Finite Group Databasebeta

← #481872

Subgroups — #481872

Order 512.

969 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#11818C214yesmaximal, complemented
5#11814C214yesmaximal, complemented
6#11723C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic
8C22×#117C2211yescharacteristic
9#2920—21no—
10C2×#407—21no—
11#2628—21no—
12#2628—21no—
13#2628—21no—
14#2628—21no—
15#2628C412yes—
16#2628C412yes—
17#2628C2212yes—
18#2628C2212yes—
19C2×#400C2212yes—
20C2×#399C412yes—
21C2×#395—21no—
22C2×#393—21no—
23C2×#389—21no—
24C2×#382C411yescharacteristic
25C2×#380—21no—
26#1822—21no—
27#1822C2212yes—
28#1821—21no—
29#1821C2212yes—
30#1742—21no—
31#1742C2212yes—
32#1464—21no—
33#1464C2212yes—
34#1463C2212yes—
35#1462—21no—
36#1458—21no—
37#1458C2212yes—
38#1353C22116yescomplemented
39C23×Q8—21no—
40C24×C4D411yescharacteristic
41C22×M16—22no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117—41no—
53C2×#117—41no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117D411yescharacteristic
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73#432—22no—
74#432—41no—
75#431—22no—
76#431—41no—
77#423—41no—
78#407—22no—
79#407—22no—
80#407—22no—
81#407—41no—
82#407—41no—
83C2×C22⋊C8—41no—
84C2×C22⋊C8—41no—
85C4×M16—41no—
86#400—22no—
87#400—22no—
88#400C2×C414yes—
89#400C2314yes—
90#399—22no—
91#399—22no—
92#399C2×C414yes—
93#399C2×C414yes—
94#395—22no—
95#395—22no—
96#395—22no—
97#395—22no—
98#393—22no—
99#393—22no—
100#393—22no—
101#393—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#382—21no—
107#382—21no—
108#382C2×C412yes—
109#382C2×C412yes—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95C2×C412yes—
118#343—22no—
119#343—22no—
120#343—22no—
121#343C2314yes—
122#324—22no—
123#324C2314yes—
124#323—22no—
125#323—22no—
126C25C22⋊C411yescharacteristic
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—84no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—42no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C2×M16—44no—
154C2×M16—44no—
155C22×C8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172#117—41no—
173#117—41no—
174#117—41no—
175#117—41no—
176#117—41no—
177#117—41no—
178#117—41no—
179#117—41no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—84no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2411yescharacteristic
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—84no—
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C22⋊C411yescharacteristic
217C2×C22⋊C4C22⋊C411yescharacteristic
218C2×C22⋊C4C22⋊C411yescharacteristic
219C2×C22⋊C4C22×C411yescharacteristic
220C2×C22⋊C4C22×C411yescharacteristic
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C22⋊C4C2×D411yescharacteristic
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C4⋊C8—42no—
240C4⋊C8—42no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#97—42no—
248C4×C8—42no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—24no—
253#95—22no—
254#95—24no—
255#95—24no—
256#95C22×C414yes—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—41no—
261C24—42no—
262C24—42no—
263C24C2×C22⋊C411yescharacteristic
264C24C2×C22⋊C411yescharacteristic
265C24C2×C22⋊C411yescharacteristic
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—88no—
271C2×Q8—88no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4#9911yescharacteristic
319C22×C4#9911yescharacteristic
320C22×C4#9911yescharacteristic
321C22×C4#9911yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4#13714yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335M16—44no—
336M16—44no—
337M16—84no—
338M16—84no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—84no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—84no—
369C22⋊C4—84no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C23—22no—
387C23—21no—
388C23—21no—
389C23—22no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—48no—
402C23—42no—
403C23—42no—
404C23—81no—
405C23—81no—
406C23—84no—
407C23C2×#9911yescharacteristic
408C23C2×#9911yescharacteristic
409C23#41912yes—
410C23C22×C22⋊C411yescharacteristic
411Q8—88no—
412Q8—88no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—88no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C2×#9911yescharacteristic
468C2×C4C2×#9911yescharacteristic
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9911yescharacteristic
471C2×C4#41912yes—
472C2×C4#42112yes—
473C2×C4C2×#13712yes—
474C8—84no—
475C8—84no—
476C22—24no—
477C22—24no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—22no—
485C22—42no—
486C22—42no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—81no—
493C22—84no—
494C22—84no—
495C22—81no—
496C22#212312yes—
497C22#213412yes—
498C22C22×#9911yescharacteristic
499C22C2×#41911yescharacteristic
500C22C2×#42111yescharacteristic
501C4—22no—
502C4—21no—
503C4—21no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—84no—
514C4—84no—
515C4#151112yes—
516C4#151112yes—
517C2—21no—
518C2—21no—
519C2—22no—
520C2—22no—
521C2—42no—
522C2—81no—
523C2C2×#151111yescharacteristic
524C2C2×#212311yescharacteristic
525C2C2×#213411yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11818C2semidirect
5. #11814C2semidirect
6. #11723C2semidirect
38. #1353C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#206942
#173572
#118182
#118142
#117232