Finite Group Databasebeta

← #6506790

Subgroups — #6506790

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic
2#35639C211yesmaximal, characteristic
3#29306C212yesmaximal, complemented
4#29304C212yesmaximal, complemented
5#28786C214yesmaximal, complemented
6#28780C214yesmaximal, complemented
7#27957C212yesmaximal, complemented
8#26078C214yesmaximal, complemented
9#26071C214yesmaximal, complemented
10#25819C212yesmaximal, complemented
11#21946C212yesmaximal
12#20847C212yesmaximal
13#20532C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C2×#501C2214yes—
17C2×#428C2212yes—
18C22×#107C2212yes—
19C2×#408C2212yes—
20#2857C2212yes—
21#2806—22no—
22#2748—22no—
23#2732—22no—
24#2683—22no—
25#2629C2212yes—
26#2579—22no—
27#2487—22no—
28#2454—22no—
29#2453—22no—
30#2442C2214yes—
31#2441C2214yes—
32#2441C2214yes—
33#2440C2214yes—
34#2317C2211yescharacteristic
35#2316C2211yescharacteristic
36#2311C2211yescharacteristic
37#2311C2212yes—
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic
40#2085C2214yes—
41#2084C2214yes—
42#2084C2212yes—
43#2083C2212yes—
44#2083C2214yes—
45#2059C2214yes—
46#2059C2214yes—
47#2059C2214yes—
48#2058C2212yes—
49#2057C2212yes—
50#1992C2212yes—
51#1992C2214yes—
52#1991C2212yes—
53#1944C2214yes—
54#1943C2214yes—
55#1943C2214yes—
56#1942C2214yes—
57#1854C2214yes—
58#1854C2214yes—
59#1854C2214yes—
60#1854C2214yes—
61#1853C2214yes—
62#1853C2214yes—
63#1852C2214yes—
64#1852C2214yes—
65#1847C2212yes—
66#1844C2212yes—
67#1809C2214yes—
68#1807C2214yes—
69#1753C2214yes—
70#1752C2214yes—
71#1750C2212yes—
72#1749C2212yes—
73#1736C2212yes—
74#1735C2212yes—
75C22×#41C2311yescharacteristic
76C22×M16C2312yes—
77C23×C8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×#125—22no—
80C2×#125—44no—
81C2×#125C2314yes—
82C2×#125C2312yes—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—24no—
85#520—44no—
86C2×C4×Q8—21no—
87C2×C4×D4—22no—
88C2×C4×D4—21no—
89C2×#117C2312yes—
90C2×#117C2311yescharacteristic
91C2×#117C2312yes—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C42C2311yescharacteristic
101#501—48no—
102#428—44no—
103C2×#107—44no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#107C2314yes—
108C2×#106C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112#408Q8116yescomplemented
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2312yes—
116C2×#97C2314yes—
117C2×C4×C8C2312yes—
118#400—24no—
119#399—24no—
120#399—24no—
121#398—24no—
122#398—24no—
123#398—24no—
124#396—24no—
125#396—24no—
126#396—24no—
127#395—24no—
128#391—21no—
129#391—22no—
130#391—22no—
131#391D412yes—
132#390—22no—
133#390—21no—
134#390D412yes—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—24no—
139#389—24no—
140#389C2312yes—
141#389C2312yes—
142#389C2314yes—
143#389C2314yes—
144#389C2314yes—
145#387—24no—
146#387—24no—
147#386—22no—
148#386—22no—
149#386—22no—
150#386D412yes—
151#386D412yes—
152#385—21no—
153#385—21no—
154#385—22no—
155#385—24no—
156#385—24no—
157#384C2312yes—
158#384C2312yes—
159#384C2312yes—
160#382C2312yes—
161#380—22no—
162#380—22no—
163#380—24no—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C22⋊C4C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#336C2312yes—
178#336C2312yes—
179#336C2314yes—
180C2×#41—42no—
181C2×#41—41no—
182C2×#41—41no—
183C2×#41C2×Q818yes—
184C22×Q8—22no—
185C22×Q8C2×D411yescharacteristic
186C22×D4C2×D412yes—
187C22×D4C2×D411yescharacteristic
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16C2×Q818yes—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8C2×Q818yes—
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—44no—
205#125—28no—
206#125—44no—
207#125—42no—
208#125—42no—
209#125—48no—
210C22⋊Q8—88no—
211C22⋊Q8—84no—
212C4×Q8—44no—
213C4×D4—44no—
214C4×D4—48no—
215#117—24no—
216#117—41no—
217#117—41no—
218#117—41no—
219#117—41no—
220#117—42no—
221#117—42no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2411yescharacteristic
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2411yescharacteristic
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4C2×D414yes—
266C2×C22⋊C4C2×D414yes—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42C2×D412yes—
273C2×C42C2412yes—
274C2×C42C2412yes—
275#107—28no—
276#107—28no—
277#107—44no—
278#107—44no—
279#106—44no—
280#106—44no—
281#106—44no—
282#106—44no—
283C22⋊C8C2×Q8116yes—
284C22⋊C8C2×Q8116yes—
285#97—42no—
286#97—44no—
287C4×C8—42no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95C2×D414yes—
298#95C2×D414yes—
299C24C22×D412yes—
300C24C22×D411yescharacteristic
301#41—48no—
302#41—48no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—44no—
306C2×Q8—82no—
307C2×Q8—82no—
308C2×Q8C4⋊Q814yes—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4C4⋊Q818yes—
314C2×D4C4⋊Q814yes—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4C22×D412yes—
342C22×C4C22×D414yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q814yes—
347C22×C4C22×Q814yes—
348C22×C4C22×Q814yes—
349C22×C4C2511yescharacteristic
350M16—48no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—48no—
356C2×C8C22×Q818yes—
357C2×C8C22×Q818yes—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—48no—
376C4⋊C4—84no—
377C4⋊C4—82no—
378C4⋊C4—84no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C4⋊C4—82no—
386C4⋊C4H(5,𝔽2)14yes—
387C22⋊C4—44no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—84no—
394C22⋊C4—82no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C42—24no—
398C42—22no—
399C42—22no—
400C42—42no—
401C42—44no—
402C42—42no—
403C42—42no—
404C42—82no—
405C23—24no—
406C23—22no—
407C23—24no—
408C23—22no—
409C23—21no—
410C23—22no—
411C23—21no—
412C23—22no—
413C23C2×C4⋊Q812yes—
414C23C2×C4⋊Q814yes—
415C23C23×D411yescharacteristic
416Q8—44no—
417Q8—164no—
418D4—44no—
419D4—48no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—84no—
460C2×C4—164no—
461C2×C4—164no—
462C2×C4C2×C4⋊Q814yes—
463C2×C4C2×C4⋊Q812yes—
464C2×C4C23×Q812yes—
465C2×C4C2×H(5,𝔽2)12yes—
466C2×C4#58511yescharacteristic
467C2×C4#58511yescharacteristic
468C8—28no—
469C8—28no—
470C22—24no—
471C22—22no—
472C22—28no—
473C22—24no—
474C22—44no—
475C22—42no—
476C22—41no—
477C22—42no—
478C22—41no—
479C22—42no—
480C22C22×C4⋊Q811yescharacteristic
481C22#348411yescharacteristic
482C22#348511yescharacteristic
483C22#358512yes—
484C22#358612yes—
485C4—22no—
486C4—24no—
487C4—44no—
488C4—42no—
489C4—44no—
490C4—42no—
491C4—84no—
492C4—82no—
493C4—81no—
494C4—84no—
495C4—81no—
496C4—82no—
497C4#350112yes—
498C4#350112yes—
499C2—42no—
500C2—44no—
501C2#3539111yescharacteristic
502C2#6066812yes—
503C2#6164811yescharacteristic
504C2#6165211yescharacteristic
505C2#6180012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29306C2semidirect
4. #29304C2semidirect
5. #28786C2semidirect
6. #28780C2semidirect
7. #27957C2semidirect
8. #26078C2semidirect
9. #26071C2semidirect
10. #25819C2semidirect
112. #408Q8semidirect

Maximal subgroups

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

subgroupindex
#373072
#356392
#293062
#293042
#287862
#287802
#279572
#260782
#260712
#258192
#219462
#208472
#205322