Finite Group Databasebeta

← #195973

Subgroups — #195973

Order 512.

1439 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2C2×#2853C211yesmaximal, characteristic
3#20682C211yesmaximal, characteristic, complemented
4C2×#1468C212yesmaximal
5#9086C214yesmaximal, complemented
6#9084C214yesmaximal, complemented
7#7913C212yesmaximal, complemented
8C22×#123C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432—22no—
11C2×#432—28no—
12C2×#432C2214yes—
13C22×C4⋊C8—21no—
14C2×#407C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97—21no—
17#2853—22no—
18#2853—21no—
19#2853—21no—
20#2853—22no—
21#2853—21no—
22#2853—21no—
23C2×#388C412yes—
24C2×#388C2212yes—
25C2×#387C412yes—
26C2×#387C2212yes—
27C2×#386C411yescharacteristic
28C2×#386C411yescharacteristic
29C2×#385C411yescharacteristic
30C2×#385C411yescharacteristic
31C2×#381C2211yescharacteristic
32#1817—22no—
33#1817C2214yes—
34#1728—21no—
35#1728—21no—
36#1468—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39#1280C2218yes—
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C23×C8—22no—
46C2×#123—21no—
47C2×#123—21no—
48C2×#123—21no—
49C2×#123—21no—
50C2×#123—24no—
51C2×#123—24no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2311yescharacteristic
63C22×C42—24no—
64C22×C42—24no—
65C22×C42C2×C412yes—
66C22×C42D412yes—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D412yes—
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2312yes—
73#432—216no—
74#432—216no—
75C2×C4⋊C8—28no—
76C2×C4⋊C8—24no—
77#407—24no—
78#407—24no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8D418yes—
83C2×C22⋊C8C2314yes—
84C2×C22⋊C8C2314yes—
85C2×#97—28no—
86C2×#97—24no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
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#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#386—21no—
104#386—22no—
105#386—22no—
106#386—21no—
107#386—21no—
108#386—21no—
109#385—21no—
110#385—22no—
111#385—22no—
112#385—21no—
113#385—21no—
114#385—21no—
115#381—21no—
116#381—21no—
117#381—21no—
118#381—21no—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2312yes—
125#336—24no—
126#336—24no—
127#324—28no—
128#323—28no—
129C25C22×C411yescharacteristic
130C25C22×C411yescharacteristic
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—24no—
134C22×D4—42no—
135C22×D4—42no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D414yes—
161C23×C4#4114yes—
162C23×C4C2411yescharacteristic
163C2×M16—44no—
164C2×M16—44no—
165C2×M16—44no—
166C22×C8—28no—
167C22×C8—24no—
168C22×C8—24no—
169C22×C8—44no—
170C22×C8—44no—
171#123—42no—
172#123—42no—
173#123—44no—
174#123—44no—
175#123—42no—
176#123—42no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C42—28no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—28no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42C22⋊C412yes—
227C2×C42C22⋊C412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C4⋊C8—216no—
233C22⋊C8—28no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—44no—
238C22⋊C8C2×D418yes—
239#97—216no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252C24—24no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—22no—
257C24—24no—
258C24—42no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24C2×C22⋊C411yescharacteristic
263C24C2×C22⋊C411yescharacteristic
264C24C4×D414yes—
265C24C23×C411yescharacteristic
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—88no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—416no—
319C22×C4—416no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C4×D414yes—
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4#12114yes—
337C22×C4#12114yes—
338C22×C4#12312yes—
339C22×C4#12312yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C2×#4112yes—
344M16—88no—
345C2×C8—28no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—416no—
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—42no—
372C22⋊C4—41no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C42—22no—
377C42—22no—
378C42—24no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C23—22no—
387C23—21no—
388C23—21no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—28no—
396C23—28no—
397C23—44no—
398C23—48no—
399C23—48no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—44no—
406C23—82no—
407C23—84no—
408C23—84no—
409C23—82no—
410C23#38612yes—
411C23#38612yes—
412C23#41012yes—
413C23#41012yes—
414C23C22×C22⋊C411yescharacteristic
415C23C2×C4×D412yes—
416C23#56814yes—
417D4—84no—
418D4—84no—
419C2×C4—28no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#38612yes—
474C2×C4#38612yes—
475C2×C4#41314yes—
476C2×C4C2×#12011yescharacteristic
477C2×C4C2×#12112yes—
478C2×C4C2×#12311yescharacteristic
479C8—48no—
480C8—84no—
481C22—22no—
482C22—24no—
483C22—24no—
484C22—22no—
485C22—24no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—416no—
493C22—42no—
494C22—42no—
495C22—81no—
496C22—82no—
497C22—81no—
498C22—82no—
499C22—81no—
500C22—164no—
501C22#189914yes—
502C22C2×#38611yescharacteristic
503C22#287912yes—
504C22C2×#41011yescharacteristic
505C22C2×#41011yescharacteristic
506C22C2×#41312yes—
507C22#293014yes—
508C4—21no—
509C4—22no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—24no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—82no—
519C4—81no—
520C4—82no—
521C4—81no—
522C2—24no—
523C2—24no—
524C2—42no—
525C2—161no—
526C2#1739512yes—
527C2#1740212yes—
528C2#2089611yescharacteristic
529C2C2×#189911yescharacteristic
530C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#432C2semidirect
3. #20682C2semidirect
5. #9086C2semidirect
6. #9084C2semidirect
7. #7913C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#28532
#206822
C2×#14682
#90862
#90842
#79132