Finite Group Databasebeta

← #6594497

Subgroups — #6594497

Order 512.

1423 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#36210C212yesmaximal
2#33263C211yesmaximal, characteristic, complemented
3#33210C211yesmaximal, characteristic, complemented
4#33207C211yesmaximal, characteristic, complemented
5#29176C212yesmaximal, complemented
6#28002C214yesmaximal, complemented
7#27962C212yesmaximal, complemented
8#27875C212yesmaximal, complemented
9#27850C214yesmaximal, complemented
10#27665C214yesmaximal
11#22620C214yesmaximal, complemented
12#21727C211yesmaximal, characteristic, complemented
13#20824C211yesmaximal, characteristic, complemented
14#20797C211yesmaximal, characteristic, complemented
15C4×#435C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#435C2212yes—
18C2×#427C2214yes—
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2859C2212yes—
22#2857C2212yes—
23#2857C2212yes—
24#2562—22no—
25#2514—22no—
26#2509C2214yes—
27#2495—22no—
28#2494—22no—
29#2494C2214yes—
30#2485C2214yes—
31#2421—22no—
32#2415—22no—
33#2370—22no—
34#2362—22no—
35#2347C2214yes—
36C4×C4⋊Q8C2212yes—
37#2285C2212yes—
38#2285C2212yes—
39C42×D4C2211yescharacteristic
40#2101C2218yes—
41#2088C2218yes—
42#2077C2218yes—
43#1990C2212yes—
44#1990C2212yes—
45#1990C2212yes—
46#1944C2214yes—
47#1944C2214yes—
48#1943C2214yes—
49#1943C2214yes—
50#1941C2214yes—
51#1920C2218yes—
52#1920C2218yes—
53#1846C2214yes—
54#1845C2211yescharacteristic
55#1845C2211yescharacteristic
56#1844C2212yes—
57#1844C2211yescharacteristic
58#1834C2212yes—
59#1834C2212yes—
60#1825C2218yes—
61#1809C2214yes—
62#1809C2214yes—
63#1807C2214yes—
64#1776C2214yes—
65#1776C2214yes—
66#1771C2212yes—
67C4×C4⋊C8C2211yescharacteristic
68#1753C2212yes—
69#1752C2212yes—
70C4×C22⋊C8C2211yescharacteristic
71C4×C22⋊C8C2211yescharacteristic
72C4×#97C2211yescharacteristic
73C22×M16C2312yes—
74C23×C8C2312yes—
75C2×C4⋊Q8C2312yes—
76C2×#125—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×D4D412yes—
81C2×C4×D4C2311yescharacteristic
82C2×#117—22no—
83C2×#117C2314yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#498—24no—
87#435—28no—
88#427—216no—
89C2×#107C2314yes—
90C2×#107C2314yes—
91C2×#106C2314yes—
92C2×#106C2314yes—
93C2×#106C2318yes—
94C2×C4⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C4×M16—24no—
98C2×#97C2312yes—
99C2×#97C2312yes—
100C2×#97C2312yes—
101C2×C4×C8—24no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2312yes—
106C2×C4×C8C2311yescharacteristic
107#401—24no—
108#400—24no—
109#400—24no—
110#397—24no—
111#397—24no—
112#394—24no—
113#391—22no—
114#391—22no—
115#391—22no—
116#389—24no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389C2314yes—
123#389C2314yes—
124#389C2314yes—
125#389C2314yes—
126#389C2314yes—
127#388—24no—
128#388—24no—
129#388—24no—
130#388—24no—
131#387—24no—
132#387—24no—
133#387—24no—
134#384C2312yes—
135#384C2312yes—
136#384C2312yes—
137#383—24no—
138#382—22no—
139#382—22no—
140#382C2314yes—
141#380—22no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380—22no—
146#380C2314yes—
147#380C2314yes—
148#380C2314yes—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D412yes—
155C4×C4⋊C4C2314yes—
156C4×C4⋊C4C2314yes—
157C4×C4⋊C4C2311yescharacteristic
158C4×C22⋊C4D412yes—
159C4×C22⋊C4D412yes—
160C4×C22⋊C4C2311yescharacteristic
161C4×C22⋊C4C2311yescharacteristic
162C43C2311yescharacteristic
163#340C2318yes—
164#340C2318yes—
165#340C2318yes—
166#340C2318yes—
167#340C2318yes—
168#336C2312yes—
169#336C2312yes—
170#336C2312yes—
171C22×Q8—22no—
172C22×D4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C2×M16—24no—
178C2×M16—24no—
179C2×M16—24no—
180C22×C8—24no—
181C22×C8—24no—
182C22×C8#4118yes—
183C22×C8C2412yes—
184C22×C8C2412yes—
185C22×C8C2412yes—
186C22×C8C2412yes—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189#125—48no—
190C22⋊Q8—48no—
191C22⋊Q8—84no—
192C4×Q8—84no—
193C4×D4—24no—
194C4×D4—44no—
195#117—216no—
196#117—84no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C4⋊C4C2414yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C22⋊C4C2×D411yescharacteristic
221C2×C22⋊C4C2×D412yes—
222C2×C22⋊C4C2×D412yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2411yescharacteristic
235C2×C42C2411yescharacteristic
236C2×C42C2411yescharacteristic
237C2×C42C2411yescharacteristic
238C2×C42C2411yescharacteristic
239#107—28no—
240#107—28no—
241#106—216no—
242#106—28no—
243#106—28no—
244C4⋊C8—24no—
245C22⋊C8—24no—
246C22⋊C8—24no—
247#97—24no—
248#97—24no—
249#97—24no—
250C4×C8—24no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8#4114yes—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95C2×D412yes—
268#95C2×D412yes—
269#95C2×D412yes—
270C24C22×D411yescharacteristic
271C24C22×D411yescharacteristic
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×D4—22no—
276C2×D4—42no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C2×#4114yes—
303C22×C4C2511yescharacteristic
304M16—48no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—216no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8C2×#4114yes—
315C2×C8C2×#4114yes—
316C2×C8C2×#4114yes—
317C2×C82−1+414yes—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—21no—
350C42—21no—
351C42—22no—
352C42—22no—
353C42—21no—
354C42—21no—
355C42—48no—
356C42—84no—
357C42—84no—
358C42C2×#4112yes—
359C422−1+412yes—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—22no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—22no—
369C23#53212yes—
370C23C23×D411yescharacteristic
371Q8—88no—
372D4—44no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4#53212yes—
410C2×C4#54714yes—
411C2×C4#57514yes—
412C2×C4#57714yes—
413C2×C4C22×#4111yescharacteristic
414C2×C4C22×#4112yes—
415C2×C4C2×2−1+412yes—
416C2×C4C2×2−1+411yescharacteristic
417C8—24no—
418C8—42no—
419C8—42no—
420C8#55218yes—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22C2×#53211yescharacteristic
432C22C2×#54711yescharacteristic
433C22#349911yescharacteristic
434C22C2×#57512yes—
435C22C2×#57712yes—
436C4—22no—
437C4—42no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—41no—
443C4—84no—
444C4—84no—
445C4—84no—
446C4—84no—
447C4#335018yes—
448C4C2×#55212yes—
449C4#350912yes—
450C2—42no—
451C2—81no—
452C2#3618111yescharacteristic
453C2#6069311yescharacteristic
454C2#6071611yescharacteristic
455C2C2×#335012yes—
456C2#6248312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33263C2semidirect
3. #33210C2semidirect
4. #33207C2semidirect
5. #29176C2semidirect
6. #28002C2semidirect
7. #27962C2semidirect
8. #27875C2semidirect
9. #27850C2semidirect
11. #22620C2semidirect
12. #21727C2semidirect
13. #20824C2semidirect
14. #20797C2semidirect

Maximal subgroups

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

subgroupindex
#362102
#332632
#332102
#332072
#291762
#280022
#279622
#278752
#278502
#276652
#226202
#217272
#208242
#207972
C4×#4352