Finite Group Databasebeta

← #200416

Subgroups — #200416

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic, complemented
2#35328C211yesmaximal, characteristic
3#20632C211yesmaximal, characteristic, complemented
4#17574C212yesmaximal, complemented
5#9384C218yesmaximal, complemented
6#8026C212yesmaximal, complemented
7C2×#537—21no—
8C2×#532C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2988—24no—
11#2982—21no—
12#2972—21no—
13#2972C2214yes—
14C8×#41—24no—
15C2×#431—22no—
16C2×#431C2212yes—
17C2×#410—21no—
18#2850—22no—
19#2621C412yescomplemented
20#2619C412yescomplemented
21#2543—22no—
22#2543C2214yes—
23#2536—21no—
24#2536C412yescomplemented
25#2535—21no—
26#2535C412yescomplemented
27#2311—21no—
28#2285C2212yes—
29C2×#382C2211yescharacteristic
30#1799—22no—
31#1799C2214yes—
32#1740—21no—
33#1519C2218yes—
34#1517—24no—
35#1502—24no—
36#1294C2218yes—
37C22×#41—21no—
38C22×#41D411yescharacteristic
39#537—44no—
40#537—44no—
41#532—22no—
42#532—22no—
43#532—22no—
44#532—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#124—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×#121—22no—
51C2×#121—24no—
52C2×#121—22no—
53C2×#121—21no—
54C2×#121C2312yes—
55C4×#41—22no—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—22no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D411yescharacteristic
66C2×C4×D4C2312yes—
67C2×#117—22no—
68C2×#117C2×C412yes—
69C2×#117D411yescharacteristic
70C22×C4⋊C4—21no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C8×Q8—28no—
74C8×D4—28no—
75C8×D4—216no—
76#431—24no—
77#431—24no—
78#431—22no—
79#431—24no—
80#431—22no—
81#431D414yescomplemented
82#431D414yescomplemented
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85#410—44no—
86C2×#99—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8C2314yes—
89C2×C4×C8—24no—
90C2×C4×C8—22no—
91#398—22no—
92#398C2×C414yes—
93#397—22no—
94#397C2×C414yes—
95#394—22no—
96#394C2×C414yes—
97#392—22no—
98#392C2×C414yes—
99#391—22no—
100#390—22no—
101#388—24no—
102#388—24no—
103#386—24no—
104#386—24no—
105#386—24no—
106#384—22no—
107#382—21no—
108#382—21no—
109#382—22no—
110#382C2×C412yes—
111#382C2×C412yes—
112#380—22no—
113#380—22no—
114#380—22no—
115#380C2314yes—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C2×#95—21no—
119C2×#95C2312yes—
120#342—24no—
121#327—216no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—24no—
125C2×#41—42no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—44no—
130C2×#41D814yes—
131C2×#41SD1614yes—
132C22×Q8—21no—
133C22×Q8C22⋊C411yescharacteristic
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22⋊C411yescharacteristic
139C22×D4C22×C412yes—
140C22×D4C2×D412yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—28no—
150C22×C8—24no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153#127—22no—
154#127—22no—
155#124—41no—
156#124—41no—
157#124—41no—
158#124—41no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161#121—24no—
162#121—24no—
163#121—48no—
164#121—48no—
165#121—48no—
166#121—48no—
167#121—42no—
168#121—42no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—42no—
184C4×D4—42no—
185#117—22no—
186#117—22no—
187#117—24no—
188#117—24no—
189#117—44no—
190#117—44no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22×C412yes—
214C2×C22⋊C4C2×D412yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—42no—
220C2×C42C22⋊C412yes—
221C2×C42C22⋊C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4112yes—
230C2×C42#4112yes—
231C4⋊C8—28no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234#99—48no—
235C22⋊C8—28no—
236C22⋊C8—24no—
237C22⋊C8C2×D418yes—
238C4×C8—28no—
239C4×C8—24no—
240C4×C8—24no—
241#95—22no—
242#95—22no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—24no—
247#95—42no—
248#95C22×C414yes—
249C24—22no—
250C24C2×C22⋊C412yes—
251C24C22×D411yescharacteristic
252#41—28no—
253#41—28no—
254#41—28no—
255#41—28no—
256#41—84no—
257#41—84no—
258#41—84no—
259#41—84no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8#10214yes—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—48no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—84no—
285C2×D4—84no—
286C2×D4#10214yes—
287C2×D4C2×D814yes—
288C2×D4C2×SD1614yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4#10218yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4C2×D812yes—
323C22×C4C2×SD1612yes—
324C22×C4#13514yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4112yes—
329C2×C8—28no—
330C2×C8—84no—
331C2×C8—88no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—48no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—84no—
364C42C4×D412yes—
365C42C4×D412yes—
366C42C4×D412yes—
367C42C4×D412yes—
368C42#12011yescharacteristic
369C42#12012yes—
370C42#12011yescharacteristic
371C42#12311yescharacteristic
372C42#12311yescharacteristic
373C42#12312yes—
374C23—22no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23C2×#10214yes—
388C23C22×C22⋊C411yescharacteristic
389C23C2×#12112yes—
390C23C22×D811yescharacteristic
391C23C22×SD1611yescharacteristic
392C23C2×#13512yes—
393Q8—28no—
394Q8—82no—
395Q8—82no—
396D4—28no—
397D4—28no—
398D4—28no—
399D4—84no—
400D4—84no—
401D4—84no—
402D4—84no—
403D4—82no—
404D4—82no—
405D4—168no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—88no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—162no—
437C2×C4—162no—
438C2×C4#38611yescharacteristic
439C2×C4#38611yescharacteristic
440C2×C4#38612yes—
441C2×C4#38611yescharacteristic
442C2×C4#38611yescharacteristic
443C2×C4#38612yes—
444C2×C4C2×#10214yes—
445C2×C4C2×#10212yes—
446C2×C4C2×#10212yes—
447C2×C4#41612yes—
448C2×C4#41612yes—
449C2×C4#46514yes—
450C2×C4#46514yes—
451C2×C4#46614yes—
452C2×C4#46614yes—
453C2×C4C2×C4×D412yes—
454C2×C4C2×#12011yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C8—48no—
457C22—21no—
458C22—22no—
459C22—21no—
460C22—22no—
461C22—24no—
462C22—41no—
463C22—41no—
464C22—42no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22C2×#38611yescharacteristic
477C22C22×#10211yescharacteristic
478C22C2×#41611yescharacteristic
479C22C2×#46512yes—
480C22C2×#46612yes—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—21no—
485C4—21no—
486C4—22no—
487C4—24no—
488C4—42no—
489C4—42no—
490C4—84no—
491C4—82no—
492C4—164no—
493C4#189312yes—
494C4#189312yes—
495C4#189314yes—
496C2—21no—
497C2—22no—
498C2—21no—
499C2—22no—
500C2—44no—
501C2—162no—
502C2#2094711yescharacteristic
503C2C2×#189311yescharacteristic
504C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35328C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60276C2semidirect
3. #20632C2semidirect
4. #17574C2semidirect
5. #9384C2semidirect
6. #8026C2semidirect
19. #2621C4semidirect
20. #2619C4semidirect
24. #2536C4semidirect
26. #2535C4semidirect
81. #431D4semidirect
82. #431D4semidirect

Maximal subgroups

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

subgroupindex
#602762
#353282
#206322
#175742
#93842
#80262