C411yescharacteristic
50C2×#124C2×C411yescharacteristic
51C2×#123—21no—
52C2×#123—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121C2×C411yescharacteristic
56C2×#121C2×C411yescharacteristic
57#519—44no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2311yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C2×#117D412yes—
69C2×#117C2311yescharacteristic
70#446—44no—
71C8×Q8—44no—
72#432—24no—
73#431—24no—
74#431C2314yes—
75#431C2314yes—
76#398—22no—
77#398—22no—
78#394—22no—
79#394—22no—
80#394—22no—
81#394—22no—
82#393—22no—
83#393—22no—
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#390—22no—
89#390—22no—
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#390C2×C411yescharacteristic
94#388—22no—
95#388—22no—
96#388—21no—
97#388—21no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#387—21no—
103#387—21no—
104#387—22no—
105#387—22no—
106#386—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113#382C2×C411yescharacteristic
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122D4⋊C8—44no—
123D4⋊C8—44no—
124#324—24no—
125#324—22no—
126#324—22no—
127#324—24no—
128#323—22no—
129#323—22no—
130C22×Q8—24no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8C22⋊C412yes—
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C411yescharacteristic
141C22×D4C22×C411yescharacteristic
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C22⋊C412yes—
145C23×C4C22×C411yescharacteristic
146#127—42no—
147#125—42no—
148#124—24no—
149#124—24no—
150#124—42no—
151#124—42no—
152#123—44no—
153#123—44no—
154#121—44no—
155#121—44no—
156#121—42no—
157#121—42no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—44no—
161C4×Q8—44no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×D4—24no—
165C4×D4—24no—
166#117—24no—
167#117—44no—
168#117—44no—
169#117—42no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C2×D411yescharacteristic
202C2×C22⋊C4C2×D411yescharacteristic
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222#97—44no—
223C4×C8—24no—
224C4×C8—24no—
225C4×C8—44no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238C24—21no—
239C24—21no—
240C24C2×C22⋊C411yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—416no—
250C2×Q8#13114yes—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—44no—
292C2×C8—44no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C42—22no—
327C42—24no—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—81no—
337C42—81no—
338C42H(5,𝔽2)14yes—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C22×C22⋊C411yescharacteristic
355Q8—28no—
356Q8—48no—
357Q8—44no—
358D4—44no—
359D4—44no—
360D4—162no—
361D4—162no—
362D4—162no—
363D4—162no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
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—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—162no—
401C2×C4—162no—
402C2×C4#40814yes—
403C2×C4#51812yes—
404C2×C4#53412yes—
405C2×C4#53412yes—
406C2×C4C2×#13112yes—
407C2×C4#56812yes—
408C8—84no—
409C8—84no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—81no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22#230811yescharacteristic
428C22C2×#40811yescharacteristic
429C22#287911yescharacteristic
430C4—22no—
431C4—22no—
432C4—24no—
433C4—42no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4#297614yes—
446C2—21no—
447C2—21no—
448C2—81no—
449C2—161no—
450C2—161no—
451C2#2174111yescharacteristic
452C2#2178511yescharacteristic
453C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#540C22
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. #17371C2semidirect
3. #17352C2semidirect
4. #9106C2semidirect
5. #9102C2semidirect
6. #8964C2semidirect
7. #8960C2semidirect
8. #8732C2semidirect

Maximal subgroups

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

subgroupindex
#356782
#173712
#173522
#91062
#91022
#89642
#89602
#87322
#83262