Finite Group Databasebeta

← #192774

Subgroups — #192774

Order 512.

1108 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#34182C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21964C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8892C212yesmaximal, complemented
8#8888C212yesmaximal, complemented
9#8884C212yesmaximal, complemented
10#8042C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C2×#519—21no—
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#2351—22no—
18#2351—22no—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4—21no—
21Q8×C4⋊C4C2212yes—
22Q8×C4⋊C4C2212yes—
23#2316—22no—
24#2316—21no—
25#2285—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1949—22no—
30#1948—21no—
31#1948C2212yes—
32#1947—21no—
33#1947C2212yes—
34#1854—22no—
35#1845C2211yescharacteristic
36#1844C2211yescharacteristic
37#1834C2211yescharacteristic
38#1834C2211yescharacteristic
39#1823—21no—
40#1809—22no—
41#1803C2212yes—
42#1803C2212yes—
43#1787—21no—
44#1763—22no—
45#1487C418yescomplemented
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C23×Q8D411yescharacteristic
53C22×M16—21no—
54#519—48no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C414yes—
64C2×C4×Q8D412yes—
65C2×C4×Q8Q814yes—
66C2×C4×Q8C2312yes—
67C2×#117—24no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#431—22no—
78#431C2×C414yes—
79C2×#107—22no—
80C2×#107—22no—
81C2×#106—22no—
82C2×#106—22no—
83#423—22no—
84#423C2×C414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×#97—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#389—22no—
97#389—22no—
98#389—24no—
99#389—24no—
100#389—22no—
101#389—22no—
102#389—24no—
103#389—24no—
104#384—22no—
105#384—22no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—24no—
113#382—21no—
114#382—21no—
115#382—21no—
116#380—24no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380D412yes—
122#380D412yes—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2312yes—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138Q8⋊C8C2×C418yes—
139Q8⋊C8C2×C418yes—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8C4⋊C414yes—
145C22×Q8C2×D412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—24no—
152C2×M16—81no—
153C2×M16—81no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—48no—
166C4×Q8—48no—
167C4×Q8—44no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×Q8C4⋊C418yes—
172C4×Q8C22×C414yes—
173C4×Q8C2×Q814yes—
174#117—416no—
175#117—44no—
176#117—44no—
177#117—82no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C4⋊C414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—41no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—41no—
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×Q812yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228#107—82no—
229#107—82no—
230#106—82no—
231#106—82no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8C22×C414yes—
238#103—44no—
239#103—44no—
240C22⋊C8—24no—
241C22⋊C8—42no—
242#97—42no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8C22×C412yes—
246C4×C8C22×C412yes—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—41no—
254#95—41no—
255C24#12011yescharacteristic
256C2×Q8—24no—
257C2×Q8—216no—
258C2×Q8—24no—
259C2×Q8—216no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8C2×C4⋊C414yes—
265C2×Q8AGL(1,ℤ/8)14yes—
266C2×Q8#13714yes—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C2×C4⋊C412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4#12312yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—84no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—28no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4C2×C4⋊C414yes—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—82no—
351C42—24no—
352C42—24no—
353C42—28no—
354C42—48no—
355C42—48no—
356C42—44no—
357C42—42no—
358C42—42no—
359C42—82no—
360C42C2×C4⋊C412yes—
361C42C4×D412yes—
362C42C4×D412yes—
363C42C22⋊Q812yes—
364C42C22⋊Q812yes—
365C42#12312yes—
366C42C23×C411yescharacteristic
367C42C22×Q811yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23#38512yes—
377C23C2×#12011yescharacteristic
378Q8—216no—
379Q8—24no—
380Q8—24no—
381Q8—24no—
382Q8#42818yes—
383C2×C4—216no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#38512yes—
426C2×C4#38512yes—
427C2×C4#38512yes—
428C2×C4C22×C4⋊C411yescharacteristic
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)12yes—
435C2×C4C2×#13712yes—
436C2×C4#57711yescharacteristic
437C2×C4#57711yescharacteristic
438C2×C4#57811yescharacteristic
439C2×C4#57811yescharacteristic
440C8—42no—
441C8—42no—
442C8—44no—
443C8—82no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22C2×#38511yescharacteristic
452C22#300311yescharacteristic
453C22#300411yescharacteristic
454C22#301311yescharacteristic
455C22#301311yescharacteristic
456C22#302111yescharacteristic
457C22#302211yescharacteristic
458C4—28no—
459C4—22no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—22no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—161no—
470C4—161no—
471C4C2×#42812yes—
472C4#294911yescharacteristic
473C4#295011yescharacteristic
474C4#308211yescharacteristic
475C4#308211yescharacteristic
476C4#309212yes—
477C2—42no—
478C2#1885411yescharacteristic
479C2#1885511yescharacteristic
480C2#1886211yescharacteristic
481C2#1886311yescharacteristic
482C2#2084411yescharacteristic
483C2#2086511yescharacteristic
484C2#2088011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ 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
6. #8896C2semidirect
7. #8892C2semidirect
8. #8888C2semidirect
9. #8884C2semidirect
45. #1487C4semidirect

Maximal subgroups

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

subgroupindex
#341822
#219842
#219642
#207402
C2×#14872
#88962
#88922
#88882
#88842
#80422
#79002