Finite Group Databasebeta

← #188770

Subgroups — #188770

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2#17803C212yesmaximal, complemented
3#17357C211yesmaximal, characteristic
4C2×#1458C211yesmaximal, characteristic
5#8603C214yesmaximal, complemented
6#8602C212yesmaximal, complemented
7#8601C212yesmaximal, complemented
8#7863C212yesmaximal, complemented
9C22×#117C2211yescharacteristic
10#2924—22no—
11#2924—22no—
12#2924—24no—
13#2920C2212yes—
14C2×#433C412yes—
15C2×#433C2212yes—
16C2×#432C411yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C411yescharacteristic
19C2×#431C2212yes—
20C2×#431C2211yescharacteristic
21C2×#424C412yes—
22C2×#407C411yescharacteristic
23C22×C22⋊C8C411yescharacteristic
24C2×#405C2212yes—
25#1572—24no—
26#1571—22no—
27#1570—22no—
28#1564C2214yescomplemented
29#1564C2214yescomplemented
30#1464—21no—
31#1464—21no—
32#1463—21no—
33#1462—21no—
34#1458—21no—
35#1458—21no—
36#1458C2211yescharacteristic
37#1458C2212yes—
38#1458C2211yescharacteristic
39#1458C2212yes—
40#1458C2212yes—
41C2×#324—21no—
42C2×#323—21no—
43#1275C2214yescomplemented
44#1275C2214yescomplemented
45C24×C4C2×C411yescharacteristic
46C22×M16C2×C412yes—
47C23×C8C2×C412yes—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C2×#117Q811yescharacteristic
57C2×#117Q811yescharacteristic
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4Q811yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4Q811yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2311yescharacteristic
66#433—24no—
67#433—24no—
68#433—24no—
69#433—24no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#431—21no—
77#431—21no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431D412yes—
84#431D412yes—
85#431D414yes—
86#431C2314yes—
87#431C2312yes—
88#431C2312yes—
89#424—24no—
90#424—24no—
91#423—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C412yes—
94C2×C4⋊C8C2×C412yes—
95C2×C4⋊C8C2×C412yes—
96C2×C4⋊C8C2×C412yes—
97#407—22no—
98#407—22no—
99#407—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8C2×C411yescharacteristic
105C2×C22⋊C8C2×C412yes—
106C2×C22⋊C8C2×C411yescharacteristic
107C2×C22⋊C8C2×C412yes—
108C2×C22⋊C8C2×C411yescharacteristic
109C2×C22⋊C8C2×C411yescharacteristic
110#405—24no—
111#405—24no—
112C4×M16—22no—
113C2×#97C2×C412yes—
114C2×#97C2312yes—
115C2×C4×C8—22no—
116C2×C4×C8C2×C412yes—
117C2×C4×C8C2312yes—
118#335—24no—
119#335—24no—
120#334—24no—
121#334—24no—
122#324—22no—
123#324—22no—
124#324—21no—
125#324—21no—
126#324—22no—
127#323—22no—
128#323—22no—
129#323—21no—
130#323—21no—
131#323—22no—
132C25C4⋊C411yescharacteristic
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—42no—
139C23×C4C4⋊C411yescharacteristic
140C23×C4C4⋊C411yescharacteristic
141C23×C4C4⋊C411yescharacteristic
142C23×C4C4⋊C411yescharacteristic
143C23×C4C4⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×Q811yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—24no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8C22×C412yes—
159C22×C8C22×C412yes—
160C22×C8C22×C412yes—
161C22×C8C22×C412yes—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C4⋊C411yescharacteristic
183C2×C4⋊C4C4⋊C411yescharacteristic
184C2×C4⋊C4C4⋊C411yescharacteristic
185C2×C4⋊C4C4⋊C411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×Q811yescharacteristic
189C2×C4⋊C4C2×Q811yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C4⋊C411yescharacteristic
195C2×C22⋊C4C4⋊C411yescharacteristic
196C2×C22⋊C4C4⋊C411yescharacteristic
197C2×C22⋊C4C4⋊C411yescharacteristic
198C2×C22⋊C4C2×D411yescharacteristic
199C2×C22⋊C4C2×D411yescharacteristic
200C2×C22⋊C4C2×Q811yescharacteristic
201C2×C22⋊C4C2×Q811yescharacteristic
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C4⋊C411yescharacteristic
207C2×C42C4⋊C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q811yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8C22⋊C414yes—
228C4⋊C8C22⋊C414yes—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8C22⋊C411yescharacteristic
242C22⋊C8C22⋊C411yescharacteristic
243C22⋊C8C22⋊C412yes—
244C22⋊C8C22⋊C412yes—
245C22⋊C8C22⋊C412yes—
246#97—24no—
247#97—24no—
248#97—42no—
249C4×C8—22no—
250C4×C8—22no—
251C4×C8—24no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8—42no—
255C4×C8C2×D414yes—
256C4×C8C2×D414yes—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—41no—
261C24—42no—
262C24—81no—
263C24C2×C4⋊C411yescharacteristic
264C24C2×C4⋊C411yescharacteristic
265C24C2×C4⋊C411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C4⋊C411yescharacteristic
302C22×C4C2×C4⋊C411yescharacteristic
303C22×C4C2×C4⋊C411yescharacteristic
304C22×C4C2×C4⋊C411yescharacteristic
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4C2×C4⋊C411yescharacteristic
309C22×C4C2×C4⋊C411yescharacteristic
310C22×C4#11711yescharacteristic
311C22×C4#11711yescharacteristic
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4C22⋊Q812yes—
315C22×C4C22⋊Q812yes—
316C22×C4#13114yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×Q811yescharacteristic
320M16—48no—
321M16—84no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—84no—
338C2×C8C2×C22⋊C412yes—
339C2×C8C2×C22⋊C412yes—
340C2×C8C2×C22⋊C414yes—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42C2×C22⋊C412yes—
366C42#11712yes—
367C42#12112yes—
368C42#12112yes—
369C42C22⋊Q812yes—
370C42C22⋊Q812yes—
371C42C22×D412yes—
372C42C2×#4112yes—
373C23—22no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—81no—
391C23—81no—
392C23—84no—
393C23—84no—
394C23#38011yescharacteristic
395C23#38011yescharacteristic
396C23#42412yes—
397C23C22×C4⋊C411yescharacteristic
398C2×C4—24no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#38011yescharacteristic
440C2×C4#38011yescharacteristic
441C2×C4#38012yes—
442C2×C4#38012yes—
443C2×C4#42412yes—
444C2×C4C22×C22⋊C411yescharacteristic
445C2×C4C2×#11711yescharacteristic
446C2×C4C2×#12112yes—
447C2×C4C2×C22⋊Q812yes—
448C2×C4C2×#13112yes—
449C2×C4#56812yes—
450C8—22no—
451C8—22no—
452C8—24no—
453C8—42no—
454C8—42no—
455C8—84no—
456C8—82no—
457C8—82no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—21no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—84no—
476C22—81no—
477C22—81no—
478C22—84no—
479C22#181212yes—
480C22C2×#38011yescharacteristic
481C22C2×#42411yescharacteristic
482C22#290511yescharacteristic
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4#288012yes—
496C4#292412yes—
497C2—21no—
498C2—21no—
499C2—22no—
500C2—41no—
501C2—42no—
502C2—81no—
503C2#2062811yescharacteristic
504C2C2×#181211yescharacteristic
505C2#2067311yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#380
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#432 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17803C2semidirect
5. #8603C2semidirect
6. #8602C2semidirect
7. #8601C2semidirect
8. #7863C2semidirect
28. #1564C22semidirect
29. #1564C22semidirect
43. #1275C22semidirect
44. #1275C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
#178032
#173572
C2×#14582
#86032
#86022
#86012
#78632