Finite Group Databasebeta

← #361376

Subgroups — #361376

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20881C212yesmaximal
2#20763C211yesmaximal, characteristic, complemented
3C2×#1764C211yesmaximal, characteristic
4#20348C211yesmaximal, characteristic
5#11276C212yesmaximal, complemented
6#11270C212yesmaximal, complemented
7#10862C211yesmaximal, characteristic, complemented
8#10827C211yesmaximal, characteristic, complemented
9#10818C211yesmaximal, characteristic, complemented
10#10781C212yesmaximal
11#10774C211yesmaximal, characteristic
12C22×#131C2211yescharacteristic
13C2×#520C2212yes—
14C22×#107C2212yes—
15C22×#104C2211yescharacteristic
16C2×#417C2212yes—
17C2×#416C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2317C2212yes—
21#1863—22no—
22#1863—22no—
23#1862—22no—
24#1862—22no—
25#1836—22no—
26#1835—21no—
27#1834—21no—
28#1764C418yescomplemented
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#341C2211yescharacteristic
32C2×#341C2211yescharacteristic
33#1736—22no—
34#1736C2212yes—
35#1736C2212yes—
36#1736C2212yes—
37#1735—21no—
38#1735C2211yescharacteristic
39C2×#340—21no—
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2211yescharacteristic
44#1729C2212yes—
45C4⋊#95C2212yes—
46C22×#41C2311yescharacteristic
47C2×#131—22no—
48C2×#131—24no—
49C2×#131—42no—
50C2×#131C2×C414yes—
51C22×M16—21no—
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8—21no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58#520—24no—
59#520—24no—
60#520—24no—
61#520—42no—
62C2×C4×Q8—22no—
63C2×C4×D4—22no—
64C2×C4×D4—42no—
65C2×#117—22no—
66C2×#117C2312yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—24no—
75C2×#107—42no—
76C2×#104—24no—
77C2×#104C2×C414yes—
78#417—24no—
79#417—24no—
80#416C2×C414yes—
81#416Q814yes—
82C2×#103—22no—
83C2×#103D411yescharacteristic
84C2×#103D411yescharacteristic
85C2×#102—22no—
86C2×#102D411yescharacteristic
87C2×#102D411yescharacteristic
88#405—22no—
89#405C2×C414yescomplemented
90C2×#97—21no—
91C2×C4×C8—21no—
92C2×C4×C8—42no—
93C2×C4×C8C2×C414yescomplemented
94#389—22no—
95#389—22no—
96#385—42no—
97#384—22no—
98#382—22no—
99C2×#95—42no—
100#343—24no—
101#343—42no—
102#341—22no—
103#341C2×C414yescomplemented
104#340—21no—
105#340—21no—
106#340—21no—
107#340—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340D411yescharacteristic
115#340D411yescharacteristic
116#340D412yes—
117#339—42no—
118#339C2×C414yescomplemented
119#336—22no—
120#336—22no—
121C2×#41—22no—
122C2×#41—21no—
123C2×#41—22no—
124C2×#41—21no—
125C2×#41C22×C412yes—
126C2×#41C2×Q812yes—
127C2×#41#4114yes—
128C2×#41#4114yes—
129C22×Q8C2×D411yescharacteristic
130C22×D4—21no—
131C22×D4C2×D411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—42no—
137C23×C4C2411yescharacteristic
138#131—28no—
139#131—44no—
140#131—48no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—24no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—44no—
155C2×M16C22×C412yes—
156C2×M16C22×C412yes—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—24no—
164C22×C8—22no—
165C22×C8—44no—
166C22×C8C22×C412yes—
167C22×C8C22×C412yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×Q812yes—
173C22×C8#4112yes—
174C4×Q8—44no—
175C4×Q8—44no—
176C4×D4—44no—
177C4×D4—48no—
178C4×D4—48no—
179C4×D4—44no—
180#117—21no—
181#117—21no—
182#117—28no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—44no—
188#117C22×C412yes—
189#117C2×Q812yes—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—42no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—42no—
220C2×C42C22×C412yes—
221#107—44no—
222#107—44no—
223#104—28no—
224#103—44no—
225#103C4⋊C414yes—
226#103C4⋊C414yes—
227#102—44no—
228#102C4⋊C414yes—
229#102C4⋊C414yes—
230#97—24no—
231C4×C8—24no—
232C4×C8—44no—
233#95—42no—
234#95—42no—
235#95—84no—
236C24—21no—
237C24H(5,𝔽2)11yescharacteristic
238#41—22no—
239#41—22no—
240#41—24no—
241#41—24no—
242#41—24no—
243#41—24no—
244#41—24no—
245#41#11714yes—
246#41#11714yes—
247#41#12518yes—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8C2×C4⋊C412yes—
251C2×Q8#12414yes—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4C2×C4⋊C412yes—
259C2×D4#12414yes—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—162no—
286C22×C4—162no—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+411yescharacteristic
297M16—22no—
298M16—22no—
299M16—22no—
300M16—22no—
301M16—48no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—48no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8C2×C4⋊C412yes—
327C2×C8C2×C4⋊C412yes—
328C2×C8C4×D412yes—
329C2×C8C4×D412yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—48no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—162no—
346C4⋊C4—162no—
347C4⋊C4C2×C4⋊C412yes—
348C4⋊C4C2×C4⋊C412yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—48no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—48no—
359C42—42no—
360C42—42no—
361C42—42no—
362C23—22no—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—24no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23#52011yescharacteristic
371C23#53812yes—
372C23D4211yescharacteristic
373C23#55112yes—
374C23#55411yescharacteristic
375Q8—22no—
376Q8—24no—
377Q8#38214yes—
378D4—24no—
379D4—24no—
380D4—28no—
381D4—22no—
382D4—24no—
383D4#38214yes—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—162no—
414C2×C4—168no—
415C2×C4—162no—
416C2×C4C22×C4⋊C411yescharacteristic
417C2×C4C2×#11711yescharacteristic
418C2×C4C2×#11711yescharacteristic
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C4×#4111yescharacteristic
421C2×C4#51811yescharacteristic
422C2×C4#52512yes—
423C2×C4C2×#12412yes—
424C2×C4C2×#12512yes—
425C2×C4#52912yes—
426C2×C4#53912yes—
427C2×C4#54612yes—
428C2×C4#54612yes—
429C2×C4#54711yescharacteristic
430C2×C4D4×Q811yescharacteristic
431C2×C4#55112yes—
432C2×C4#55212yes—
433C2×C4#55412yes—
434C8—22no—
435C8—22no—
436C8—22no—
437C8—22no—
438C8—42no—
439C8—42no—
440C8—82no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—24no—
450C22—44no—
451C22—44no—
452C22—42no—
453C22—42no—
454C22#231811yescharacteristic
455C22#234311yescharacteristic
456C22D4×C4⋊C411yescharacteristic
457C22#247212yes—
458C22#250512yes—
459C22#329012yes—
460C22#329211yescharacteristic
461C22#329311yescharacteristic
462C4—24no—
463C4—44no—
464C4—42no—
465C4—42no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4—162no—
470C4C2×#38211yescharacteristic
471C4#229011yescharacteristic
472C4#234211yescharacteristic
473C4#234911yescharacteristic
474C4#247312yes—
475C4#248812yes—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—44no—
480C2#1389611yescharacteristic
481C2#2269011yescharacteristic
482C2#2269211yescharacteristic
483C2#2728612yes—
484C2#2728712yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20763C2semidirect
5. #11276C2semidirect
6. #11270C2semidirect
7. #10862C2semidirect
8. #10827C2semidirect
9. #10818C2semidirect
28. #1764C4semidirect
89. #405C2×C4semidirect
93. C2×C4×C8C2×C4semidirect
103. #341C2×C4semidirect
118. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
#208812
#207632
C2×#17642
#203482
#112762
#112702
#108622
#108272
#108182
#107812
#107742