Finite Group Databasebeta

← #236054

Subgroups — #236054

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic
4#21385C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10075C214yesmaximal, complemented
8#10059C214yesmaximal, complemented
9#7901C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8—21no—
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3229—24no—
15#3229—24no—
16C2×#498C2212yes—
17C2×#487C2212yes—
18C2×#483—22no—
19#3078—24no—
20#3078—24no—
21C2×#479—22no—
22C2×#477C2212yes—
23C2×#475—22no—
24C2×#475C2212yes—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#428—21no—
28C2×#423C2211yescharacteristic
29C2×#419—21no—
30#2546—22no—
31#2546—22no—
32#2546—22no—
33#2546C2214yes—
34C2×#398—22no—
35C2×#391—21no—
36C2×#389—22no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1987C2212yes—
39#1981C2212yes—
40#1872—22no—
41C4×#103—22no—
42#1546—24no—
43C2×#332C2214yes—
44#1479—24no—
45C2×Q8⋊C8C2214yes—
46#1279C2214yes—
47C23×Q8—21no—
48C23×Q8D411yescharacteristic
49C22×M16—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533D414yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—44no—
63C2×C4⋊Q8—44no—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2312yes—
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—21no—
78C2×C4×Q8—22no—
79C2×C4×Q8D412yes—
80C2×C4×Q8C2312yes—
81C2×#117—21no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4D412yes—
87C22×C4⋊C4D411yescharacteristic
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#498—24no—
92#498—24no—
93#487—24no—
94#487—24no—
95#483—216no—
96#479—216no—
97#477—24no—
98#477—24no—
99#475—216no—
100#475—24no—
101#475—24no—
102#431—22no—
103#431—22no—
104#431—22no—
105#431D414yes—
106#428—28no—
107C2×#107—22no—
108C2×#106—24no—
109C2×#106—22no—
110#423—22no—
111#423D414yes—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116#419—28no—
117C2×#103—22no—
118C2×#103—22no—
119C2×#103—24no—
120C2×#103—24no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2312yes—
125#398—28no—
126#398—28no—
127#391—24no—
128#391—24no—
129#389—28no—
130#389—28no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D412yes—
134C4×C4⋊C4C2312yes—
135C2×#95—24no—
136C2×#95—21no—
137#340—24no—
138#332—28no—
139Q8⋊C8—28no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8SD1614yes—
157C22×Q8C2×D412yes—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×M16—24no—
166C2×M16—42no—
167C2×M16—42no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—22no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—48no—
184C4⋊Q8—416no—
185C4⋊Q8—48no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C4⋊Q8—42no—
189C4⋊Q8—42no—
190C4⋊Q8C2×D412yes—
191C4⋊Q8C2×D412yes—
192#125—24no—
193#125—42no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8—42no—
212#117—24no—
213#117—42no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—41no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—41no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42#4112yes—
258C2×C42C2411yescharacteristic
259#107—28no—
260#106—28no—
261#106—216no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8C2×D414yes—
268#103—28no—
269#103—28no—
270#103—216no—
271#103—48no—
272C22⋊C8—24no—
273C22⋊C8—42no—
274C4×C8—22no—
275C4×C8—42no—
276C4×C8C2×D414yes—
277#95—216no—
278#95—22no—
279#95—22no—
280C24#12011yescharacteristic
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—28no—
290C2×Q8—28no—
291C2×Q8—44no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—42no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8C2×SD1614yes—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4#12011yescharacteristic
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12712yes—
341C22×C4C2×SD1612yes—
342C22×C4#13512yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4#13712yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348M16—48no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—44no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—44no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—432no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4#13718yes—
382C22⋊C4—44no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—44no—
388C42—44no—
389C42—44no—
390C42#12012yes—
391C42#12012yes—
392C42#12112yes—
393C42#12112yes—
394C42#12112yes—
395C42#12712yes—
396C42C22×D411yescharacteristic
397C42C22×D411yescharacteristic
398C42C22×D411yescharacteristic
399C42C2×#4111yescharacteristic
400C23—22no—
401C23—22no—
402C23#39212yes—
403C23#45012yes—
404C23#45212yes—
405C23C2×#12011yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—44no—
409Q8—42no—
410Q8—48no—
411Q8—42no—
412Q8—44no—
413Q8—82no—
414Q8—82no—
415Q8—82no—
416Q8—82no—
417Q8—88no—
418Q8—82no—
419Q8—82no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—416no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—88no—
446C2×C4#39212yes—
447C2×C4#39212yes—
448C2×C4#39212yes—
449C2×C4#44814yes—
450C2×C4#44914yes—
451C2×C4#45012yes—
452C2×C4#45212yes—
453C2×C4#46514yes—
454C2×C4#46814yes—
455C2×C4C2×#12011yescharacteristic
456C2×C4C2×#12011yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12111yescharacteristic
460C2×C4C2×#12711yescharacteristic
461C2×C4C22×SD1611yescharacteristic
462C2×C4C2×#13511yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4C2×#13714yes—
465C2×C4C2×#13711yescharacteristic
466C8—44no—
467C8—44no—
468C8—82no—
469C22—24no—
470C22—24no—
471C22—22no—
472C22—22no—
473C22#200014yes—
474C22C2×#39211yescharacteristic
475C22C2×#45011yescharacteristic
476C22C2×#45211yescharacteristic
477C22#300412yes—
478C22#302012yes—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—22no—
483C4—44no—
484C4—48no—
485C4—84no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4#200014yes—
490C4C2×#44811yescharacteristic
491C4C2×#44911yescharacteristic
492C4#303912yes—
493C4C2×#46511yescharacteristic
494C4C2×#46811yescharacteristic
495C4#315212yes—
496C2—24no—
497C2#1876112yes—
498C2#1877712yes—
499C2C2×#200011yescharacteristic
500C2#2340311yescharacteristic
501C2#2343211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10075C2semidirect
8. #10059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C2×#30782
C2×#25462
#213852
C2×#15462
C2×#14792
#100752
#100592
#79012