Finite Group Databasebeta

← #188303

Subgroups — #188303

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic
2#21955C212yesmaximal, complemented
3C2×#1464C212yesmaximal
4#8560C212yesmaximal, complemented
5#8559C212yesmaximal
6#8558C212yesmaximal, complemented
7#8557C212yesmaximal
8#7912C212yesmaximal, complemented
9C22×#117C411yescharacteristic
10C22×#117C2211yescharacteristic
11C23×C4⋊C4C411yescharacteristic
12C2×#432C2214yes—
13#2274—28no—
14C2×C4×C4⋊C4C411yescharacteristic
15C2×C4×C4⋊C4C411yescharacteristic
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C411yescharacteristic
23C2×C4×C22⋊C4C411yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1940—28no—
27#1797C2214yes—
28#1797C2214yes—
29#1464—28no—
30C2×#324C2214yes—
31C2×#323C2214yes—
32#1280C2214yes—
33#1280C2214yes—
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—24no—
40C2×#117—24no—
41C2×#117—24no—
42C2×#117D414yes—
43C2×#117Q814yes—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#432—28no—
72#432—28no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C22⋊C8C2318yes—
77C2×#97—24no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4D418yes—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—24no—
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2311yescharacteristic
97C2×#95C2311yescharacteristic
98#336—24no—
99#336—24no—
100#324—28no—
101#323—28no—
102C25C22×C411yescharacteristic
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—81no—
115C23×C4—81no—
116C23×C4—81no—
117C23×C4—81no—
118C23×C4C4⋊C414yes—
119C23×C4M1614yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—28no—
135#117—232no—
136#117—216no—
137#117—216no—
138C2×C4⋊C4—232no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
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⋊C4C22⋊C414yes—
159C2×C4⋊C4C22⋊C414yes—
160C2×C4⋊C4C4⋊C414yes—
161C2×C4⋊C4M1614yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×Q812yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4C4⋊C414yes—
174C2×C22⋊C4M1614yes—
175C2×C22⋊C4C2×D412yes—
176C2×C22⋊C4C2×Q812yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—81no—
191C2×C42—81no—
192C2×C42—81no—
193C2×C42—81no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42—81no—
197C2×C42—81no—
198C2×C42C4⋊C414yes—
199C2×C42M1614yes—
200C2×C42C2×D414yes—
201C2×C42C2×D414yes—
202C2×C42C2×D412yes—
203C2×C42C2×D414yes—
204C2×C42C2×Q812yes—
205C2×C42#4114yes—
206C4⋊C8—48no—
207C4⋊C8—84no—
208C4⋊C8—84no—
209C22⋊C8—28no—
210C22⋊C8—28no—
211#97—48no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95C22⋊C414yes—
219#95C22⋊C414yes—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24C2×C4⋊C412yes—
229C24C2×M1612yes—
230C24C2×M1612yes—
231C24C23×C411yescharacteristic
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—28no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C4⋊C412yes—
304C22×C4C2×C4⋊C412yes—
305C22×C4C2×C4⋊C412yes—
306C22×C4C2×C4⋊C412yes—
307C22×C4C2×C4⋊C412yes—
308C22×C4#11712yes—
309C22×C4#11712yes—
310C22×C4C2×M1612yes—
311C22×C4C2×M1612yes—
312C22×C4C2×M1612yes—
313C22×C4C2×M1612yes—
314C22×C4C2×M1612yes—
315C22×C4C2×M1612yes—
316C22×C4C2×M1612yes—
317C22×C4C2×M1612yes—
318C22×C4C2×M1612yes—
319C22×C4C2×M1612yes—
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×Q811yescharacteristic
323C22×C4C2×#4112yes—
324C2×C8—48no—
325C2×C8—48no—
326C2×C8—48no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—28no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—28no—
340C22⋊C4—28no—
341C42—28no—
342C42—28no—
343C42—28no—
344C42—44no—
345C42—44no—
346C42—82no—
347C42—82no—
348C42—82no—
349C42—82no—
350C42—82no—
351C42—82no—
352C42#12118yes—
353C42C22⋊Q818yes—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23#42314yes—
381C23C22×C22⋊C411yescharacteristic
382C23C22×C4⋊C411yescharacteristic
383C23C2×#11711yescharacteristic
384C23C22×M1611yescharacteristic
385C23C22×M1611yescharacteristic
386C23C22×M1611yescharacteristic
387C23C22×M1611yescharacteristic
388C2×C4—232no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—88no—
459C2×C4—88no—
460C2×C4#38014yes—
461C2×C4#38014yes—
462C2×C4#40714yes—
463C2×C4#40714yes—
464C2×C4#42314yes—
465C2×C4C2×#12112yes—
466C2×C4C2×C22⋊Q812yes—
467C8—88no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—24no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22C2×#38011yescharacteristic
492C22C2×#40711yescharacteristic
493C22C2×#40711yescharacteristic
494C22#288311yescharacteristic
495C22#288411yescharacteristic
496C22#288811yescharacteristic
497C22#288911yescharacteristic
498C22C2×#42311yescharacteristic
499C22C2×#42311yescharacteristic
500C22#292011yescharacteristic
501C22#292011yescharacteristic
502C4—24no—
503C4—24no—
504C4—22no—
505C4—22no—
506C4—24no—
507C4—24no—
508C4—24no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—42no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—44no—
522C2#1736711yescharacteristic
523C2#1736811yescharacteristic
524C2#1737111yescharacteristic
525C2#1737211yescharacteristic
526C2#2062611yescharacteristic
527C2#2063411yescharacteristic
528C2#2064011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21955C2semidirect
4. #8560C2semidirect
6. #8558C2semidirect
8. #7912C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
#219552
C2×#14642
#85602
#85592
#85582
#85572
#79122