Finite Group Databasebeta

← #6355342

Subgroups — #6355342

Order 512.

1593 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic
2#60560C212yesmaximal, complemented
3C2×#2983C211yesmaximal, characteristic, complemented
4#19443C214yesmaximal, complemented
5#19442C214yesmaximal, complemented
6#19437C214yesmaximal, complemented
7#19436C214yesmaximal, complemented
8#19287C212yesmaximal, complemented
9#19286C212yesmaximal, complemented
10#19285C214yesmaximal, complemented
11#17815C211yesmaximal, characteristic, complemented
12#17716C212yesmaximal, complemented
13#3444C2212yes—
14C22×C4×Q8C2211yescharacteristic
15#3250—22no—
16#3231C2212yes—
17#3228C2212yes—
18#3210C2212yes—
19#3207C2212yes—
20#3191—22no—
21#3170C2211yescharacteristic
22#3169C2211yescharacteristic
23#3167C2212yes—
24#3117—24no—
25#3113C2214yes—
26#3112C2214yes—
27#3108—24no—
28#3104C2214yes—
29#3103C2214yes—
30#3102—24no—
31#3092—22no—
32#3078C2214yes—
33#3037C2214yes—
34#3004—22no—
35#2983—24no—
36#2983C2214yes—
37#2983C2214yes—
38C2×#446—22no—
39C2×#446C2212yes—
40C2×#446C2212yes—
41C4×#137—24no—
42#2920C2212yes—
43C2×#432—22no—
44C2×#431C2211yescharacteristic
45C2×#431C2211yescharacteristic
46#1704C2218yescomplemented
47#1702C2218yescomplemented
48#1700C2218yescomplemented
49#1698C2218yescomplemented
50#1696C2214yescomplemented
51#1694C2214yescomplemented
52#1692C2218yescomplemented
53#1690C2214yescomplemented
54#1688C2214yescomplemented
55#1686C2218yescomplemented
56#1572C2214yes—
57#1572C2214yes—
58#1567C2212yes—
59#1566C2212yes—
60#1554C2214yes—
61#1553C2214yes—
62#1550C2214yes—
63#1547C2214yes—
64#1544C2214yes—
65#1543C2214yes—
66C23×Q8D411yescharacteristic
67C2×#137—24no—
68#533C2312yes—
69#533C2312yes—
70#533C2312yes—
71#532C2312yes—
72#532C2312yes—
73#532C2312yes—
74#529—22no—
75#529—22no—
76#529—22no—
77#529—22no—
78#529—22no—
79#529—22no—
80#525—22no—
81#525—22no—
82#525—22no—
83C4×#41—22no—
84C4×#41—22no—
85C4×#41—22no—
86C2×C4×Q8—21no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—21no—
90C2×C4×Q8D414yes—
91C2×C4×Q8D412yes—
92C2×C4×Q8C2312yes—
93C2×C4×Q8C2314yes—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D411yescharacteristic
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98#500C2311yescharacteristic
99#500C2311yescharacteristic
100#498C2312yes—
101#494C2311yescharacteristic
102#493C2311yescharacteristic
103#492C2312yes—
104#487C2312yes—
105#487C2312yes—
106#486C2312yes—
107#486C2312yes—
108#485—24no—
109#484—24no—
110#483—24no—
111#482—24no—
112#481—24no—
113#480—24no—
114#479—28no—
115#479—24no—
116#478—24no—
117#477C2318yes—
118#475C2318yes—
119#467—22no—
120#466—22no—
121#465—24no—
122#464—28no—
123#464—24no—
124#463—24no—
125#462C2318yes—
126#460C2318yes—
127#452—24no—
128#451—24no—
129#451—24no—
130#449—24no—
131#448—24no—
132#448—24no—
133#446—28no—
134#446—28no—
135#446—22no—
136#446—22no—
137#446—24no—
138#446C2318yes—
139#446C2314yes—
140#446C2314yes—
141#441—28no—
142#440—28no—
143C4×Q16—28no—
144C4×Q16—22no—
145C4×SD16—28no—
146C4×SD16—24no—
147C4×D8—22no—
148#432—22no—
149#432—28no—
150#432—28no—
151#432C2314yes—
152#431—22no—
153#431—22no—
154#431D414yes—
155#431C2314yes—
156#431C2314yes—
157#431C2312yes—
158#431C2312yes—
159#428—24no—
160#423C2314yes—
161C2×C4⋊C8—22no—
162C2×C4⋊C8—22no—
163C2×C4⋊C8D412yes—
164C2×C4⋊C8C2312yes—
165#419—24no—
166#417—24no—
167#407—22no—
168C2×C22⋊C8—21no—
169C2×C22⋊C8D412yes—
170C4×M16—22no—
171C2×#97—22no—
172C2×#97—22no—
173C2×C4×C8—21no—
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176#335C2318yes—
177#334C2314yes—
178#334C2314yes—
179#333C2318yes—
180#332C2318yes—
181#332C2318yes—
182#331C2318yes—
183C2×#41—22no—
184C2×#41—22no—
185C2×#41—22no—
186C22×Q8—22no—
187C22×Q8—21no—
188C22×Q8—44no—
189C22×Q8C2×D412yes—
190C22×Q8C2×D414yes—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196#137—416no—
197C2×Q16—28no—
198C2×Q16—22no—
199C2×SD16—28no—
200C2×SD16—24no—
201C2×D8—22no—
202C2×M16—24no—
203C22×C8—22no—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C4⋊Q8C2412yes—
207C4⋊Q8C2412yes—
208C4⋊Q8C2412yes—
209#127C2412yes—
210#126—42no—
211#126—42no—
212#125—22no—
213#125—22no—
214#125—22no—
215#124—22no—
216#124—22no—
217#124—22no—
218#123—42no—
219#123—42no—
220#123—42no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227#121—22no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#120—42no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—24no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8C2×D418yes—
248C4×Q8C2×D414yes—
249C4×Q8C2414yes—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262#117—22no—
263#117—22no—
264#117—22no—
265#117—42no—
266#117—42no—
267#117—42no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42C2×D412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42#4112yes—
286C2×C42#4111yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2412yes—
289#107—22no—
290#107—22no—
291#107—28no—
292#106—24no—
293#106—28no—
294C4⋊C8—24no—
295C4⋊C8—28no—
296C4⋊C8—24no—
297C4⋊C8—22no—
298C4⋊C8—24no—
299C4⋊C8C2×D414yes—
300C4⋊C8C2×D414yes—
301C4⋊C8C2414yes—
302C4⋊C8C2418yes—
303#103—28no—
304#103—28no—
305#103—24no—
306#103—24no—
307#103—28no—
308#102—24no—
309#102—24no—
310#102—28no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8C2×D414yes—
315C22⋊C8C2×D414yes—
316#97—24no—
317#97—28no—
318#97—24no—
319#97—24no—
320C4×C8—24no—
321C4×C8—24no—
322C4×C8C2×D412yes—
323C4×C8C2×D412yes—
324C4×C8C2411yescharacteristic
325C4×C8C2411yescharacteristic
326C4×C8C2412yes—
327C24#12111yescharacteristic
328#41—48no—
329#41—84no—
330#41—84no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—28no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—48no—
338C2×Q8—44no—
339C2×Q8—48no—
340C2×Q8C22×D414yes—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—81no—
358C22×C4—82no—
359C22×C4#12112yes—
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12112yes—
363C22×C4#12111yescharacteristic
364C22×C4#12112yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4111yescharacteristic
371Q16—416no—
372Q16—82no—
373SD16—416no—
374SD16—84no—
375D8—82no—
376M16—48no—
377C2×C8—24no—
378C2×C8—28no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—44no—
382C2×C8C22×D412yes—
383C2×C8C22×D414yes—
384C2×C8C22×D412yes—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4C22×D414yes—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C42—28no—
413C42—24no—
414C42—24no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42#12112yes—
419C42#12112yes—
420C42#12112yes—
421C42C22×D412yes—
422C42C22×D412yes—
423C42C22×D411yescharacteristic
424C42C22×D411yescharacteristic
425C42C2×#4111yescharacteristic
426C42C2×#4112yes—
427C42C2511yescharacteristic
428C23—22no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23C2×#12112yes—
433C23C2×#12111yescharacteristic
434Q8—28no—
435Q8—48no—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—82no—
440D4—44no—
441D4—44no—
442D4—82no—
443D4—84no—
444D4—84no—
445D4—84no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4C2×#12112yes—
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12112yes—
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12112yes—
474C2×C4C2×#12111yescharacteristic
475C2×C4#57514yes—
476C2×C4C23×D411yescharacteristic
477C2×C4C23×D411yescharacteristic
478C2×C4C22×#4111yescharacteristic
479C8—22no—
480C8—24no—
481C8—22no—
482C8—44no—
483C8—48no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—44no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22#305412yes—
494C22C22×#12111yescharacteristic
495C4—22no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—21no—
500C4—21no—
501C4—44no—
502C4—44no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4#305412yes—
507C4C2×#57512yes—
508C4#358612yes—
509C2—22no—
510C2—42no—
511C2—162no—
512C2#6042411yescharacteristic
513C2C2×#305411yescharacteristic
514C2#6046911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60944C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60560C2semidirect
3. C2×#2983C2semidirect
4. #19443C2semidirect
5. #19442C2semidirect
6. #19437C2semidirect
7. #19436C2semidirect
8. #19287C2semidirect
9. #19286C2semidirect
10. #19285C2semidirect
11. #17815C2semidirect
12. #17716C2semidirect
46. #1704C22semidirect
47. #1702C22semidirect
48. #1700C22semidirect
49. #1698C22semidirect
50. #1696C22semidirect
51. #1694C22semidirect
52. #1692C22semidirect
53. #1690C22semidirect
54. #1688C22semidirect
55. #1686C22semidirect

Maximal subgroups

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

subgroupindex
#609442
#605602
C2×#29832
#194432
#194422
#194372
#194362
#192872
#192862
#192852
#178152
#177162