Finite Group Databasebeta

← #493031

Subgroups — #493031

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34199C211yesmaximal, characteristic
2C22×#352C212yesmaximal
3#20398C211yesmaximal, characteristic
4#20392C211yesmaximal, characteristic
5#12621C214yesmaximal, complemented
6C4×#352C214yesmaximal, complemented
7#12574C212yesmaximal
8C22×#123C2211yescharacteristic
9C2×#518C2211yescharacteristic
10C22×#117C411yescharacteristic, complemented
11C22×#100C2212yes—
12C22×#99C2212yes—
13#2873—21no—
14#2359—22no—
15#2359—22no—
16#2344—21no—
17#2344—21no—
18#2342—21no—
19#2342—21no—
20#2322C412yescomplemented
21#2322C2212yes—
22#2315C411yescharacteristic, complemented
23#2308C2211yescharacteristic
24C4×#123C412yescomplemented
25C4×#123C2212yes—
26C2×#381C411yescharacteristic, complemented
27#2273—21no—
28#2273C411yescharacteristic, complemented
29C2×#352C4116yescomplemented
30C2×#352C2214yes—
31C2×#352C2214yes—
32C4×#100—22no—
33C4×#100C2214yes—
34C4×#99—22no—
35C4×#99C2214yes—
36#1742—21no—
37C2×#342—21no—
38#1390C2218yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C22×M16—22no—
42C2×#123—21no—
43C2×#123—21no—
44C2×#123—24no—
45C2×#123C2×C418yes—
46C2×#123C2×C411yescharacteristic
47C2×#123C2×C411yescharacteristic
48C2×#123C2311yescharacteristic
49C2×#123C2311yescharacteristic
50#518—44no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4D412yes—
53C2×C4×D4D412yes—
54C2×C4×D4C2312yes—
55C2×#117—21no—
56C2×#117—22no—
57C2×#117—21no—
58C2×#117—44no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C412yescomplemented
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42C2×C412yescomplemented
73C2×#100—22no—
74C2×#100—24no—
75C2×#100C2×C418yes—
76C2×#100C2314yes—
77C2×#99—22no—
78C2×#99—24no—
79C2×#99C2×C418yes—
80C2×#99C2314yes—
81#405—44no—
82C4×M16—24no—
83#389—22no—
84#389—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#386—21no—
94#386—21no—
95#386C2×C412yes—
96#386C2312yes—
97#385—22no—
98#385—22no—
99#385—22no—
100#385C2×C412yescomplemented
101#385C2×C412yescomplemented
102#381—21no—
103#381—21no—
104#381—21no—
105#381C2×C412yescomplemented
106#381C2×C412yescomplemented
107#380—21no—
108#380—21no—
109#379—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—44no—
119C4×C22⋊C4C2×C412yescomplemented
120C4×C22⋊C4C2×C412yescomplemented
121C2×#95—22no—
122#352C2×C4116yescomplemented
123#352C2×C4116yescomplemented
124#343—24no—
125#342—24no—
126C25C22×C411yescharacteristic
127C25C2×D411yescharacteristic
128C22×D4—42no—
129C22×D4—42no—
130C22×D4C22×C411yescharacteristic
131C22×D4C22×C414yes—
132C22×D4C2×D411yescharacteristic
133C22×D4C2×D411yescharacteristic
134C22×D4#4114yes—
135C22×D4C2411yescharacteristic
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4C4214yescomplemented
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C2×M16—28no—
154C2×M16—24no—
155C2×M16—44no—
156C2×M16—42no—
157C22×C8—42no—
158#123—22no—
159#123—22no—
160#123—28no—
161#123—28no—
162#123—22no—
163#123—22no—
164#123C4214yescomplemented
165#123C4214yescomplemented
166#123C22×C414yes—
167#123C22×C414yes—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171C4×D4—44no—
172#117—24no—
173#117—42no—
174#117—44no—
175#117—48no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4C4214yescomplemented
212C2×C22⋊C4C22⋊C411yescharacteristic
213C2×C22⋊C4C22⋊C411yescharacteristic
214C2×C22⋊C4C22⋊C411yescharacteristic
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C411yescharacteristic
218C2×C22⋊C4C22×C411yescharacteristic
219C2×C22⋊C4C22×C411yescharacteristic
220C2×C22⋊C4C22×C412yes—
221C2×C22⋊C4C22×C412yes—
222C2×C22⋊C4C2×D412yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42C22⋊C412yes—
230C2×C42C2×D412yes—
231#100—28no—
232#100—24no—
233#100C22×C418yes—
234#99—28no—
235#99—24no—
236#99C22×C418yes—
237#97—44no—
238C4×C8—44no—
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—
248#95—22no—
249#95—44no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—82no—
255C24—82no—
256C24C2×C4212yes—
257C24C2×C22⋊C411yescharacteristic
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C411yescharacteristic
260C24#11712yes—
261C24C4×D414yes—
262C24C23×C411yescharacteristic
263C24C22×D411yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4C2×C4212yes—
277C2×D4#11712yes—
278C2×D4C4×D414yes—
279C2×D4C4×D414yes—
280C2×D4C23×C412yes—
281C2×D4C2×#4112yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—48no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—48no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—48no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—164no—
321C22×C4C2×C4212yes—
322C22×C4C2×C4212yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C4×D414yes—
329C22×C4C22×D411yescharacteristic
330M16—28no—
331M16—44no—
332M16—48no—
333C2×C8—44no—
334C2×C8—48no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—28no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—48no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4#9914yes—
369C22⋊C4C2×C4212yes—
370C22⋊C4C2×C4212yes—
371C42—24no—
372C42—42no—
373C42—44no—
374C42—44no—
375C42—82no—
376C42—82no—
377C42—164no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—84no—
396C23—84no—
397C23—82no—
398C23—82no—
399C23—84no—
400C23—84no—
401C23—162no—
402C23C4×C22⋊C412yes—
403C23C4×C22⋊C412yes—
404C23C4×C22⋊C412yes—
405C23C2×#9912yes—
406C23#41012yes—
407C23C22×C4211yescharacteristic
408C23C22×C22⋊C411yescharacteristic
409C23C2×#11711yescharacteristic
410C23C2×C4×D412yes—
411D4—42no—
412D4—42no—
413D4—44no—
414D4—44no—
415D4—42no—
416D4—42no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—48no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—48no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—88no—
457C2×C4—84no—
458C2×C4—162no—
459C2×C4—162no—
460C2×C4—162no—
461C2×C4—168no—
462C2×C4C4×C22⋊C412yes—
463C2×C4C2×#9914yes—
464C2×C4C2×C4×D412yes—
465C8—48no—
466C22—22no—
467C22—24no—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—21no—
472C22—22no—
473C22—42no—
474C22—42no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—41no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—84no—
487C22—81no—
488C22—81no—
489C22—84no—
490C22—84no—
491C22—84no—
492C22C4×#9914yes—
493C22C2×C4×C22⋊C411yescharacteristic
494C22C22×#9911yescharacteristic
495C22C2×#41011yescharacteristic
496C4—24no—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—88no—
508C4—82no—
509C2—21no—
510C2—21no—
511C2—24no—
512C2—42no—
513C2—41no—
514C2—42no—
515C2—41no—
516C2—42no—
517C2C2×C4×#9911yescharacteristic
518C2#3335012yes—
subgroupG/H
center, socleC22C22×#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12621C2semidirect
6. C4×#352C2semidirect
10. C22×#117C4semidirect
20. #2322C4semidirect
22. #2315C4semidirect
24. C4×#123C4semidirect
26. C2×#381C4semidirect
28. #2273C4semidirect
29. C2×#352C4semidirect
62. C22×C4⋊C4C2×C4semidirect
72. C22×C42C2×C4semidirect
100. #385C2×C4semidirect
101. #385C2×C4semidirect
105. #381C2×C4semidirect
106. #381C2×C4semidirect
119. C4×C22⋊C4C2×C4semidirect
120. C4×C22⋊C4C2×C4semidirect
122. #352C2×C4semidirect
123. #352C2×C4semidirect
144. C23×C4C42semidirect
164. #123C42semidirect
165. #123C42semidirect
211. C2×C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
#341992
C22×#3522
#203982
#203922
#126212
C4×#3522
#125742