Finite Group Databasebeta

← #6854368

Subgroups — #6854368

Order 512.

1374 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#36563C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#33312C211yesmaximal, characteristic
4#30948C218yesmaximal
5#30307C212yesmaximal
6#29206C211yesmaximal, characteristic
7#29178C212yesmaximal
8#29141C211yesmaximal, characteristic
9#28515C212yesmaximal
10#26047C214yesmaximal
11#25063C214yesmaximal
12#22221C212yesmaximal
13C4×#487C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×#500C2214yes—
16C2×#487C2211yescharacteristic
17C2×#479C2218yes—
18#2868C2211yescharacteristic
19#2868C2211yescharacteristic
20#2859C2212yes—
21#2859C2212yes—
22#2582—21no—
23#2582C2212yes—
24#2567—21no—
25#2567—21no—
26#2552—21no—
27#2552C2212yes—
28#2550—22no—
29#2550C2214yes—
30#2507—22no—
31#2507—22no—
32#2451—22no—
33#2451C2214yes—
34#2369—21no—
35#2369—21no—
36#2355—21no—
37#2355C2212yes—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C42×Q8C2211yescharacteristic
45#2097C2218yes—
46#2086C2218yes—
47#2085C2218yes—
48#2059C2218yes—
49#2058C2214yes—
50#2024C2214yes—
51#1990C2212yes—
52#1990C2211yescharacteristic
53#1984C2212yes—
54#1956C2211yescharacteristic
55#1956C2212yes—
56#1948C2214yes—
57#1939C2212yes—
58#1933C2214yes—
59#1928C2218yes—
60#1918C2218yes—
61#1916C2214yes—
62#1863C2218yes—
63#1845C2212yes—
64#1845C2212yes—
65#1845C2214yes—
66#1844C2212yes—
67#1834C2211yescharacteristic
68#1834C2212yes—
69#1803C2214yes—
70C4×C4⋊C8C2211yescharacteristic
71C4×C4⋊C8C2212yes—
72C4×#103C2214yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×#125—22no—
84C2×#125C2314yes—
85C2×C4×Q8—21no—
86C2×C4×Q8—22no—
87C2×C4×Q8D411yescharacteristic
88C2×C4×Q8D411yescharacteristic
89C2×C4×Q8C2312yes—
90C2×C4×Q8C2314yes—
91C2×C4×Q8C2311yescharacteristic
92#500—48no—
93#487Q818yescomplemented
94#479—416no—
95C2×#107C2318yes—
96C2×#107C2318yes—
97C2×#106C2318yes—
98C2×C4⋊C8C2314yes—
99C2×C4⋊C8C2312yes—
100C2×#103C2314yes—
101C2×#103C2318yes—
102C2×#103C2314yes—
103C2×C4×C8C2312yes—
104C2×C4×C8C2312yes—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2312yes—
107C2×C4×C8C2311yescharacteristic
108#398—24no—
109#398—24no—
110#398—24no—
111#398—24no—
112#395—24no—
113#395—24no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391—21no—
118#391—24no—
119#391—24no—
120#391C2314yes—
121#391C2312yes—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—24no—
137#389—24no—
138#389C2312yes—
139#389C2314yes—
140#389C2314yes—
141#389C2314yes—
142#387—24no—
143#387—24no—
144#384—22no—
145#384—22no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#384C2312yes—
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#382—22no—
157#382C2314yes—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2312yes—
174C43C2312yes—
175C43C2311yescharacteristic
176#340C2318yes—
177#340C2318yes—
178#340C2314yes—
179#340C2314yes—
180#336C2312yes—
181#336C2314yes—
182#336C2314yes—
183#336C2314yes—
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8C2×D411yescharacteristic
188C22×Q8C2412yes—
189C22×C8C2414yes—
190C22×C8C2412yes—
191C22×C8C2412yes—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196C4⋊Q8—44no—
197C4⋊Q8—44no—
198C4⋊Q8—82no—
199C4⋊Q8—82no—
200C4⋊Q8C2×Q814yes—
201C4⋊Q8C2×Q814yes—
202#125—48no—
203#125—84no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—28no—
208C4×Q8—48no—
209C4×Q8—44no—
210C4×Q8—84no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2412yes—
260C2×C42C2412yes—
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263C2×C42C2411yescharacteristic
264#107—216no—
265#107—48no—
266#106—48no—
267C4⋊C8—28no—
268C4⋊C8—24no—
269#103—28no—
270#103—48no—
271#103C2×Q8116yes—
272C4×C8—24no—
273C4×C8—24no—
274C4×C8—22no—
275C4×C8—24no—
276C4×C8C2×Q814yes—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95C2×D412yes—
290#95C2×D412yes—
291#95C2×D412yes—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—82no—
304C2×Q8—82no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×Q8C22×Q814yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4C22×D412yes—
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2511yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8C22×Q814yes—
333C2×C8H(5,𝔽2)14yes—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4C22×Q814yes—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—22no—
375C42—22no—
376C42—24no—
377C42—22no—
378C42—24no—
379C42—24no—
380C42—44no—
381C42—44no—
382C42—84no—
383C42—84no—
384C42#13514yes—
385C42#13714yes—
386C42C22×Q812yes—
387C422−1+414yes—
388C422−1+412yes—
389C23C23×D411yescharacteristic
390Q8—28no—
391Q8—48no—
392Q8—44no—
393Q8—44no—
394Q8—164no—
395Q8—164no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4#53612yes—
433C2×C4#53612yes—
434C2×C4D4×Q812yes—
435C2×C4D4×Q812yes—
436C2×C4C2×#13512yes—
437C2×C4C2×#13512yes—
438C2×C4C2×#13512yes—
439C2×C4C2×#13712yes—
440C2×C4C2×#13712yes—
441C2×C4C2×#13712yes—
442C2×C4#57612yes—
443C2×C4#57812yes—
444C2×C4C23×Q811yescharacteristic
445C2×C4C2×H(5,𝔽2)12yes—
446C2×C4C2×2−1+412yes—
447C2×C4C2×2−1+411yescharacteristic
448C8—24no—
449C8—24no—
450C8—44no—
451C22C2×#53611yescharacteristic
452C22C2×D4×Q811yescharacteristic
453C22#349811yescharacteristic
454C22C22×#13511yescharacteristic
455C22C22×#13711yescharacteristic
456C22C2×#57611yescharacteristic
457C22C2×#57811yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—42no—
462C4—44no—
463C4—44no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—82no—
468C4—84no—
469C4—84no—
470C4—82no—
471C4#323114yes—
472C4#338314yes—
473C4#338514yes—
474C4#350312yes—
475C4#354412yes—
476C2#3961711yescharacteristic
477C2C2×#323111yescharacteristic
478C2#6088011yescharacteristic
479C2C2×#338311yescharacteristic
480C2C2×#338511yescharacteristic
481C2#6241711yescharacteristic
482C2#6242411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
93. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#333122
#309482
#303072
#292062
#291782
#291412
#285152
#260472
#250632
#222212
C4×#4872
#213392