Finite Group Databasebeta

← #356133

Subgroups — #356133

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2343C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3C2×#1740C211yesmaximal, characteristic
4#13799C212yesmaximal
5#10912C214yesmaximal
6#10801C212yesmaximal
7#10736C212yesmaximal
8C4×#342C212yesmaximal
9C22×#117C2212yes—
10C22×#117C2211yescharacteristic
11#2343—22no—
12#2343—22no—
13#2343—24no—
14C2×#387C412yescomplemented
15C2×#387C412yescomplemented
16C2×#385C2212yes—
17C2×#383C411yescharacteristic, complemented
18C2×#382C411yescharacteristic, complemented
19C2×#381C2212yes—
20C2×#381C2211yescharacteristic
21C2×C4×C4⋊C4C411yescharacteristic, complemented
22C2×C4×C22⋊C4C2214yes—
23C2×C4×C22⋊C4C2212yes—
24C2×#376C411yescharacteristic, complemented
25C22×#95C2211yescharacteristic
26#1820C418yescomplemented
27#1740C418yescomplemented
28C2×#342C2214yes—
29C2×#342C2214yes—
30C2×#342C2211yescharacteristic
31C2×#342C2212yes—
32C2×#342C2212yes—
33C2×#342C2211yescharacteristic
34C2×#342C2212yes—
35#1436—24no—
36#1434—22no—
37C22⋊#95C2214yes—
38C4×#95—22no—
39C24×C4C2311yescharacteristic
40C24×C4C2312yes—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—28no—
46C2×#117—44no—
47C2×#117C2×C418yes—
48C2×#117C2×C414yes—
49C2×#117C2×C414yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4C2312yes—
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2314yes—
58C22×C22⋊C4C2311yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2312yes—
61C22×C42—22no—
62C22×C42—24no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C412yes—
66C22×C42C2×C412yes—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#385—24no—
72#385—24no—
73#383—22no—
74#383—22no—
75#382—22no—
76#382—22no—
77#381—22no—
78#381—22no—
79#381—24no—
80#381C2×C418yes—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—21no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—28no—
87C4×C22⋊C4—44no—
88#376—21no—
89#376—22no—
90#376—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C412yes—
106C2×#95D412yes—
107C2×#95D412yes—
108C2×#95D414yes—
109#342—28no—
110#342—44no—
111#342C2×C418yes—
112#342C2×C418yes—
113#342C2×C414yes—
114#342C2×C418yes—
115#342C2×C414yes—
116C25C2411yescharacteristic
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—44no—
127C23×C4—82no—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C22×C412yes—
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C412yes—
140C23×C4C2×D414yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D414yes—
144C23×C4#4114yes—
145#117—28no—
146#117—28no—
147#117—28no—
148#117—28no—
149#117—216no—
150#117—48no—
151#117—416no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C4214yescomplemented
164C2×C4⋊C4C4214yescomplemented
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4C22×C414yes—
184C2×C22⋊C4C22×C418yes—
185C2×C22⋊C4C22×C412yes—
186C2×C22⋊C4C22×C412yes—
187C2×C22⋊C4C22×C412yes—
188C2×C22⋊C4C22×C412yes—
189C2×C22⋊C4C22×C414yes—
190C2×C22⋊C4#4118yes—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—28no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—48no—
203C2×C42—44no—
204C2×C42—48no—
205C2×C42—48no—
206C2×C42—84no—
207C2×C42—82no—
208C2×C42C4214yescomplemented
209C2×C42C4214yescomplemented
210#95—21no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—24no—
215#95—24no—
216#95—21no—
217#95—21no—
218#95—24no—
219#95—24no—
220#95—28no—
221#95—28no—
222#95—48no—
223#95—48no—
224#95—84no—
225#95C22⋊C411yescharacteristic
226#95C22⋊C412yes—
227#95C22⋊C411yescharacteristic
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C414yes—
231#95C22⋊C414yes—
232C24—42no—
233C24—41no—
234C24—41no—
235C24C23×C411yescharacteristic
236C24C23×C411yescharacteristic
237C24C23×C411yescharacteristic
238C24C22×D412yes—
239C24C2×#4112yes—
240C24C2×#4114yes—
241C24H(5,𝔽2)12yes—
242C242−1+412yes—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—28no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—84no—
316C22×C4—88no—
317C22×C4—84no—
318C22×C4C2×C4212yes—
319C22×C4C2×C4212yes—
320C22×C4C2×C4212yes—
321C22×C4C2×C4212yes—
322C22×C4C2×C4212yes—
323C22×C4C2×C4212yes—
324C22×C4C2×C22⋊C414yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4#11712yes—
331C22×C4#11712yes—
332C22×C4C4×D414yes—
333C22×C4C4×D414yes—
334C22×C4C4×D414yes—
335C22×C4C4×D414yes—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—416no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—84no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—28no—
348C22⋊C4—24no—
349C22⋊C4—28no—
350C22⋊C4—24no—
351C22⋊C4—28no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4#11718yes—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—416no—
363C42—44no—
364C42—44no—
365C42—82no—
366C42—82no—
367C42—816no—
368C23—22no—
369C23—24no—
370C23—21no—
371C23—22no—
372C23—21no—
373C23—22no—
374C23—44no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23C22×C4211yescharacteristic
387C23C22×C22⋊C411yescharacteristic
388C23C2×#11711yescharacteristic
389C23C2×#11712yes—
390C23C2×C4×D412yes—
391C23C2×C4×D412yes—
392C23C4×#4112yes—
393C23C4×#4112yes—
394C23#51811yescharacteristic
395C23#51911yescharacteristic
396C23#52012yes—
397C23#52012yes—
398C23#54614yes—
399C23#54714yes—
400C23#55114yes—
401C23#55214yes—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—84no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4C4×C22⋊C412yes—
479C2×C4C4×C22⋊C412yes—
480C2×C4C4×C22⋊C414yes—
481C2×C4#41014yes—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—21no—
490C22—24no—
491C22—22no—
492C22—22no—
493C22—24no—
494C22—24no—
495C22—22no—
496C22—22no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22C2×C4×C22⋊C411yescharacteristic
506C22C4×#11711yescharacteristic
507C22#227411yescharacteristic
508C22#233912yes—
509C22#234212yes—
510C22#234312yes—
511C22#234412yes—
512C22#235914yes—
513C22C2×#41012yes—
514C22#288311yescharacteristic
515C22#288411yescharacteristic
516C4—24no—
517C4—22no—
518C4—24no—
519C4—22no—
520C4—48no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—44no—
527C4—84no—
528C4—88no—
529C4—82no—
530C4—82no—
531C2—22no—
532C2—24no—
533C2—22no—
534C2—22no—
535C2—21no—
536C2—21no—
537C2#1379911yescharacteristic
538C2C4×#41011yescharacteristic
539C2#2039311yescharacteristic
540C2#2247712yes—
541C2#2247812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, prime radical, cosocleC25C24
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. C2×#387C4semidirect
15. C2×#387C4semidirect
17. C2×#383C4semidirect
18. C2×#382C4semidirect
21. C2×C4×C4⋊C4C4semidirect
24. C2×#376C4semidirect
26. #1820C4semidirect
27. #1740C4semidirect
163. C2×C4⋊C4C42semidirect
164. C2×C4⋊C4C42semidirect
208. C2×C42C42semidirect
209. C2×C42C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23432
C2×#18202
C2×#17402
#137992
#109122
#108012
#107362
C4×#3422