Finite Group Databasebeta

← #362541

Subgroups — #362541

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1941C212yesmaximal
2C2×C4×#106C211yesmaximal, characteristic
3#13933C211yesmaximal, characteristic
4#13918C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11292C212yesmaximal
7#10861C212yesmaximal
8#10861C211yesmaximal, characteristic
9#10807C211yesmaximal, characteristic
10#10806C211yesmaximal, characteristic
11C8×#95C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4×C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#382—21no—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1941—28no—
23C4×#106C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1442—22no—
34#1442—22no—
35C4⋊#95—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42—21no—
55C22×C42—21no—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×#106—24no—
59C2×#106—24no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106C2×C414yes—
63C2×#106C2×C414yes—
64C2×#106D414yes—
65C2×#106Q814yes—
66C2×C4×C8—24no—
67C2×C4×C8—24no—
68C2×C4×C8—24no—
69C2×C4×C8—24no—
70C2×C4×C8C2×C414yes—
71#389—24no—
72#389—24no—
73#389—42no—
74#389—42no—
75#382—44no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2×C414yes—
79C4×C4⋊C4C2×C414yes—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95D411yescharacteristic
93C2×#95D412yes—
94C2×#95D411yescharacteristic
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—42no—
100#340C2×C414yes—
101#340C2×C414yes—
102#336—24no—
103#336—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8C22×C412yes—
126C22×C8C22×C412yes—
127C22×C8C2×D412yes—
128C22×C8C2×Q812yes—
129C22×C8#4114yes—
130C22×C8#4114yes—
131C22×C8#4114yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
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—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×Q812yes—
169C2×C4⋊C4C2×Q812yes—
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—82no—
183C2×C42C22×C412yes—
184C2×C42#4112yes—
185#106—28no—
186#106—48no—
187#106C4⋊C4116yes—
188C4×C8—28no—
189C4×C8—48no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—42no—
198#95—44no—
199#95—42no—
200#95—82no—
201#95—82no—
202#95—81no—
203#95—81no—
204#95SD1614yes—
205C24C22×D411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
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—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
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×C4C4×D412yes—
261C22×C4C4×D412yes—
262C22×C4C2×SD1614yes—
263C22×C4C2×SD1612yes—
264C22×C4#13514yes—
265C22×C4#13512yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4C2×#4111yescharacteristic
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C42−1+412yes—
275C22×C42−1+411yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8C2×C4⋊C414yes—
289C2×C8#11714yes—
290C2×C8#11714yes—
291C2×C8#12418yes—
292C2×C8#12518yes—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
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—
317C42—22no—
318C42—22no—
319C42—24no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—48no—
325C42—84no—
326C23C2×C4×D411yescharacteristic
327C23#53212yes—
328C23D4211yescharacteristic
329C23#54712yes—
330C23D4×Q811yescharacteristic
331C23C22×SD1611yescharacteristic
332C23C2×#13511yescharacteristic
333C23C2×#13512yes—
334C23#57712yes—
335C23#57711yescharacteristic
336C23#57711yescharacteristic
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
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—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
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×C4C4×SD1614yes—
397C2×C4C22×C4⋊C411yescharacteristic
398C2×C4C2×#11711yescharacteristic
399C2×C4C2×#11711yescharacteristic
400C2×C4C4×#4111yescharacteristic
401C2×C4#51911yescharacteristic
402C2×C4#52011yescharacteristic
403C2×C4C2×#12412yes—
404C2×C4C2×#12512yes—
405C2×C4#52912yes—
406C2×C4#54112yes—
407C2×C4#54212yes—
408C2×C4#54711yescharacteristic
409C2×C4#54812yes—
410C2×C4#55212yes—
411C2×C4#55212yes—
412C2×C4#55411yescharacteristic
413C2×C4#55512yes—
414C2×C4#55712yes—
415C8—42no—
416C8—44no—
417C8#38218yescomplemented
418C22#229311yescharacteristic
419C22#234411yescharacteristic
420C22D4×C4⋊C411yescharacteristic
421C22#249112yes—
422C22#250612yes—
423C22C2×C4×SD1611yescharacteristic
424C22C4×#13511yescharacteristic
425C22#294911yescharacteristic
426C22#295211yescharacteristic
427C22#316712yes—
428C22#316812yes—
429C22#317312yes—
430C22#317412yes—
431C22D4×SD1611yescharacteristic
432C22#328212yes—
433C22#328411yescharacteristic
434C22#333512yes—
435C22#333712yes—
436C22#333712yes—
437C22#334012yes—
438C22Q8×SD1611yescharacteristic
439C22#338112yes—
440C22#338111yescharacteristic
441C4—24no—
442C4—42no—
443C4—44no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4—82no—
448C4—82no—
449C4—162no—
450C4—162no—
451C4C2×#38211yescharacteristic
452C4#232011yescharacteristic
453C4#234611yescharacteristic
454C4#235111yescharacteristic
455C4#248912yes—
456C4#250812yes—
457C2#1391811yescharacteristic
458C2#2127411yescharacteristic
459C2#2128011yescharacteristic
460C2#2260911yescharacteristic
461C2#2261011yescharacteristic
462C2C4⋊C4×SD1611yescharacteristic
463C2#2268311yescharacteristic
464C2#2778712yes—
465C2#2778812yes—
466C2#2799012yes—
467C2#2799112yes—
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
417. C8#382semidirect

Maximal subgroups

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

subgroupindex
C2×#19412
C2×C4×#1062
#139332
#139182
#112932
#112922
#108612
#108072
#108062
C8×#952