Finite Group Databasebeta

← #192360

Subgroups — #192360

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21395C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic
5#13854C211yesmaximal, characteristic, complemented
6#8870C214yesmaximal, complemented
7#8869C214yesmaximal, complemented
8#8022C211yesmaximal, characteristic, complemented
9#7912C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14#2367—22no—
15#2366—21no—
16#2365—21no—
17#2354—21no—
18#2352—22no—
19D4×C4⋊C4—21no—
20#2325C2214yes—
21#2311—21no—
22C4×#121—22no—
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1987—21no—
28#1982—21no—
29#1948—21no—
30#1947—21no—
31#1874C2212yes—
32#1874C2212yes—
33#1853C2212yes—
34#1852C2212yes—
35#1776—22no—
36#1762—22no—
37#1505C418yescomplemented
38C2×#327C2214yes—
39#1442—21no—
40#1442—21no—
41#1442C2212yes—
42#1442C2212yes—
43C4×#95—22no—
44#1294C2214yes—
45#1280C2214yes—
46C22×#41—21no—
47#532—22no—
48#532—22no—
49#532C2×C414yes—
50#532Q814yes—
51C2×C4⋊Q8D411yescharacteristic
52C2×#127D411yescharacteristic
53C2×#124—21no—
54C2×#121—21no—
55C2×#121C2312yes—
56#520—48no—
57C2×C4×Q8—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4D412yes—
63C2×#117—22no—
64C2×#117—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42D412yes—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#432—22no—
77#432C2×C414yescomplemented
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×#102—22no—
83C2×#102—22no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×#97—21no—
87C2×#97—21no—
88#391—22no—
89#390—22no—
90#390—22no—
91#389—22no—
92#388—24no—
93#388—24no—
94#387—24no—
95#386—24no—
96#386—24no—
97#385—24no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#382—21no—
101#380—22no—
102#380—22no—
103#380—21no—
104#380C2312yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4D412yes—
116C4×C22⋊C4—24no—
117#376—22no—
118#376C2×C414yescomplemented
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#327C2×C4116yescomplemented
132C2×#41—24no—
133C2×#41—24no—
134C2×#41—44no—
135C2×#41—44no—
136C22×Q8—21no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C2×D412yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8C4⋊C414yes—
155#127—22no—
156#127C4⋊C414yes—
157#124—24no—
158#124—24no—
159#121—24no—
160#121—24no—
161#121—22no—
162#121—22no—
163#121C22×C414yes—
164#121C2×Q814yes—
165C4×Q8—48no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—48no—
170C4×D4—48no—
171C4×D4—48no—
172C4×D4—84no—
173C4×D4C4⋊C418yes—
174#117—48no—
175#117—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q812yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223#107—44no—
224#106—44no—
225C4⋊C8—24no—
226C4⋊C8—42no—
227#102—44no—
228#102—44no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8C22×C414yes—
233#97—24no—
234#97—42no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—41no—
247#95—41no—
248#95—44no—
249#95C22×C414yes—
250C24—22no—
251C24C22×D411yescharacteristic
252#41—416no—
253#41—416no—
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4C2×C4⋊C414yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—28no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—48no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4C22⋊Q812yes—
319C22×C4C22⋊Q812yes—
320C22×C4AGL(1,ℤ/8)14yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C2×C8—24no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4C2×C4⋊C414yes—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—28no—
358C22⋊C4—48no—
359C22⋊C4—44no—
360C22⋊C4—84no—
361C22⋊C4—84no—
362C42—48no—
363C42—44no—
364C42—48no—
365C42—44no—
366C42—82no—
367C42—82no—
368C42—82no—
369C42C2×C4⋊C412yes—
370C42C2×C4⋊C412yes—
371C42C4×D412yes—
372C42C4×D412yes—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C42#12312yes—
376C42#12312yes—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—24no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23C2×C4×D411yescharacteristic
389C23C2×#12011yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23C2×AGL(1,ℤ/8)12yes—
392C23#57512yes—
393Q8—48no—
394D4—44no—
395D4—48no—
396D4—48no—
397D4—48no—
398D4—48no—
399D4—82no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—416no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—168no—
456C2×C4#38512yes—
457C2×C4#38512yes—
458C2×C4#38512yes—
459C2×C4#38512yes—
460C2×C4#45412yes—
461C2×C4#45512yes—
462C2×C4C22×C4⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C8—44no—
467C8—82no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22C2×#38511yescharacteristic
484C22C4×AGL(1,ℤ/8)12yes—
485C22#300311yescharacteristic
486C22#300511yescharacteristic
487C22C2×#45511yescharacteristic
488C22C2×#45411yescharacteristic
489C22#302511yescharacteristic
490C22#302511yescharacteristic
491C22#307712yes—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—44no—
500C4—44no—
501C4—84no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4#186014yes—
508C2—22no—
509C2—22no—
510C2—84no—
511C2#1817911yescharacteristic
512C2#1818211yescharacteristic
513C2#1818611yescharacteristic
514C2#1819111yescharacteristic
515C2C2×#186011yescharacteristic
516C2#2087311yescharacteristic
517C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21365C2semidirect
5. #13854C2semidirect
6. #8870C2semidirect
7. #8869C2semidirect
8. #8022C2semidirect
9. #7912C2semidirect
37. #1505C4semidirect
77. #432C2×C4semidirect
118. #376C2×C4semidirect
131. #327C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#213952
#213652
C2×#15052
#138542
#88702
#88692
#80222
#79122