Finite Group Databasebeta

← #361082

Subgroups — #361082

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1919C212yesmaximal
2C2×C4×#106C211yesmaximal, characteristic
3#13918C211yesmaximal, characteristic
4#13818C211yesmaximal, characteristic
5#11327C212yesmaximal
6#11220C212yesmaximal
7#10908C212yesmaximal
8#10861C211yesmaximal, characteristic
9#10807C211yesmaximal, characteristic
10#10806C211yesmaximal, characteristic
11#10797C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#382—21no—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1919—28no—
23C4×#106C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1442—22no—
35#1436—22no—
36C4⋊#95—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—21no—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42—21no—
55C22×C42—21no—
56C22×C42—22no—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#106—24no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—42no—
63C2×#106C2×C414yes—
64C2×#106C2×C414yes—
65C2×#106D414yes—
66C2×#106Q814yes—
67C2×C4×C8—24no—
68C2×C4×C8C2×C414yes—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—42no—
73#382—44no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—42no—
78C4×C4⋊C4C2×C414yes—
79C4×C4⋊C4C2×C414yes—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95D411yescharacteristic
92C2×#95D412yes—
93C2×#95D411yescharacteristic
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340C2×C414yes—
100#340C2×C414yes—
101#336—24no—
102#336—24no—
103#336—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C22×C412yes—
128C22×C8C22×C412yes—
129C22×C8C2×D412yes—
130C22×C8C2×Q812yes—
131C22×C8#4114yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×Q812yes—
167C2×C4⋊C4C2×Q812yes—
168C2×C4⋊C4#4114yes—
169C2×C4⋊C4#4114yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—82no—
185C2×C42—82no—
186C2×C42C22×C412yes—
187C2×C42#4112yes—
188#106—28no—
189#106—48no—
190#106C4⋊C4116yescomplemented
191C4×C8—28no—
192#95—28no—
193#95—24no—
194#95—24no—
195#95—22no—
196#95—42no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—42no—
202#95—42no—
203#95—81no—
204#95—81no—
205#95SD1614yes—
206C24C22×D411yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C4×D412yes—
263C22×C4C4×D412yes—
264C22×C4C2×SD1614yes—
265C22×C4C2×SD1612yes—
266C22×C4#13512yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C42−1+412yes—
276C22×C42−1+411yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8C2×C4⋊C414yes—
293C2×C8#11714yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—88no—
314C4⋊C4—88no—
315C4⋊C4C2×C4⋊C414yes—
316C4⋊C4C2×C4⋊C414yes—
317C4⋊C4#11714yes—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—48no—
326C42—84no—
327C42—88no—
328C23C2×C4×D411yescharacteristic
329C23#52512yes—
330C23D4211yescharacteristic
331C23#54712yes—
332C23D4×Q811yescharacteristic
333C23C22×SD1611yescharacteristic
334C23C2×#13511yescharacteristic
335C23#57512yes—
336C23#57712yes—
337C23#57711yescharacteristic
338C23#57711yescharacteristic
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4C4×SD1614yes—
399C2×C4C22×C4⋊C411yescharacteristic
400C2×C4C2×#11711yescharacteristic
401C2×C4C2×#11711yescharacteristic
402C2×C4C4×#4111yescharacteristic
403C2×C4#51911yescharacteristic
404C2×C4#52011yescharacteristic
405C2×C4#52912yes—
406C2×C4#53212yes—
407C2×C4#53312yes—
408C2×C4#54112yes—
409C2×C4#54212yes—
410C2×C4#54711yescharacteristic
411C2×C4#54812yes—
412C2×C4#55112yes—
413C2×C4#55212yes—
414C2×C4#55212yes—
415C2×C4#55411yescharacteristic
416C2×C4#55712yes—
417C8—24no—
418C8—24no—
419C8—82no—
420C22#229011yescharacteristic
421C22#234411yescharacteristic
422C22D4×C4⋊C411yescharacteristic
423C22#246112yes—
424C22#246212yes—
425C22C2×C4×SD1611yescharacteristic
426C22#294311yescharacteristic
427C22#294911yescharacteristic
428C22#295211yescharacteristic
429C22#310212yes—
430C22#310312yes—
431C22#312312yes—
432C22#312512yes—
433C22#327512yes—
434C22D4×SD1611yescharacteristic
435C22#328411yescharacteristic
436C22#333512yes—
437C22#333712yes—
438C22#334812yes—
439C22#334812yes—
440C22Q8×SD1611yescharacteristic
441C22#338111yescharacteristic
442C22#338812yes—
443C4—24no—
444C4—42no—
445C4—44no—
446C4—44no—
447C4—44no—
448C4—44no—
449C4—82no—
450C4—82no—
451C4—84no—
452C4—162no—
453C4#228511yescharacteristic
454C4#232011yescharacteristic
455C4#234311yescharacteristic
456C4#235111yescharacteristic
457C4#251912yes—
458C4#252012yes—
459C2#1388911yescharacteristic
460C2#2111511yescharacteristic
461C2#2113311yescharacteristic
462C2#2260911yescharacteristic
463C2#2261911yescharacteristic
464C2#2267311yescharacteristic
465C2C4⋊C4×SD1611yescharacteristic
466C2#2678812yes—
467C2#2679612yes—
468C2#2686012yes—
469C2#2686812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. C4×#106C4semidirect
190. #106C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19192
C2×C4×#1062
#139182
#138182
#113272
#112202
#109082
#108612
#108072
#108062
#107972