Finite Group Databasebeta

← #197862

Subgroups — #197862

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3116C211yesmaximal, characteristic
2C2×#2988C211yesmaximal, characteristic
3#21835C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic, complemented
6#9218C212yesmaximal, complemented
7#9217C212yesmaximal, complemented
8#9216C212yesmaximal, complemented
9#9215C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#7878C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#3116—24no—
15#3116C418yescomplemented
16C2×#485—21no—
17C2×#477C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×#461C2211yescharacteristic
21C2×#452—21no—
22#2988—24no—
23#2988—24no—
24C2×C8×Q8C2212yes—
25C2×#441—21no—
26C2×#440—21no—
27C2×C8×D4—22no—
28C2×C8×D4C2212yes—
29C2×#432C2211yescharacteristic
30C2×#431—21no—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C22×C4⋊C8—21no—
34C22×C4⋊C8C2211yescharacteristic
35#2285C2211yescharacteristic
36#1990C2212yes—
37#1990C2212yes—
38#1989—21no—
39#1989—21no—
40#1825—22no—
41#1825—21no—
42#1803C2212yes—
43#1802C2212yes—
44#1788—21no—
45C4×#103—21no—
46C4×#102—21no—
47#1542—24no—
48C2×#333C2212yes—
49C2×#332C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1277C2212yes—
53#1277C2212yes—
54C22×#41D411yescharacteristic
55C22×Q16—21no—
56C22×SD16—21no—
57C23×C8—21no—
58C23×C8—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533C2×C414yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41C2×C414yes—
71C4×#41D414yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117D411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#485—28no—
87#477—22no—
88#477C2×C414yes—
89#476—22no—
90#476C2×C414yes—
91#462—22no—
92#462C2×C414yes—
93#461—22no—
94#461C2×C414yes—
95#452—28no—
96C8×Q8—24no—
97C8×Q8—24no—
98#441—28no—
99#440—28no—
100C8×D4—216no—
101C8×D4—24no—
102C8×D4—24no—
103#432—22no—
104#432C2×C414yes—
105#431—28no—
106#431—22no—
107#431—22no—
108#431—22no—
109#431—22no—
110C2×#107—21no—
111C2×#106—21no—
112C2×C4⋊C8—28no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8D411yescharacteristic
117C2×C4⋊C8D411yescharacteristic
118C2×C4⋊C8D414yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2312yes—
123C2×#103—21no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—22no—
127C2×#102—21no—
128C2×#102—22no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—22no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8D412yes—
133C2×#97—21no—
134C2×C4×C8—22no—
135C2×C4×C8—21no—
136C2×C4×C8C2312yes—
137#384C2312yes—
138#382—21no—
139#380—21no—
140#380—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#333—24no—
150#332—24no—
151C2×#41—24no—
152C2×#41—24no—
153C2×#41—22no—
154C2×#41C22⋊C414yes—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8C2×D411yescharacteristic
158C22×D4—21no—
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4111yescharacteristic
166C2×Q16—24no—
167C2×Q16—44no—
168C2×SD16—24no—
169C2×SD16—44no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—28no—
174C22×C8—24no—
175C22×C8—22no—
176C22×C8—84no—
177C22×C8—82no—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187C4⋊Q8C22×C412yes—
188#125—24no—
189#125—42no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C4×Q8—22no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8C22⋊C414yes—
202C4×Q8C22×C412yes—
203C4×Q8C2×D412yes—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—22no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4C22⋊C414yes—
212C4×D4C22×C412yes—
213C4×D4C2×D412yes—
214#117—22no—
215#117—24no—
216#117—24no—
217#117—24no—
218#117—42no—
219#117—82no—
220#117C22⋊C414yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42C22×C412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42#4111yescharacteristic
252C2×C42#4112yes—
253C2×C42C2411yescharacteristic
254#107—24no—
255#106—24no—
256C4⋊C8—28no—
257C4⋊C8—22no—
258C4⋊C8—24no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8C22×C412yes—
266C4⋊C8C22×C412yes—
267C4⋊C8C2×D414yes—
268#103—24no—
269#103—24no—
270#103—24no—
271#103—44no—
272#102—24no—
273#102—44no—
274C22⋊C8—28no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C22⋊C8—42no—
278#97—24no—
279C4×C8—28no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—42no—
283#95—41no—
284#95—41no—
285C24—21no—
286C24#12111yescharacteristic
287#41—216no—
288#41—28no—
289#41—28no—
290C2×Q8—21no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—81no—
299C2×Q8—81no—
300C2×Q8C2×C22⋊C412yes—
301C2×Q8#13514yes—
302C2×D4—28no—
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4C2×C22⋊C412yes—
307C2×D4#13514yes—
308C22×C4—28no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12012yes—
333C22×C4#12112yes—
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12312yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341Q16—48no—
342SD16—48no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—44no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—84no—
357C2×C8—88no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—82no—
378C4⋊C4C2×C22⋊C412yes—
379C4⋊C4C2×C22⋊C412yes—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—24no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—24no—
390C42—44no—
391C42—82no—
392C42C2×C22⋊C412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42#12012yes—
396C42#12112yes—
397C42#12312yes—
398C42C23×C411yescharacteristic
399C42C22×D411yescharacteristic
400C42C22×D411yescharacteristic
401C42C2×#4111yescharacteristic
402C23—22no—
403C23—22no—
404C23—24no—
405C23—22no—
406C23—21no—
407C23—21no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23#38612yes—
412C23C2×#12111yescharacteristic
413Q8—22no—
414Q8—24no—
415Q8—42no—
416Q8—42no—
417Q8—82no—
418Q8—82no—
419Q8—82no—
420Q8#41614yes—
421D4—216no—
422D4—22no—
423D4—24no—
424D4#41614yes—
425C2×C4—216no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—84no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4#38612yes—
460C2×C4#38612yes—
461C2×C4#38612yes—
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×#12011yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C2×C4C2×#13512yes—
469C2×C4C2×#13512yes—
470C2×C4#57712yes—
471C2×C4#57812yes—
472C8—42no—
473C8—42no—
474C8—42no—
475C8—44no—
476C8—42no—
477C8—82no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—24no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—44no—
488C22C2×#38611yescharacteristic
489C22#304612yes—
490C22#305012yes—
491C22#306811yescharacteristic
492C22#306911yescharacteristic
493C4—22no—
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—24no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4—82no—
503C4—162no—
504C4C2×#41611yescharacteristic
505C4C2×#41611yescharacteristic
506C4#294712yes—
507C4#300911yescharacteristic
508C4#301111yescharacteristic
509C4#305012yes—
510C2—42no—
511C2—44no—
512C2#1941812yes—
513C2#1941912yes—
514C2#2093111yescharacteristic
515C2#2096711yescharacteristic
516C2#2100411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21835C2semidirect
5. C2×#1542C2semidirect
6. #9218C2semidirect
7. #9217C2semidirect
8. #9216C2semidirect
9. #9215C2semidirect
10. #8044C2semidirect
11. #7878C2semidirect
15. #3116C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31162
C2×#29882
#218352
#207372
C2×#15422
#92182
#92172
#92162
#92152
#80442
#78782