Finite Group Databasebeta

← #6495084

Subgroups — #6495084

Order 512.

2004 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36644C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic, complemented
3#33236C211yesmaximal, characteristic, complemented
4#33235C211yesmaximal, characteristic, complemented
5#33232C211yesmaximal, characteristic, complemented
6#33231C211yesmaximal, characteristic, complemented
7#30225C214yesmaximal, complemented
8#28757C214yesmaximal, complemented
9#28750C214yesmaximal, complemented
10#25862C214yesmaximal, complemented
11#25857C214yesmaximal, complemented
12#21812C214yesmaximal, complemented
13C4×#432C211yesmaximal, characteristic, complemented
14C2×#532C2212yes—
15C2×#532C2212yes—
16C2×#532C2212yes—
17C2×#491C2214yes—
18C2×#490C2214yes—
19C2×#487C2214yes—
20C2×#486C2214yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2867C2211yescharacteristic
24#2864C2212yes—
25#2863—22no—
26#2862C2211yescharacteristic
27#2859C2211yescharacteristic
28#2858C2212yes—
29#2858C2212yes—
30#2858C2212yes—
31#2857C2212yes—
32#2856—22no—
33#2856—22no—
34#2855—22no—
35#2851C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×#127—22no—
38C4×#127—22no—
39C4×#125C2212yes—
40C4×C22⋊Q8—22no—
41#2295—22no—
42#2283C2212yes—
43C2×C43C2211yescharacteristic
44#2042C22116yes—
45#2042C22116yes—
46#1983C22116yes—
47#1961C2214yes—
48#1961C2214yes—
49#1961C2214yes—
50#1960C2214yes—
51#1960C2214yes—
52#1960C2214yes—
53#1957C2214yes—
54#1956C2214yes—
55#1788C22116yes—
56#1767C2212yes—
57C4×C4⋊C8C2212yes—
58#1749C2212yes—
59C4×C22⋊C8C2212yes—
60#1745C2212yes—
61C4×#97C2211yescharacteristic
62C4×#97C2211yescharacteristic
63C22×#41C2314yes—
64#532—216no—
65#532—48no—
66#532—48no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2312yes—
71C2×#127—22no—
72C2×#127—22no—
73C2×#127C2312yes—
74C2×#127C2312yes—
75C2×#127C2312yes—
76C2×#125C2312yes—
77C2×#124C2314yes—
78C2×#124C2314yes—
79C2×#124C2314yes—
80C2×C22⋊Q8—22no—
81C2×#121—24no—
82C2×#121—24no—
83C2×#121—22no—
84C2×#121—22no—
85C2×C4×Q8—22no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C22×C42D412yes—
91C22×C42D412yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#491—48no—
96#490—48no—
97#487—48no—
98#486—48no—
99#432—28no—
100C2×C4⋊C8C2314yes—
101C2×#103C23116yes—
102C2×#103C23116yes—
103C2×#102C23116yes—
104C2×#102C23116yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2312yes—
107C2×#97C2312yes—
108C2×#97C2312yes—
109C2×#97C2312yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112#391—22no—
113#391—22no—
114#391—22no—
115#391—22no—
116#390—24no—
117#390—24no—
118#390—24no—
119#390—24no—
120#390—24no—
121#390—24no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389C2314yes—
127#389C2314yes—
128#387—24no—
129#387—24no—
130#387—24no—
131#387—24no—
132#385—24no—
133#384—22no—
134#384—22no—
135#384C2312yes—
136#383—22no—
137#382C2312yes—
138#382C2312yes—
139#382C2312yes—
140#381—22no—
141#380—22no—
142#380C2314yes—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4C2314yes—
146C4×C22⋊C4—22no—
147#376—22no—
148C43D412yes—
149C43D412yes—
150C43C2311yescharacteristic
151C43C2311yescharacteristic
152C43C2311yescharacteristic
153C43C2311yescharacteristic
154#336C2314yes—
155#336C2314yes—
156C2×#41—216no—
157C2×#41—48no—
158C2×#41—48no—
159C22×Q8—22no—
160C22×Q8C2414yes—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4C2414yes—
167C23×C4—24no—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8C2414yes—
174C22×C8C2414yes—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179#127—28no—
180#127—28no—
181#127—28no—
182#127—48no—
183#127—48no—
184#125—28no—
185#124—216no—
186#124—48no—
187#124—48no—
188C22⋊Q8—48no—
189#121—416no—
190#121—416no—
191#121—48no—
192#121—48no—
193C4×Q8—48no—
194C4×D4—416no—
195C4×D4—416no—
196C4×D4—48no—
197C4×D4—48no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4118yes—
233C2×C42C2411yescharacteristic
234C2×C42C2411yescharacteristic
235C2×C42C2411yescharacteristic
236C2×C42C2411yescharacteristic
237C2×C42C2411yescharacteristic
238C2×C42C2411yescharacteristic
239C4⋊C8—28no—
240#103—416no—
241#103—416no—
242#102—416no—
243#102—416no—
244C22⋊C8—28no—
245#97—24no—
246#97—24no—
247#97—24no—
248#97—24no—
249C4×C8—24no—
250C4×C8—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260C24—24no—
261C24C22×D411yescharacteristic
262#41—832no—
263C2×Q8—28no—
264C2×Q8—28no—
265C2×Q8—28no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×D4—28no—
270C2×D4—28no—
271C2×D4—28no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—48no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4#12414yes—
302C22×C4#12414yes—
303C22×C4C22×D412yes—
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D412yes—
311C22×C4C22×D412yes—
312C22×C4C2×#4114yes—
313C22×C4C2511yescharacteristic
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C22⋊C4—416no—
333C22⋊C4—416no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—22no—
344C42—22no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—48no—
350C42#12414yes—
351C42#12414yes—
352C42C2×#4114yes—
353C42C2×#4114yes—
354C42H(5,𝔽2)14yes—
355C42H(5,𝔽2)14yes—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—22no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—48no—
364C23—48no—
365C23C2×#12412yes—
366C23C23×D411yescharacteristic
367Q8—416no—
368Q8—88no—
369D4—416no—
370D4—816no—
371D4—88no—
372D4—88no—
373D4—816no—
374D4—816no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—28no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—816no—
404C2×C4C2×#12412yes—
405C2×C4C2×#12412yes—
406C2×C4C2×#12412yes—
407C2×C4C2×#12412yes—
408C2×C4C2×#12412yes—
409C2×C4#53514yes—
410C2×C4#53514yes—
411C2×C4#57514yes—
412C2×C4#57514yes—
413C2×C4C22×#4112yes—
414C2×C4C2×H(5,𝔽2)12yes—
415C2×C4C2×H(5,𝔽2)12yes—
416C8—44no—
417C8—44no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—88no—
424C22—88no—
425C22—84no—
426C22—84no—
427C22—84no—
428C22C22×#12411yescharacteristic
429C22C2×#53511yescharacteristic
430C22C2×#53511yescharacteristic
431C22C2×#57512yes—
432C22C2×#57512yes—
433C4—24no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—42no—
441C4—84no—
442C4—84no—
443C4—84no—
444C4#321918yes—
445C4#350814yes—
446C2—42no—
447C2—164no—
448C2#3528511yescharacteristic
449C2#6061811yescharacteristic
450C2#6061811yescharacteristic
451C2C2×#321912yes—
452C2#6080112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36644C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36635C2semidirect
3. #33236C2semidirect
4. #33235C2semidirect
5. #33232C2semidirect
6. #33231C2semidirect
7. #30225C2semidirect
8. #28757C2semidirect
9. #28750C2semidirect
10. #25862C2semidirect
11. #25857C2semidirect
12. #21812C2semidirect
13. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366442
#366352
#332362
#332352
#332322
#332312
#302252
#287572
#287502
#258622
#258572
#218122
C4×#4322