Finite Group Databasebeta

← #6461787

Subgroups — #6461787

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29529C218yesmaximal, complemented
4#29521C214yesmaximal, complemented
5#29513C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7C2×#2094C211yesmaximal, characteristic
8#23551C211yesmaximal, characteristic, complemented
9#23454C211yesmaximal, characteristic
10#21155C212yesmaximal, complemented
11#21091C212yesmaximal
12#21065C212yesmaximal
13#21038C212yesmaximal, complemented
14#20368C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C2×#427C2212yes—
17C22×#106C2212yes—
18C2×#418C2212yes—
19C22×#102C2212yes—
20C2×#405C2212yes—
21#2860C2214yes—
22#2644C2214yes—
23#2513—22no—
24#2512—22no—
25#2480—24no—
26#2360—24no—
27#2355—22no—
28Q8×C4⋊C4—22no—
29#2330C2214yescomplemented
30#2317C2212yes—
31#2317C2211yescharacteristic
32#2316C2211yescharacteristic
33#2309C2211yescharacteristic
34#2309C2211yescharacteristic
35C4×C4⋊Q8C2214yescomplemented
36#2296C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2274C2212yes—
40C4×#117C2211yescharacteristic
41#2100C2214yescomplemented
42#2099C2214yescomplemented
43#2094C2212yes—
44#2094C2212yes—
45#2094C2214yes—
46#2059C2214yescomplemented
47#2058C2212yescomplemented
48#2057C2212yescomplemented
49#2025C2214yes—
50#2023C2214yes—
51#1943C2218yescomplemented
52#1941C2218yescomplemented
53#1936C2218yes—
54#1934C2218yes—
55#1926C2218yes—
56#1925C2218yes—
57#1920C2218yescomplemented
58#1919C2218yescomplemented
59#1746C2214yescomplemented
60#1745C2212yescomplemented
61#1744C2212yescomplemented
62#1736C2212yes—
63#1736C2212yes—
64#1736C2212yes—
65#1735C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C23×D4C2311yescharacteristic
70C22×M16C2312yes—
71C23×C8C2312yes—
72#533—28no—
73C2×C4⋊Q8C2312yes—
74C2×#126—24no—
75C2×#125C2312yes—
76C2×C22⋊Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—24no—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2312yes—
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C4⋊C4—22no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4—22no—
88C22×C42C2311yescharacteristic
89#427—28no—
90C2×#107C2314yes—
91C2×#106—28no—
92C2×#106C2314yes—
93C2×#106C2314yes—
94C2×#106C2314yes—
95#418—28no—
96C2×#102—28no—
97C2×#102C2314yes—
98C2×#102C2314yes—
99C2×#102C2318yes—
100#405—28no—
101C2×#97C2314yes—
102C2×C4×C8C2314yes—
103#397—24no—
104#397—24no—
105#397—24no—
106#396—24no—
107#391—24no—
108#390—21no—
109#390—22no—
110#390—21no—
111#390—22no—
112#390C2314yes—
113#390C2314yes—
114#390C2312yes—
115#390C2312yes—
116#389—22no—
117#389—24no—
118#389—28no—
119#389C2318yes—
120#389C2312yes—
121#389C2312yes—
122#389C2314yes—
123#389C2314yes—
124#388—24no—
125#388—24no—
126#388—24no—
127#387—28no—
128#385—22no—
129#385—24no—
130#385—22no—
131#384—22no—
132#384—21no—
133#384—22no—
134#384C2312yes—
135#384C2312yes—
136#382—22no—
137#382—21no—
138#380—22no—
139#379—22no—
140#379—22no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4D412yes—
145C4×C4⋊C4D412yes—
146C4×C4⋊C4C2318yes—
147C4×C22⋊C4—24no—
148C4×C22⋊C4D414yes—
149#376C2312yes—
150C43C2312yes—
151#340C2314yes—
152#340C2314yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#336C2311yescharacteristic
157#336C2311yescharacteristic
158#336C2312yes—
159#336C2312yes—
160#336C2312yes—
161#336C2314yes—
162#336C2314yes—
163C25—21no—
164C22×Q8—22no—
165C22×D4—21no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4C2412yes—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4C2×D412yes—
174C23×C4C2411yescharacteristic
175C2×M16—24no—
176C2×M16—24no—
177C2×M16—24no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8#4118yes—
181C22×C8C2414yes—
182C22×C8C2412yes—
183C22×C8C2412yes—
184C4⋊Q8—28no—
185#126—88no—
186#125—28no—
187C22⋊Q8—216no—
188C4×Q8—216no—
189C4×Q8—88no—
190#117—24no—
191#117—24no—
192#117—22no—
193#117—22no—
194#117—24no—
195#117—22no—
196#117—22no—
197#117#4118yes—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—84no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2412yes—
218C2×C4⋊C4C2412yes—
219C2×C4⋊C4C2412yes—
220C2×C4⋊C4C2414yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—84no—
231C2×C22⋊C4C2×D412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—28no—
238C2×C42C2×D412yes—
239C2×C42C2×D414yes—
240C2×C42C2×D414yes—
241C2×C42C2412yes—
242C2×C42C2412yes—
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245#107—28no—
246#106—28no—
247#106—28no—
248#106—28no—
249#102—28no—
250#102—28no—
251#102—216no—
252#97—28no—
253C4×C8—28no—
254#95—28no—
255#95—28no—
256#95—22no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95C2×D414yes—
261#95C2×D414yes—
262C24—22no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—82no—
267C24—82no—
268C24C22×D411yescharacteristic
269C2×Q8—28no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—48no—
280C2×D4—48no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—82no—
292C22×C4—84no—
293C22×C4C22×D412yes—
294C22×C4C22×D414yes—
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4114yes—
300C22×C4C2×#4114yes—
301C22×C42−1+412yes—
302C22×C42−1+412yes—
303C22×C4C2511yescharacteristic
304M16—48no—
305C2×C8—216no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8C2×#4114yes—
314C2×C8C2×#4114yes—
315C4⋊C4—216no—
316C4⋊C4—28no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—48no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—88no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C4⋊C4C2×#4118yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—48no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C42—24no—
349C42—28no—
350C42—28no—
351C42—24no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—44no—
357C42—42no—
358C42—42no—
359C42—88no—
360C23—22no—
361C23—22no—
362C23—44no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—44no—
367C23—88no—
368C23—84no—
369C23—162no—
370C23—162no—
371C23—164no—
372C23#52512yes—
373C23#53212yes—
374C23#53612yes—
375C23C23×D411yescharacteristic
376Q8—416no—
377D4—44no—
378D4—44no—
379D4—84no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—28no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—162no—
407C2×C4—164no—
408C2×C4#52512yes—
409C2×C4#53212yes—
410C2×C4#53612yes—
411C2×C4#55218yes—
412C2×C4#55618yes—
413C2×C4#57714yes—
414C2×C4C22×#4112yes—
415C2×C4C22×#4112yes—
416C2×C4C2×2−1+411yescharacteristic
417C2×C4C2×2−1+411yescharacteristic
418C8—24no—
419C8—24no—
420C8—44no—
421C22—22no—
422C22—22no—
423C22—24no—
424C22—24no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—84no—
429C22—84no—
430C22—168no—
431C22#260414yes—
432C22C2×#52511yescharacteristic
433C22C2×#53211yescharacteristic
434C22C2×#53611yescharacteristic
435C22C2×#57712yes—
436C22#358512yes—
437C4—24no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—48no—
442C4—44no—
443C4—82no—
444C4—84no—
445C4—82no—
446C4—82no—
447C4#260414yes—
448C4C2×#55212yes—
449C4C2×#55612yes—
450C2—24no—
451C2—162no—
452C2C2×#260411yescharacteristic
453C2#6058812yes—
454C2#6071211yescharacteristic
455C2#6071911yescharacteristic
456C2#6089212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #29529C2semidirect
4. #29521C2semidirect
5. #29513C2semidirect
6. #23745C2semidirect
8. #23551C2semidirect
10. #21155C2semidirect
13. #21038C2semidirect
14. #20368C2semidirect
29. #2330C22semidirect
35. C4×C4⋊Q8C22semidirect
41. #2100C22semidirect
42. #2099C22semidirect
46. #2059C22semidirect
47. #2058C22semidirect
48. #2057C22semidirect
51. #1943C22semidirect
52. #1941C22semidirect
57. #1920C22semidirect
58. #1919C22semidirect
59. #1746C22semidirect
60. #1745C22semidirect
61. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342142
#295292
#295212
#295132
#237452
C2×#20942
#235512
#234542
#211552
#210912
#210652
#210382
#203682