Finite Group Databasebeta

← #242535

Subgroups — #242535

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21985C211yesmaximal, characteristic, complemented
2#21975C211yesmaximal, characteristic, complemented
3#21810C211yesmaximal, characteristic
4#21803C211yesmaximal, characteristic
5C2×#1572C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic
7#10599C212yesmaximal, complemented
8#10595C212yesmaximal, complemented
9#10591C212yesmaximal, complemented
10#10587C212yesmaximal, complemented
11#8042C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#2285C2211yescharacteristic
18#2283C2211yescharacteristic
19#1982—21no—
20#1982—21no—
21#1982C2212yes—
22#1982C2212yes—
23#1961—21no—
24#1961—21no—
25#1957C2212yes—
26#1957C2212yes—
27#1948C2212yescomplemented
28#1947C2212yescomplemented
29#1939C2212yescomplemented
30#1939C2212yescomplemented
31#1854—22no—
32#1851—21no—
33#1851—21no—
34#1846—21no—
35#1843—21no—
36#1835—21no—
37#1832—21no—
38#1786—21no—
39#1786—21no—
40#1786—22no—
41#1572—24no—
42C2×#335C2211yescharacteristic, complemented
43C2×#335C2211yescharacteristic, complemented
44C2×#334C2211yescharacteristic, complemented
45C2×#334C2211yescharacteristic, complemented
46#1545—24no—
47C2×#331C2212yes—
48C2×#331C2212yes—
49#1296C2212yescomplemented
50#1296C2212yescomplemented
51C23×D4—21no—
52C23×D4—21no—
53C23×D4—21no—
54C22×M16—21no—
55C2×#127—22no—
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127—21no—
61C2×#127Q814yes—
62C2×#127C2312yes—
63C2×#117—21no—
64C22×C42C2311yescharacteristic
65#432—22no—
66#432—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431D414yes—
71C2×#107—22no—
72C2×#106—22no—
73#423—22no—
74#423—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8D411yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8D411yescharacteristic
96C2×C22⋊C8D411yescharacteristic
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#384C2312yes—
102#383—21no—
103#382—21no—
104#382D411yescharacteristic
105#382D411yescharacteristic
106#381—21no—
107#380—21no—
108#380—21no—
109#380D412yes—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4—21no—
114#376—21no—
115#340—22no—
116#340—22no—
117#336—22no—
118#336—22no—
119#335—22no—
120#335—22no—
121#334—22no—
122#334—22no—
123#331—24no—
124#331—24no—
125C25—41no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—41no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4—42no—
138C22×D4—81no—
139C22×D4—81no—
140C22×D4—82no—
141C22×D4—82no—
142C22×D4—81no—
143C22×D4—81no—
144C22×D4—84no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4#4111yescharacteristic
148C2×M16—42no—
149C2×M16—82no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159#127—24no—
160#127—24no—
161#127—24no—
162#127—22no—
163#127—22no—
164#127—22no—
165#127—42no—
166#127—42no—
167#127—42no—
168#127C2×Q814yes—
169#117—81no—
170#117—81no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D412yes—
183C2×C42C2×Q812yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#107—82no—
189#106—82no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—42no—
201C4⋊C8—82no—
202C4⋊C8—82no—
203C4⋊C8C2×D412yes—
204C4⋊C8C2×D412yes—
205#102—82no—
206#102—82no—
207#102—82no—
208#102—82no—
209#102—82no—
210#102—82no—
211#102—82no—
212#102—82no—
213C22⋊C8—24no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—42no—
217#97—42no—
218C4×C8—24no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—41no—
230#95—41no—
231C24—42no—
232C24—44no—
233C24—84no—
234C24—82no—
235C24—82no—
236C24—84no—
237C24#12111yescharacteristic
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—84no—
256C2×D4—84no—
257C2×D4—88no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4—84no—
261C2×D4—84no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—84no—
265C2×D4—84no—
266C2×D4—84no—
267C2×D4—84no—
268C2×D4—88no—
269C2×D4—88no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12612yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4111yescharacteristic
292M16—84no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—42no—
328C42—42no—
329C42—82no—
330C42#12112yes—
331C42#12112yes—
332C42C22⋊Q812yes—
333C42C22⋊Q812yes—
334C42C22⋊Q812yes—
335C42#12612yes—
336C42C22×D411yescharacteristic
337C42C22×Q811yescharacteristic
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C23—22no—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—44no—
345C23—88no—
346C23—88no—
347C23—88no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23—168no—
355C23—168no—
356C23#39612yes—
357C23C2×#12111yescharacteristic
358D4—44no—
359D4—44no—
360D4—44no—
361D4—44no—
362D4—84no—
363D4—88no—
364D4—84no—
365D4—88no—
366D4—88no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—82no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#39612yes—
395C2×C4#39612yes—
396C2×C4#39612yes—
397C2×C4C2×#12111yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×C22⋊Q811yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4C2×#12611yescharacteristic
403C2×C4#57611yescharacteristic
404C2×C4#57611yescharacteristic
405C2×C4#57612yes—
406C2×C4#57711yescharacteristic
407C2×C4#57711yescharacteristic
408C2×C4#57712yes—
409C8—42no—
410C8—42no—
411C8—41no—
412C8—41no—
413C8—41no—
414C8—41no—
415C8—81no—
416C8—81no—
417C8—82no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—84no—
424C22—84no—
425C22—84no—
426C22—1616no—
427C22—162no—
428C22—162no—
429C22—162no—
430C22—162no—
431C22C2×#39611yescharacteristic
432C22#304512yes—
433C22#305711yescharacteristic
434C22#305811yescharacteristic
435C22#306812yes—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—82no—
442C4—82no—
443C4—162no—
444C4#304511yescharacteristic
445C4#304511yescharacteristic
446C4#305711yescharacteristic
447C4#305811yescharacteristic
448C4#308111yescharacteristic
449C4#308111yescharacteristic
450C4#308112yes—
451C2—42no—
452C2—164no—
453C2#1953712yes—
454C2#1953812yes—
455C2#2360211yescharacteristic
456C2#2361211yescharacteristic
457C2#2361211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
1. #21985C2semidirect
2. #21975C2semidirect
5. C2×#1572C2semidirect
7. #10599C2semidirect
8. #10595C2semidirect
9. #10591C2semidirect
10. #10587C2semidirect
11. #8042C2semidirect
27. #1948C22semidirect
28. #1947C22semidirect
29. #1939C22semidirect
30. #1939C22semidirect
42. C2×#335C22semidirect
43. C2×#335C22semidirect
44. C2×#334C22semidirect
45. C2×#334C22semidirect
49. #1296C22semidirect
50. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#219852
#219752
#218102
#218032
C2×#15722
C2×#15452
#105992
#105952
#105912
#105872
#80422