Finite Group Databasebeta

← #360660

Subgroups — #360660

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2#20765C211yesmaximal, characteristic, complemented
3#20649C211yesmaximal, characteristic
4C2×#1759C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic, complemented
6#11079C212yesmaximal
7#11078C212yesmaximal
8#10892C212yesmaximal, complemented
9#10891C212yesmaximal, complemented
10#10857C211yesmaximal, characteristic, complemented
11#10840C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×#104C2212yes—
15C2×#417C2211yescharacteristic
16C2×#417C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2355—22no—
19#2354—22no—
20#2352—24no—
21#2314—22no—
22#2311C2212yes—
23#2285—22no—
24#2285C2211yescharacteristic
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#382C2212yes—
29#2274C2212yes—
30C2×#376C2211yescharacteristic
31#1836—22no—
32#1835—22no—
33#1805—22no—
34#1804—22no—
35#1759C418yescomplemented
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40#1736C2211yescharacteristic
41#1736C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45#1444—24no—
46C22×#41C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49C22×M16C2312yes—
50C4×#41—22no—
51C4×#41—22no—
52C4×#41C2×C414yes—
53C4×#41Q814yes—
54C2×C4×Q8D412yes—
55C2×C4×D4—22no—
56C2×C4×D4D412yes—
57C2×#117—24no—
58C2×#117—21no—
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—24no—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#423—22no—
70#423—22no—
71C2×C4⋊C8—22no—
72C2×#104—24no—
73C2×#104—24no—
74C2×#104C2×C418yes—
75#417—42no—
76#417—42no—
77C2×#103—21no—
78C2×#103—21no—
79C2×#102—21no—
80C2×#102—21no—
81C4×M16—22no—
82C4×M16C2×C414yes—
83C2×#97—21no—
84C2×C4×C8—21no—
85#391—22no—
86#390—22no—
87#388—28no—
88#387—28no—
89#386—24no—
90#384—21no—
91#384—22no—
92#384—21no—
93#384—24no—
94#384—42no—
95#384D412yes—
96#382—24no—
97#382—22no—
98#382—21no—
99#382—21no—
100#382—24no—
101#382—42no—
102#382D412yes—
103#380—24no—
104#380—22no—
105#380—24no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—22no—
110C4×C22⋊C4—24no—
111#376—22no—
112#376C2×C414yes—
113C2×#95—22no—
114C2×#95—24no—
115C2×#95—22no—
116#343—42no—
117#343—42no—
118#340—22no—
119#340—22no—
120#340—24no—
121#340—24no—
122#339—24no—
123#339—24no—
124#339C2×C418yes—
125C2×#41—21no—
126C2×#41—21no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41C22×C412yes—
132C2×#41C2×Q812yes—
133C22×Q8C2×D411yescharacteristic
134C22×D4—21no—
135C22×D4C2×D411yescharacteristic
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×D414yes—
142C23×C4C2×D412yes—
143C23×C4C2411yescharacteristic
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16C22×C414yes—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C4×Q8—24no—
162C4×Q8C4⋊C418yescomplemented
163C4×D4—28no—
164C4×D4—28no—
165C4×D4—24no—
166C4×D4C4⋊C418yescomplemented
167#117—21no—
168#117—21no—
169#117—24no—
170#117—24no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—88no—
176#117C22×C412yes—
177#117C2×Q812yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—84no—
199C2×C4⋊C4—88no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—28no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—28no—
215C2×C42C22×C414yes—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q812yes—
221C2×C42#4114yes—
222C2×C42#4114yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C4⋊C8—44no—
226#104—216no—
227#103—42no—
228#103—42no—
229#102—42no—
230#102—42no—
231#97—24no—
232C4×C8—24no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—48no—
238#95—42no—
239#95—42no—
240C24—21no—
241C24C22×D411yescharacteristic
242#41—24no—
243#41—24no—
244#41—24no—
245#41—24no—
246#41—42no—
247#41—42no—
248#41—42no—
249#41—42no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8C2×C4⋊C412yes—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4C2×C4⋊C412yes—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C4×D414yes—
294C22×C4#12012yes—
295C22×C4#12012yes—
296C22×C4C22⋊Q814yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C42−1+412yes—
307M16—24no—
308M16—24no—
309M16—42no—
310M16—42no—
311M16—42no—
312M16—42no—
313M16—42no—
314M16—42no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—22no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—84no—
342C4⋊C4—816no—
343C4⋊C4C2×C4⋊C414yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—28no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—88no—
350C42—22no—
351C42—24no—
352C42—22no—
353C42—24no—
354C42—24no—
355C42—28no—
356C42—42no—
357C42—42no—
358C42—88no—
359C42C2×C4⋊C414yes—
360C42#11718yescomplemented
361C42C4×D412yes—
362C42C4×D412yes—
363C42C22⋊Q812yes—
364C42C22⋊Q812yes—
365C42#12314yes—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23C2×C4×D411yescharacteristic
374C23C2×#12011yescharacteristic
375C23C2×C22⋊Q811yescharacteristic
376C23#54612yes—
377C23#54812yes—
378Q8—22no—
379Q8—22no—
380Q8—41no—
381Q8—41no—
382D4—22no—
383D4—22no—
384D4—44no—
385D4—44no—
386D4—41no—
387D4—41no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—416no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—44no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4#38514yes—
422C2×C4#38514yes—
423C2×C4C22×C4⋊C411yescharacteristic
424C2×C4C2×#11712yes—
425C2×C4C2×C4×D411yescharacteristic
426C2×C4#52012yes—
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×#12311yescharacteristic
429C2×C4#53211yescharacteristic
430C2×C4#53211yescharacteristic
431C2×C4#53312yes—
432C2×C4#53511yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4#54412yes—
435C2×C4#54712yes—
436C2×C4#55012yes—
437C2×C4#55212yes—
438C2×C4#55312yes—
439C2×C4#55512yes—
440C2×C4#55612yes—
441C8—44no—
442C8—81no—
443C8—81no—
444C22—24no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—44no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22C2×#38511yescharacteristic
455C22#235212yes—
456C22#243111yescharacteristic
457C22#243211yescharacteristic
458C22#243912yes—
459C22#301912yes—
460C22#302012yes—
461C4—24no—
462C4—24no—
463C4—48no—
464C4—42no—
465C4—44no—
466C4—84no—
467C4—84no—
468C4—161no—
469C4—161no—
470C4#228511yescharacteristic
471C4#231711yescharacteristic
472C4#235412yes—
473C4#244112yes—
474C4#249911yescharacteristic
475C4#250011yescharacteristic
476C2—22no—
477C2—22no—
478C2—82no—
479C2#1387911yescharacteristic
480C2#2086412yes—
481C2#2559311yescharacteristic
482C2#2559411yescharacteristic
483C2#2566912yes—
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
2. #20765C2semidirect
5. #14354C2semidirect
8. #10892C2semidirect
9. #10891C2semidirect
10. #10857C2semidirect
11. #10840C2semidirect
35. #1759C4semidirect
162. C4×Q8C4⋊C4semidirect
166. C4×D4C4⋊C4semidirect
360. C42#117semidirect

Maximal subgroups

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

subgroupindex
#341882
#207652
#206492
C2×#17592
#143542
#110792
#110782
#108922
#108912
#108572
#108402