Finite Group Databasebeta

← #196521

Subgroups — #196521

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#97C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1532C212yesmaximal, complemented
5#9121C214yesmaximal, complemented
6#9119C214yesmaximal, complemented
7#8036C212yesmaximal, complemented
8C24×C8—21no—
9C2×#529C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432C2212yes—
12C22×#97—21no—
13C22×#97—22no—
14C22×#97—28no—
15C22×#97C2214yes—
16C22×#97C2212yes—
17#2855C411yescharacteristic
18#2854C411yescharacteristic
19#2853—21no—
20#2852—22no—
21#2851C2212yes—
22#2850C2212yes—
23#2849—21no—
24#2848C411yescharacteristic
25#2847C411yescharacteristic
26C4×C22⋊Q8C412yes—
27C4×#123—21no—
28C4×#121C412yes—
29C4×#120—21no—
30#2290—22no—
31#2283C2211yescharacteristic
32#1817—21no—
33#1797—22no—
34#1779—21no—
35#1749C2214yes—
36#1532—28no—
37C2×#330C2214yescomplemented
38C2×#329C2214yescomplemented
39#1295C2218yescomplemented
40C24×C4C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C23×C8—28no—
44C23×C8—28no—
45C23×C8—28no—
46C23×C8—24no—
47C23×C8—22no—
48#529—24no—
49#529—24no—
50C2×#126—21no—
51C2×#125C2311yescharacteristic
52C2×#124C2311yescharacteristic
53C2×#123—21no—
54C2×#123—21no—
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8C2×C412yes—
57C2×#121C2×C411yescharacteristic
58C2×#121C2×C412yes—
59C2×#120—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×#117—21no—
65C22×C42—24no—
66C22×C42—24no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C412yes—
69C22×C42D412yes—
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D412yes—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#432—24no—
76#432—24no—
77C2×C4⋊C8—22no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—22no—
80C2×#97—28no—
81C2×#97—216no—
82C2×#97—24no—
83C2×#97—22no—
84C2×#97—24no—
85C2×#97—216no—
86C2×#97D418yescomplemented
87C2×#97C2314yes—
88C2×#97C2314yes—
89#391C2×C412yes—
90#390C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386—22no—
102#386—21no—
103#386—22no—
104#386—21no—
105#386—22no—
106#386—22no—
107#385—22no—
108#385—21no—
109#385—22no—
110#385—21no—
111#385—22no—
112#385—22no—
113#383—21no—
114#382C2312yes—
115#381—21no—
116#381—21no—
117#380C2×C412yes—
118#380C2×C411yescharacteristic
119#380C2×C412yes—
120#380C2×C411yescharacteristic
121#379—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127#376—21no—
128#336—24no—
129#330—28no—
130#329—28no—
131C25C22⋊C411yescharacteristic
132C22×Q8C22×C411yescharacteristic
133C22×D4—21no—
134C22×D4—22no—
135C22×D4C22×C411yescharacteristic
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C22×C8—264no—
158C22×C8—216no—
159C22×C8—28no—
160C22×C8—28no—
161C22×C8—216no—
162C22×C8—216no—
163C22×C8—28no—
164C22×C8—24no—
165C22×C8—24no—
166#126—44no—
167#125—22no—
168#125—22no—
169#124—22no—
170#124—22no—
171#123—44no—
172#123—42no—
173#123—42no—
174C22⋊Q8—24no—
175C22⋊Q8—44no—
176#121—24no—
177#121—44no—
178#120—44no—
179C4×Q8—24no—
180C4×D4—24no—
181C4×D4—48no—
182C4×D4—44no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—28no—
214C2×C42—24no—
215C2×C42—28no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42C22⋊C412yes—
222C2×C42C22⋊C412yes—
223C2×C42C22⋊C412yes—
224C2×C42C22×C412yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D414yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4114yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—44no—
234C22⋊C8—48no—
235C22⋊C8—44no—
236#97—216no—
237#97—28no—
238#97—24no—
239#97—24no—
240#97—24no—
241#97C2×D418yes—
242#95—21no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253C24—22no—
254C24—24no—
255C24—22no—
256C24—24no—
257C24—21no—
258C24#10414yes—
259C24C2×C22⋊C411yescharacteristic
260C24C2×C22⋊C412yes—
261C2×Q8—22no—
262C2×Q8—42no—
263C2×D4—22no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—44no—
271C22×C4—216no—
272C22×C4—216no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—416no—
309C22×C4—416no—
310C22×C4—42no—
311C22×C4#10414yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C4×D414yes—
319C22×C4#12012yes—
320C22×C4#12114yes—
321C22×C4#12312yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C2×C8—264no—
326C2×C8—28no—
327C2×C8—28no—
328C2×C8—232no—
329C2×C8—28no—
330C2×C8—48no—
331C2×C8—44no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—44no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C42—24no—
359C42—22no—
360C42—24no—
361C42—22no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—44no—
366C42—42no—
367C42C4×D414yes—
368C42#12012yes—
369C42#12114yes—
370C42#12312yes—
371C42C22×D412yes—
372C42C2×#4112yes—
373C23—28no—
374C23—28no—
375C23—216no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—28no—
382C23—22no—
383C23—216no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—41no—
389C23#38612yes—
390C23#41314yes—
391C23C2×#10412yes—
392C23C2×#10412yes—
393C23C2×#10414yes—
394C23#42114yes—
395C23C22×C22⋊C411yescharacteristic
396Q8—82no—
397D4—82no—
398D4—84no—
399D4—88no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—216no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—84no—
444C2×C4#38612yes—
445C2×C4#38612yes—
446C2×C4#38612yes—
447C2×C4C2×#10414yes—
448C2×C4C2×C4×D412yes—
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×#12112yes—
451C2×C4C2×#12311yescharacteristic
452C8—216no—
453C8—84no—
454C22—232no—
455C22—22no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—24no—
465C22—81no—
466C22—82no—
467C22—82no—
468C22—81no—
469C22—81no—
470C22#153218yescomplemented
471C22#189614yes—
472C22C2×#38611yescharacteristic
473C22C2×#41312yes—
474C22C22×#10411yescharacteristic
475C22C22×#10411yescharacteristic
476C22C2×#42112yes—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—24no—
483C4—22no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—82no—
489C4—82no—
490C4—81no—
491C4—81no—
492C2—24no—
493C2—28no—
494C2—161no—
495C2C2×#153212yes—
496C2#1764612yes—
497C2#2090211yescharacteristic
498C2C2×#189611yescharacteristic
499C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2847C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#97C2semidirect
3. #20715C2semidirect
4. C2×#1532C2semidirect
5. #9121C2semidirect
6. #9119C2semidirect
7. #8036C2semidirect
37. C2×#330C22semidirect
38. C2×#329C22semidirect
39. #1295C22semidirect
86. C2×#97D4semidirect
470. C22#1532semidirect

Maximal subgroups

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

subgroupindex
C23×#972
#366212
#207152
C2×#15322
#91212
#91192
#80362