Finite Group Databasebeta

← #196530

Subgroups — #196530

Order 512.

1377 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1532C212yesmaximal, complemented
5#9125C214yesmaximal, complemented
6#9123C214yesmaximal, complemented
7#8036C212yesmaximal, complemented
8C2×#529C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432—22no—
11C2×#432—28no—
12C2×#432C2212yes—
13C2×#432C2214yes—
14C22×C4⋊C8—21no—
15C22×C22⋊C8—21no—
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—22no—
44#529—24no—
45#529—24no—
46C2×#126—21no—
47C2×#125C2311yescharacteristic
48C2×#124C2311yescharacteristic
49C2×#123—21no—
50C2×#123—21no—
51C2×C22⋊Q8C2×C411yescharacteristic
52C2×C22⋊Q8C2×C412yes—
53C2×#121C2×C411yescharacteristic
54C2×#121C2×C412yes—
55C2×#120—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×#117—21no—
61C22×C42—24no—
62C22×C42—24no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C412yes—
65C22×C42D412yes—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D412yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#432—216no—
72#432—24no—
73#432—24no—
74#432—216no—
75C2×C4⋊C8—28no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—22no—
78C2×C22⋊C8—28no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×#97—22no—
83C2×#97—24no—
84C2×#97D418yescomplemented
85C2×#97C2314yes—
86C2×#97C2314yes—
87#391C2×C412yes—
88#390C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#386—22no—
100#386—21no—
101#386—22no—
102#386—21no—
103#386—22no—
104#386—22no—
105#385—22no—
106#385—21no—
107#385—22no—
108#385—21no—
109#385—22no—
110#385—22no—
111#383—21no—
112#382C2312yes—
113#381—21no—
114#381—21no—
115#380C2×C412yes—
116#380C2×C411yescharacteristic
117#380C2×C412yes—
118#380C2×C411yescharacteristic
119#379—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C412yes—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—22no—
125#376—21no—
126#336—24no—
127#330—28no—
128#329—28no—
129C25C22⋊C411yescharacteristic
130C22×Q8C22×C411yescharacteristic
131C22×D4—21no—
132C22×D4—22no—
133C22×D4C22×C411yescharacteristic
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—44no—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C22×C8—28no—
156C22×C8—24no—
157C22×C8—24no—
158C22×C8—44no—
159C22×C8—44no—
160#126—44no—
161#125—22no—
162#125—22no—
163#124—22no—
164#124—22no—
165#123—44no—
166#123—42no—
167#123—42no—
168C22⋊Q8—24no—
169C22⋊Q8—44no—
170#121—24no—
171#121—44no—
172#120—44no—
173C4×Q8—24no—
174C4×D4—24no—
175C4×D4—48no—
176C4×D4—44no—
177#117—42no—
178#117—42no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
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×C42—24no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—28no—
208C2×C42—24no—
209C2×C42—28no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42C22⋊C412yes—
216C2×C42C22⋊C412yes—
217C2×C42C22⋊C412yes—
218C2×C42C22×C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D414yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4114yes—
226C2×C42C2411yescharacteristic
227C4⋊C8—216no—
228C4⋊C8—44no—
229C22⋊C8—216no—
230C22⋊C8—48no—
231C22⋊C8—44no—
232#97—28no—
233#97—24no—
234#97—24no—
235#97—24no—
236#97C2×D418yes—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248C24—22no—
249C24—24no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—42no—
254C24C2×C22⋊C412yes—
255C24C2×C22⋊C411yescharacteristic
256C2×Q8—22no—
257C2×Q8—42no—
258C2×D4—22no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—42no—
265C2×D4—42no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
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—416no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—44no—
305C22×C4—416no—
306C22×C4—42no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C4×D414yes—
314C22×C4#12012yes—
315C22×C4#12114yes—
316C22×C4#12312yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C2×C8—28no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—416no—
324C2×C8—48no—
325C2×C8—44no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—41no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C42—24no—
353C42—22no—
354C42—24no—
355C42—22no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—44no—
360C42—42no—
361C42C4×D414yes—
362C42#12012yes—
363C42#12114yes—
364C42#12312yes—
365C42C22×D412yes—
366C42C2×#4112yes—
367C23—22no—
368C23—22no—
369C23—28no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—48no—
377C23—48no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—41no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23#38612yes—
386C23#42114yes—
387C23#42112yes—
388C23#42112yes—
389C23C22×C22⋊C411yescharacteristic
390Q8—82no—
391D4—82no—
392D4—88no—
393D4—84no—
394C2×C4—28no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—84no—
438C2×C4#38612yes—
439C2×C4#38612yes—
440C2×C4#38612yes—
441C2×C4#41314yes—
442C2×C4C2×C4×D412yes—
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12112yes—
445C2×C4C2×#12311yescharacteristic
446C8—48no—
447C8—84no—
448C22—24no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—416no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—82no—
461C22—82no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22#189714yes—
466C22C2×#38611yescharacteristic
467C22C2×#41312yes—
468C22C2×#42111yescharacteristic
469C22C2×#42112yes—
470C22C2×#42111yescharacteristic
471C4—22no—
472C4—22no—
473C4—21no—
474C4—21no—
475C4—22no—
476C4—24no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—82no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4—81no—
486C2—24no—
487C2—44no—
488C2—161no—
489C2#1766812yes—
490C2#1767812yes—
491C2#2090211yescharacteristic
492C2C2×#189711yescharacteristic
493C2#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. C22×#432C2semidirect
3. #20715C2semidirect
4. C2×#1532C2semidirect
5. #9125C2semidirect
6. #9123C2semidirect
7. #8036C2semidirect
37. C2×#330C22semidirect
38. C2×#329C22semidirect
39. #1295C22semidirect
84. C2×#97D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#366212
#207152
C2×#15322
#91252
#91232
#80362