Finite Group Databasebeta

← #6474086

Subgroups — #6474086

Order 512.

1320 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#37394C212yesmaximal
2#33208C211yesmaximal, characteristic
3#33172C211yesmaximal, characteristic
4#33133C212yesmaximal
5#30202C212yesmaximal
6#29779C212yesmaximal
7#29074C214yesmaximal, complemented
8#29009C214yesmaximal, complemented
9#28023C212yesmaximal, complemented
10#28004C212yesmaximal, complemented
11#26493C212yesmaximal, complemented
12#26265C212yesmaximal, complemented
13#22197C212yesmaximal
14#21530C212yesmaximal
15#20435C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#467C2212yes—
19C2×#467C2212yes—
20C2×#441C2212yes—
21C2×#424C2211yescharacteristic
22C2×#405C2211yescharacteristic
23C2×#405C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2857C2212yes—
26#2851C2212yes—
27#2802—24no—
28#2787—24no—
29#2776—24no—
30#2770—24no—
31#2625C2214yes—
32#2514C2214yes—
33#2507C2214yes—
34#2453C2214yes—
35#2446C2214yes—
36#2323C2214yes—
37#2296C2211yescharacteristic
38#2100C2214yes—
39#2086C2214yes—
40#2074C2214yes—
41#2068C2214yes—
42#2068C2214yes—
43#2056C2214yes—
44#2043C2214yes—
45#2026C2214yes—
46#1991C2212yes—
47#1971C2212yes—
48#1971C2212yes—
49#1971C2212yes—
50#1969C2212yes—
51#1969C2212yes—
52#1956C2212yes—
53#1944C2212yes—
54#1944C2212yes—
55#1943C2212yes—
56#1942C2212yes—
57#1941C2212yes—
58#1933C2214yes—
59#1933C2214yes—
60#1933C2214yes—
61#1928C2214yes—
62#1918C2214yes—
63#1894C2214yes—
64#1894C2214yes—
65#1878C2214yes—
66#1863C2214yes—
67#1846C2211yescharacteristic
68#1845C2211yescharacteristic
69#1808C2212yes—
70#1807C2212yes—
71#1776C2212yes—
72#1771C2212yes—
73#1771C2212yes—
74C8×C4⋊C4C2212yes—
75C8×C4⋊C4C2212yes—
76C4×#103C2214yes—
77#1729C2211yescharacteristic
78#1728C2211yescharacteristic
79C22×Q16C2312yes—
80C22×Q16C2312yes—
81C22×M16C2311yescharacteristic
82C22×M16C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#125C2312yes—
87C2×C22⋊Q8C2314yes—
88C2×C4×Q8C2314yes—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#499—42no—
93#494—42no—
94#467—28no—
95#467—28no—
96#441—44no—
97C2×#107C2312yes—
98C2×#107C2312yes—
99C2×#107C2312yes—
100C2×#107C2312yes—
101C2×#106C2312yes—
102C2×#106C2312yes—
103#424—24no—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112#405—24no—
113#405—42no—
114C2×#97C2312yes—
115C2×#97C2312yes—
116C2×C4×C8—42no—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2312yes—
123#401—24no—
124#400—24no—
125#400—24no—
126#400—24no—
127#400—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#398—24no—
132#398—24no—
133#397—24no—
134#397—24no—
135#391C2314yes—
136#391C2314yes—
137#391C2314yes—
138#391C2314yes—
139#389—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#389D412yes—
144#389D412yes—
145#389C2314yes—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#388—24no—
150#388—24no—
151#387—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#387—24no—
156#387—24no—
157#387—24no—
158#385—24no—
159#385D414yes—
160#384C2312yes—
161#383—24no—
162#382C2312yes—
163#380—24no—
164#380C2314yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2312yes—
167C4×C4⋊C4C2312yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#336C2312yes—
175#336C2312yes—
176#336C2312yes—
177#336C2312yes—
178C22×Q8C2414yes—
179C23×C4C2×D412yes—
180C23×C4C2411yescharacteristic
181C2×Q16—24no—
182C2×Q16—24no—
183C2×Q16—24no—
184C2×Q16—24no—
185C2×Q16—42no—
186C2×Q16—42no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—41no—
190C2×M16—41no—
191C2×M16—41no—
192C2×M16—41no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—41no—
196C22×C8—41no—
197C22×C8#4114yes—
198C22×C8#4114yes—
199C22×C8C2412yes—
200C22×C8C2412yes—
201C22×C8C2411yescharacteristic
202C22×C8C2411yescharacteristic
203C22×C8C2411yescharacteristic
204C22×C8C2411yescharacteristic
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207#125—42no—
208#125—42no—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211C4×Q8—44no—
212C4×Q8—44no—
213#117—22no—
214#117—22no—
215#117—41no—
216#117—41no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2414yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4C2×D412yes—
237C2×C22⋊C4C2×D412yes—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42C2×D412yes—
243C2×C42C2412yes—
244C2×C42C2412yes—
245C2×C42C2411yescharacteristic
246C2×C42C2411yescharacteristic
247#107—24no—
248#107—24no—
249#107—42no—
250#107—42no—
251#106—24no—
252#106—24no—
253C4⋊C8—24no—
254C4⋊C8—24no—
255#103—28no—
256#103—28no—
257#103—44no—
258#103—44no—
259#103—44no—
260#103—44no—
261#97—24no—
262#97—24no—
263C4×C8—24no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—41no—
267C4×C8—41no—
268C4×C8—42no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95C2×D414yes—
285#95C2×D414yes—
286C24C22×D411yescharacteristic
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4C22×D412yes—
305C22×C4C22×D412yes—
306C22×C4C22×D414yes—
307C22×C4C22×D412yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D412yes—
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2511yescharacteristic
313Q16—48no—
314Q16—84no—
315M16—44no—
316M16—44no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—28no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—44no—
331C2×C8C2×#4112yes—
332C2×C8C2×#4112yes—
333C2×C8C2×#4112yes—
334C2×C8C2×#4112yes—
335C2×C8H(5,𝔽2)12yes—
336C2×C82−1+412yes—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—48no—
360C22⋊C4—84no—
361C42—22no—
362C42—22no—
363C42—21no—
364C42—21no—
365C42—22no—
366C42—21no—
367C42—21no—
368C42—22no—
369C42—42no—
370C42—84no—
371C42—84no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23#53212yes—
377C23C23×D411yescharacteristic
378Q8—44no—
379Q8—44no—
380Q8—44no—
381Q8—44no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—48no—
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—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4#53212yes—
414C2×C4#57814yes—
415C2×C4C22×#4111yescharacteristic
416C2×C4C22×#4111yescharacteristic
417C2×C4C2×H(5,𝔽2)11yescharacteristic
418C2×C4C2×2−1+411yescharacteristic
419C2×C4#58511yescharacteristic
420C2×C4#58511yescharacteristic
421C8—22no—
422C8—22no—
423C8—22no—
424C8—42no—
425C8—42no—
426C8#53914yes—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—42no—
431C22—42no—
432C22C2×#53211yescharacteristic
433C22#349611yescharacteristic
434C22#349711yescharacteristic
435C22C2×#57812yes—
436C22#358412yes—
437C4—24no—
438C4—22no—
439C4—42no—
440C4—42no—
441C4—44no—
442C4—42no—
443C4—84no—
444C4—84no—
445C4—84no—
446C4—84no—
447C4C2×#53911yescharacteristic
448C4#345511yescharacteristic
449C4#350511yescharacteristic
450C4#350711yescharacteristic
451C2—42no—
452C2#3500611yescharacteristic
453C2#6071511yescharacteristic
454C2#6071711yescharacteristic
455C2#6227912yes—
456C2#6238512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #29074C2semidirect
8. #29009C2semidirect
9. #28023C2semidirect
10. #28004C2semidirect
11. #26493C2semidirect
12. #26265C2semidirect

Maximal subgroups

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

subgroupindex
#373942
#332082
#331722
#331332
#302022
#297792
#290742
#290092
#280232
#280042
#264932
#262652
#221972
#215302
#204352