Finite Group Databasebeta

← #278071

Subgroups — #278071

Order 512.

788 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic
2#24653C212yesmaximal, complemented
3#21597C211yesmaximal, characteristic, complemented
4C2×#1688C211yesmaximal, characteristic, complemented
5C2×#1564C211yesmaximal, characteristic, complemented
6#10599C212yesmaximal, complemented
7#10474C212yesmaximal, complemented
8#9743C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9134C212yesmaximal, complemented
11#8653C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466—21no—
15C2×#460C2212yes—
16C2×#446C2211yescharacteristic
17C2×#431C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2856C2212yes—
20C4×C22⋊Q8C2211yescharacteristic
21#2034—22no—
22#2000—22no—
23#1984C2211yescharacteristic
24#1982C2212yes—
25#1970—21no—
26#1970—21no—
27#1968—22no—
28#1957C2212yes—
29#1948C2212yes—
30#1939C2211yescharacteristic
31#1934—22no—
32#1916—21no—
33#1850—21no—
34#1845C2211yescharacteristic
35#1803C2212yes—
36#1797C2211yescharacteristic
37#1771—21no—
38#1770C2211yescharacteristic
39#1752—21no—
40#1688—24no—
41#1564—24no—
42C2×#335C2212yes—
43C2×#334C2211yescharacteristic
44C2×#334C2211yescharacteristic
45C2×#331C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×SD16—22no—
50C22×D8—21no—
51C22×D8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×#127C2312yes—
55C2×C22⋊Q8D411yescharacteristic
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×C4×Q8C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#493—22no—
62#493—22no—
63#466—44no—
64#466—44no—
65#460—24no—
66#460—42no—
67#446—22no—
68#446—22no—
69#431—24no—
70#431—42no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#106—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#103—22no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×#97—21no—
89C2×#97—21no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391D411yescharacteristic
98#390—22no—
99#390—22no—
100#390—22no—
101#389D411yescharacteristic
102#387—21no—
103#385—21no—
104#384C2312yes—
105#380D411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2311yescharacteristic
108C4×C22⋊C4—21no—
109#340—22no—
110#340—22no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#335—24no—
117#334—22no—
118#334D414yescomplemented
119#331—24no—
120#331—42no—
121C22×Q8C2×D411yescharacteristic
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—42no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×SD16—44no—
130C2×SD16—44no—
131C2×SD16—82no—
132C2×D8—42no—
133C2×D8—42no—
134C2×D8—42no—
135C2×D8—42no—
136C2×D8—42no—
137C2×D8—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—41no—
147C22×C8SD1614yescomplemented
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152#127—22no—
153#127—22no—
154#127—22no—
155#127—22no—
156C22⋊Q8—42no—
157#121—44no—
158#121—44no—
159#121—82no—
160#121—82no—
161#121—84no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×Q8—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42C2411yescharacteristic
190#107—42no—
191#107—42no—
192#106—82no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8C2×D414yes—
203#103—82no—
204#102—44no—
205#102—44no—
206#102—44no—
207#102—44no—
208#102—82no—
209C22⋊C8—44no—
210C22⋊C8—42no—
211#97—42no—
212#97—42no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—44no—
218C4×C8C2×D412yes—
219C4×C8#4112yes—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225C24—42no—
226C24#12111yescharacteristic
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—84no—
248C2×D4—84no—
249C2×D4—84no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4C2×SD1612yes—
271C22×C4AGL(1,ℤ/8)12yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275SD16—88no—
276D8—84no—
277D8—84no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—28no—
283C2×C8—21no—
284C2×C8—24no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8#12112yes—
300C2×C8#12112yes—
301C2×C8C2×SD1612yes—
302C2×C8C2×SD1612yes—
303C2×C8#13512yes—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C22⋊C4—84no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42C22×D411yescharacteristic
333C42C2×#4111yescharacteristic
334C42H(5,𝔽2)11yescharacteristic
335C422−1+411yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23#46012yes—
350C23C2×#12111yescharacteristic
351Q8—42no—
352Q8—42no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—162no—
362D4—162no—
363D4—168no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#46012yes—
397C2×C4#47212yes—
398C2×C4C2×#12111yescharacteristic
399C2×C4#53711yescharacteristic
400C2×C4D4211yescharacteristic
401C2×C4#54711yescharacteristic
402C2×C4#54811yescharacteristic
403C2×C4#55311yescharacteristic
404C2×C4C22×SD1611yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4#57512yes—
408C2×C4#57511yescharacteristic
409C2×C4#57612yes—
410C8—22no—
411C8—22no—
412C8—44no—
413C8—41no—
414C8—81no—
415C8—82no—
416C8—81no—
417C8#46514yes—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—162no—
426C22—162no—
427C22—162no—
428C22—162no—
429C22—162no—
430C22—162no—
431C22—162no—
432C22#241611yescharacteristic
433C22C2×#46011yescharacteristic
434C22#303511yescharacteristic
435C22#305411yescharacteristic
436C22#305811yescharacteristic
437C22C2×#47211yescharacteristic
438C22#306711yescharacteristic
439C4—22no—
440C4—21no—
441C4—21no—
442C4—22no—
443C4—41no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—162no—
449C4C2×#46511yescharacteristic
450C4#305511yescharacteristic
451C4#324611yescharacteristic
452C4#324711yescharacteristic
453C4#326812yes—
454C4#334712yes—
455C2—42no—
456C2—322no—
457C2#1914611yescharacteristic
458C2#1916011yescharacteristic
459C2#1920311yescharacteristic
460C2#1922511yescharacteristic
461C2#2475311yescharacteristic
462C2#2478411yescharacteristic
463C2#2480311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#33171C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #24653C2semidirect
3. #21597C2semidirect
4. C2×#1688C2semidirect
5. C2×#1564C2semidirect
6. #10599C2semidirect
7. #10474C2semidirect
8. #9743C2semidirect
9. #9621C2semidirect
10. #9134C2semidirect
11. #8653C2semidirect
118. #334D4semidirect
147. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#331712
#246532
#215972
C2×#16882
C2×#15642
#105992
#104742
#97432
#96212
#91342
#86532