Finite Group Databasebeta

← #374577

Subgroups — #374577

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3080C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#21325C211yesmaximal, characteristic, complemented
4C2×#1759C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic, complemented
6#11217C214yesmaximal, complemented
7#10857C211yesmaximal, characteristic, complemented
8#10840C211yesmaximal, characteristic, complemented
9#10781C214yesmaximal
10C22×#125C2211yescharacteristic
11C2×#520C2212yes—
12C2×C4×#41C2211yescharacteristic
13#3080—24no—
14#3080C418yescomplemented
15C2×#479—22no—
16C2×#478—22no—
17C2×#428C2212yes—
18C2×#423C2211yescharacteristic
19C22×#104C2212yes—
20C2×#417C2212yes—
21C2×C4×M16C2211yescharacteristic
22Q8×C4⋊C4—22no—
23D4×C4⋊C4—22no—
24D4×C4⋊C4—24no—
25#2317—22no—
26#2317C2212yes—
27#2285—21no—
28#2285C2211yescharacteristic
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1992—21no—
33#1992—21no—
34#1991—22no—
35#1759—24no—
36C2×#343C2214yes—
37#1736C2214yes—
38C2×#339C2212yes—
39C2×#339C2214yes—
40#1441—24no—
41C4⋊#95C2214yes—
42C22×#41C2311yescharacteristic
43C22×M16C2312yes—
44C22×M16C2312yes—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125C2×C414yescomplemented
52#520—28no—
53#520—44no—
54C4×#41—22no—
55C4×#41—22no—
56C4×#41C2×C414yes—
57C4×#41D414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8D412yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4D412yes—
64C2×#117—22no—
65C2×#117—21no—
66C2×#117C2312yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#479—216no—
78#478—216no—
79#428—24no—
80#428C2×C418yescomplemented
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423C2×C414yescomplemented
85C2×C4⋊C8—22no—
86C2×#104—24no—
87C2×#104—24no—
88C2×#104Q818yes—
89#417—24no—
90#417C2×C418yes—
91C2×#103—22no—
92C2×#102—22no—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97—21no—
96C2×C4×C8—21no—
97#389—28no—
98#389—24no—
99#385—28no—
100#385—24no—
101#384—24no—
102#384—22no—
103#384—21no—
104#384—22no—
105#384—21no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#382—22no—
109#382D412yes—
110#380—24no—
111#380—22no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—22no—
114#376—22no—
115#376—22no—
116C2×#95—22no—
117C2×#95—28no—
118#343—44no—
119#340—24no—
120#340—24no—
121#339—24no—
122#339—44no—
123C2×#41—21no—
124C2×#41—21no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41C22×C412yes—
128C2×#41C2×D412yes—
129C2×#41C2×Q814yes—
130C2×#41#4114yes—
131C22×Q8C2×D411yescharacteristic
132C22×D4—21no—
133C22×D4C2×D411yescharacteristic
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4C2×D414yes—
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—24no—
150C2×M16C22×C414yes—
151C2×M16C2×Q814yes—
152C22×C8—22no—
153C22×C8—22no—
154#125—28no—
155#125—24no—
156#125—24no—
157#125—42no—
158#125—42no—
159#125—44no—
160C4×Q8—24no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×Q8C22⋊C418yes—
164C4×D4—28no—
165C4×D4—28no—
166C4×D4—24no—
167C4×D4—416no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4C22⋊C418yes—
171#117—21no—
172#117—21no—
173#117—24no—
174#117—24no—
175#117—24no—
176#117—24no—
177#117—48no—
178#117C22×C412yes—
179#117C2×D412yes—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—88no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C22×C412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q814yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#107—28no—
231#106—28no—
232C4⋊C8—28no—
233#104—216no—
234#103—28no—
235#102—28no—
236#97—42no—
237C4×C8—42no—
238#95—28no—
239#95—24no—
240#95—44no—
241#95—88no—
242C24—21no—
243C24C22×D411yescharacteristic
244#41—24no—
245#41—24no—
246#41—24no—
247#41—24no—
248#41—24no—
249#41—24no—
250#41C4×Q818yes—
251#41C22⋊Q818yes—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8C2×C22⋊C412yes—
256C2×Q8#12412yes—
257C2×Q8C4⋊Q812yes—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C2×C22⋊C412yes—
266C2×D4#12412yes—
267C2×D4C4⋊Q812yes—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—88no—
296C22×C4C4×D414yes—
297C22×C4#12012yes—
298C22×C4#12012yes—
299C22×C4#12114yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×Q812yes—
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+412yes—
310M16—28no—
311M16—24no—
312M16—24no—
313M16—44no—
314M16—42no—
315M16—42no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—22no—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—48no—
339C4⋊C4—84no—
340C4⋊C4—88no—
341C4⋊C4C2×C22⋊C414yes—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—28no—
345C22⋊C4—48no—
346C22⋊C4—44no—
347C42—22no—
348C42—22no—
349C42—28no—
350C42—24no—
351C42—24no—
352C42—48no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42C2×C22⋊C414yes—
358C42C4×D412yes—
359C42C4×D412yes—
360C42#12112yes—
361C42#12112yes—
362C42#12314yes—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—42no—
369C23—42no—
370C23C2×C4×D411yescharacteristic
371C23C2×#12011yescharacteristic
372C23C2×#12111yescharacteristic
373C23#54612yes—
374C23D4×Q812yes—
375Q8—22no—
376Q8—22no—
377Q8—22no—
378Q8#39114yes—
379D4—28no—
380D4—28no—
381D4—22no—
382D4—22no—
383D4—22no—
384D4#39114yes—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—416no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—48no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—88no—
417C2×C4—168no—
418C2×C4#38614yes—
419C2×C4#38614yes—
420C2×C4C22×C22⋊C411yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C4×Q812yes—
423C2×C4#52012yes—
424C2×C4C2×#12111yescharacteristic
425C2×C4C2×C22⋊Q812yes—
426C2×C4C2×#12311yescharacteristic
427C2×C4C2×#12411yescharacteristic
428C2×C4C2×C4⋊Q811yescharacteristic
429C2×C4#53511yescharacteristic
430C2×C4#53611yescharacteristic
431C2×C4#53712yes—
432C2×C4#54612yes—
433C2×C4D4×Q812yes—
434C2×C4#55212yes—
435C2×C4#55212yes—
436C2×C4#55412yes—
437C2×C4#55412yes—
438C8—44no—
439C8—44no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—24no—
445C22—24no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—44no—
450C22C2×#38611yescharacteristic
451C22#236012yes—
452C22#241712yes—
453C22#243411yescharacteristic
454C22#243511yescharacteristic
455C22#301912yes—
456C22#302012yes—
457C4—24no—
458C4—24no—
459C4—24no—
460C4—44no—
461C4—84no—
462C4—84no—
463C4—82no—
464C4—82no—
465C4—84no—
466C4C2×#39111yescharacteristic
467C4#231111yescharacteristic
468C4#236012yes—
469C4#241712yes—
470C4#250211yescharacteristic
471C4#250311yescharacteristic
472C2—22no—
473C2—22no—
474C2—44no—
475C2#1413911yescharacteristic
476C2#2094312yes—
477C2#2583212yes—
478C2#2587811yescharacteristic
479C2#2587911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21325C2semidirect
5. #14283C2semidirect
6. #11217C2semidirect
7. #10857C2semidirect
8. #10840C2semidirect
14. #3080C4semidirect
51. C2×#125C2×C4semidirect
80. #428C2×C4semidirect
84. #423C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30802
#341832
#213252
C2×#17592
#142832
#112172
#108572
#108402
#107812