Finite Group Databasebeta

← #196031

Subgroups — #196031

Order 512.

1330 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
1C2×#2983C211yesmaximal, characteristic, complemented
2C2×#2323C211yesmaximal, characteristic
3#21822C211yesmaximal, characteristic, complemented
4C2×#1464C212yesmaximal
5#9086C212yesmaximal, complemented
6#9084C212yesmaximal
7#9082C212yesmaximal, complemented
8#9080C212yesmaximal
9#7912C212yesmaximal, complemented
10C22×C22⋊Q8C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13#2983—28no—
14C2×#446—24no—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C2×#432C2212yes—
18C2×#431C2212yes—
19#2323—24no—
20#2323—24no—
21C2×#391C411yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#389C411yescharacteristic
25C2×#388C411yescharacteristic
26C2×#388C411yescharacteristic
27C2×#388C2211yescharacteristic
28C2×#388C2211yescharacteristic
29C2×#387C2211yescharacteristic
30C2×#387C2211yescharacteristic
31C2×#380C411yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1985—24no—
34#1817C2212yes—
35#1817C2212yes—
36#1464—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39#1280C2214yes—
40#1280C2214yes—
41C23×Q8C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8D414yes—
49C2×C4×Q8D414yes—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—24no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#446—232no—
78#432—24no—
79#432—28no—
80#432—24no—
81#432—24no—
82#432D418yes—
83#431—28no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8C2314yes—
89C2×C22⋊C8C2314yes—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×C4×C8—22no—
94#391—24no—
95#389—24no—
96#389—24no—
97#389—24no—
98#388—24no—
99#388—24no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#380—24no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2311yescharacteristic
120C2×#95C2311yescharacteristic
121#336—24no—
122#336—24no—
123#324—28no—
124#323—28no—
125C25C22×C411yescharacteristic
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8C22⋊C414yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—42no—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—24no—
151C22×C8—24no—
152C22⋊Q8—232no—
153C4×Q8—28no—
154C4×Q8—28no—
155C4×Q8—28no—
156C4×Q8—28no—
157#117—44no—
158#117—44no—
159#117—44no—
160#117—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—81no—
183C2×C4⋊C4—81no—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C22⋊C414yes—
187C2×C4⋊C4M1614yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—81no—
202C2×C22⋊C4—81no—
203C2×C22⋊C4M1614yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—81no—
213C2×C42—81no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D414yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42#4114yes—
222C4⋊C8—28no—
223C4⋊C8—28no—
224C4⋊C8—28no—
225C4⋊C8—44no—
226C22⋊C8—24no—
227C22⋊C8—24no—
228C22⋊C8—24no—
229C22⋊C8C2×D418yes—
230#97—28no—
231#97—28no—
232#97—44no—
233C4×C8—28no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24C2×M1612yes—
255C24C23×C411yescharacteristic
256C2×Q8—28no—
257C2×Q8—28no—
258C2×Q8—44no—
259C2×Q8—48no—
260C2×Q8—48no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—28no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—48no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C4×D414yes—
320C22×C4C4×D414yes—
321C22×C4C2×M1612yes—
322C22×C4C2×M1612yes—
323C22×C4C2×M1612yes—
324C22×C4C2×M1612yes—
325C22×C4C2×M1612yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C2×#4112yes—
330C2×C8—28no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—44no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—44no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—41no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—44no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C42—22no—
358C42—22no—
359C42—24no—
360C42—24no—
361C42—22no—
362C42—22no—
363C42—44no—
364C42—82no—
365C42—82no—
366C42#12014yes—
367C42#12114yes—
368C42#12114yes—
369C42#12314yes—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—24no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23C22×C22⋊C411yescharacteristic
392C23C2×C4×D412yes—
393C23C22×M1611yescharacteristic
394C23C22×M1611yescharacteristic
395C23#56812yes—
396Q8—416no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
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—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—416no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
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—41no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4#38614yes—
460C2×C4#38614yes—
461C2×C4#40714yes—
462C2×C4C2×#12011yescharacteristic
463C2×C4C2×#12112yes—
464C2×C4C2×#12311yescharacteristic
465C8—48no—
466C8—84no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—22no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—84no—
479C22—84no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22C2×#38611yescharacteristic
485C22C2×#40711yescharacteristic
486C22#287911yescharacteristic
487C22#288411yescharacteristic
488C22#288411yescharacteristic
489C22#288811yescharacteristic
490C22#288911yescharacteristic
491C22D4×M1612yes—
492C22D4×M1612yes—
493C4—24no—
494C4—24no—
495C4—24no—
496C4—21no—
497C4—22no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C2—22no—
510C2—22no—
511C2—82no—
512C2#1737111yescharacteristic
513C2#1737211yescharacteristic
514C2#1740911yescharacteristic
515C2#1741011yescharacteristic
516C2#2089811yescharacteristic
517C2#2098611yescharacteristic
518C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2983C2semidirect
3. #21822C2semidirect
5. #9086C2semidirect
7. #9082C2semidirect
9. #7912C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
C2×#23232
#218222
C2×#14642
#90862
#90842
#90822
#90802
#79122