Finite Group Databasebeta

← #192069

Subgroups — #192069

Order 512.

1334 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#34057C211yesmaximal, characteristic
2#21361C212yesmaximal, complemented
3C2×#1460C211yesmaximal, characteristic
4#13847C211yesmaximal, characteristic
5#8858C214yesmaximal, complemented
6#8854C212yesmaximal, complemented
7#8853C212yesmaximal, complemented
8#8013C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#431C2212yes—
11#2318—24no—
12#2315C2212yes—
13#2313—21no—
14#2312C2212yes—
15#2308—21no—
16#2290—22no—
17#2290—22no—
18#2283C2212yes—
19#2280—21no—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4—21no—
22C2×C4×C22⋊C4—21no—
23C2×C4×C22⋊C4—21no—
24C2×#376C2211yescharacteristic
25#1949—24no—
26#1853C2214yes—
27#1852C2214yes—
28C4×#107—22no—
29C4×#106—22no—
30#1460C418yescomplemented
31C2×#323C2211yescharacteristic
32C2×#323C2211yescharacteristic
33C2×#323C2212yes—
34#1445—21no—
35#1439C2212yes—
36#1439C2212yes—
37#1438—21no—
38#1294C2214yes—
39#1294C2214yes—
40C23×D4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4Q814yes—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4C2311yescharacteristic
56C22×C22⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#431—24no—
63#431C2×C418yescomplemented
64C2×#107—24no—
65C2×#106—24no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8C2312yes—
68C2×C22⋊C8C2312yes—
69C2×C4×C8—22no—
70#388—24no—
71#388—24no—
72#387—24no—
73#387—24no—
74#386—24no—
75#386—24no—
76#385—24no—
77#385—24no—
78#383—22no—
79#382D414yes—
80#381—22no—
81#381—22no—
82#380—22no—
83#380—22no—
84#380C2314yes—
85#379—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—44no—
96C4×C22⋊C4—44no—
97#376—22no—
98#376—24no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95C2312yes—
105#340—24no—
106#340—24no—
107#323C2×C414yes—
108#323C2×C414yes—
109#323C2×C418yescomplemented
110C25C2×D411yescharacteristic
111C25C2×D411yescharacteristic
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—42no—
116C22×D4C4⋊C414yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4C4⋊C414yes—
132C23×C4C4⋊C414yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C4×D4—28no—
141C4×D4—28no—
142C4×D4—28no—
143C4×D4—28no—
144#117—82no—
145#117—82no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C4⋊C414yes—
153C2×C4⋊C4C2×D414yes—
154C2×C4⋊C4C2×D414yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4C22×C412yes—
173C2×C22⋊C4C22×C412yes—
174C2×C22⋊C4C2×Q812yes—
175C2×C22⋊C4C2×Q812yes—
176C2×C42—24no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42C22×C412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×Q812yes—
193C2×C42#4114yes—
194#107—48no—
195#106—48no—
196C4⋊C8—28no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8C22×C414yes—
200C22⋊C8C22×C414yes—
201C4×C8—28no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—22no—
207#95—22no—
208#95—24no—
209#95—24no—
210#95—44no—
211#95—44no—
212#95—42no—
213#95—42no—
214C24—21no—
215C24—21no—
216C24—22no—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—41no—
222C24—41no—
223C24—42no—
224C24—42no—
225C24—41no—
226C24—41no—
227C24C2×C4⋊C412yes—
228C24C2×C4⋊C412yes—
229C24#13514yes—
230C24C22×D411yescharacteristic
231C2×D4—28no—
232C2×D4—28no—
233C2×D4—44no—
234C2×D4—48no—
235C2×D4—48no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C4⋊C412yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C2×C4⋊C412yes—
290C22×C4C2×C4⋊C412yes—
291C22×C4#12012yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12312yes—
295C22×C4#12312yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×Q811yescharacteristic
299C22×C4C2×#4112yes—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—44no—
303C2×C8—44no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—48no—
316C22⋊C4—416no—
317C22⋊C4—416no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—84no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C42—48no—
324C42—44no—
325C42—44no—
326C42—82no—
327C42—82no—
328C42—82no—
329C42—82no—
330C42C4×D414yes—
331C42C4×D414yes—
332C42C22⋊Q814yes—
333C42C22⋊Q814yes—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—24no—
338C23—22no—
339C23—24no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23#42714yes—
369C23C22×C4⋊C411yescharacteristic
370C23C2×#12011yescharacteristic
371C23C2×#12311yescharacteristic
372C23C2×#13512yes—
373C23#57512yes—
374D4—416no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—88no—
419C2×C4—88no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#38514yes—
424C2×C4#38512yes—
425C2×C4#38512yes—
426C2×C4#45312yes—
427C2×C4#45512yes—
428C2×C4C2×C4×D412yes—
429C2×C4C2×C22⋊Q812yes—
430C8—44no—
431C8—44no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—24no—
441C22—44no—
442C22—44no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22—84no—
454C22—82no—
455C22—82no—
456C22C2×#38511yescharacteristic
457C22C2×#42711yescharacteristic
458C22#291111yescharacteristic
459C22#300212yes—
460C22C2×#45311yescharacteristic
461C22C2×#45511yescharacteristic
462C22#302512yes—
463C22#309112yes—
464C4—24no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—42no—
469C4—42no—
470C4—84no—
471C4—84no—
472C4—84no—
473C4—84no—
474C4#185614yes—
475C2—22no—
476C2—21no—
477C2—21no—
478C2—44no—
479C2—82no—
480C2#1812812yes—
481C2#1812912yes—
482C2#2084311yescharacteristic
483C2C2×#185611yescharacteristic
484C2#2088811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21361C2semidirect
5. #8858C2semidirect
6. #8854C2semidirect
7. #8853C2semidirect
8. #8013C2semidirect
30. #1460C4semidirect
63. #431C2×C4semidirect
109. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340572
#213612
C2×#14602
#138472
#88582
#88542
#88532
#80132