Finite Group Databasebeta

← #195208

Subgroups — #195208

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#9044C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic
8#7907C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#483—24no—
12C2×#483—22no—
13C2×#483C418yescomplemented
14C2×#483C2214yes—
15C22×#106—21no—
16C22×#103—21no—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2368—22no—
20#2360—22no—
21#2345—22no—
22#2345—21no—
23Q8×C22⋊C4—21no—
24#2329C2214yes—
25#2315—21no—
26C4×C22⋊Q8—22no—
27C2×#381—21no—
28C2×#381—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2273—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1823—22no—
34#1823C2214yes—
35C2×#340—22no—
36#1520—24no—
37C2×#328C2214yes—
38#1445C2214yes—
39C4⋊#95—22no—
40#1294C2214yes—
41#1280C2214yes—
42C23×Q8D411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8—21no—
46C23×C8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C414yes—
52C2×C22⋊Q8D414yes—
53C2×C22⋊Q8D412yes—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4D411yescharacteristic
75C22×C42D412yes—
76#483—216no—
77#483—28no—
78#483C2×C4116yescomplemented
79C2×#106—28no—
80C2×#106—24no—
81C2×#103—28no—
82C2×#103—24no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8C2×C414yes—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91#391—22no—
92#391—24no—
93#389—22no—
94#389—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#385—24no—
99#385—24no—
100#381—44no—
101#381—44no—
102#380—22no—
103#380—22no—
104#380—21no—
105#380—21no—
106#380—22no—
107#380—22no—
108#380D412yes—
109#380C2312yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—24no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95C2312yes—
124C2×#95C2312yes—
125#340—28no—
126#328—28no—
127C25C2×D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—44no—
131C22×Q8C22⋊C414yes—
132C22×Q8C2×D412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—44no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—24no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—44no—
163C22×C8—44no—
164C22×C8—42no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8C22⋊C418yes—
173C22⋊Q8C22×C414yes—
174C22⋊Q8C2×D414yes—
175C4×Q8—84no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4SD1614yes—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—44no—
233#106—216no—
234#103—216no—
235C22⋊C8—24no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—42no—
240C22⋊C8C22×C414yes—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—48no—
252#95—48no—
253C24—42no—
254C24—42no—
255C24C4×D412yes—
256C24#12011yescharacteristic
257C24#12011yescharacteristic
258C24#12112yes—
259C24C2×SD1612yes—
260C24#13512yes—
261C24AGL(1,ℤ/8)12yes—
262C24#13712yes—
263C24C22×D411yescharacteristic
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—44no—
267C2×Q8—48no—
268C2×Q8—44no—
269C2×Q8—84no—
270C2×Q8C2×C22⋊C414yes—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C4×D412yes—
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4#12312yes—
333C22×C4#12312yes—
334C22×C4C2×SD1614yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C2×C8—24no—
342C2×C8—44no—
343C2×C8—48no—
344C2×C8—44no—
345C2×C8—42no—
346C2×C8—84no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4C2×C22⋊C414yes—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—44no—
374C22⋊C4—84no—
375C42—44no—
376C42—82no—
377C42—84no—
378C42—84no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23#38612yes—
394C23#38612yes—
395C23C4×SD1614yes—
396C23#44014yes—
397C23#44812yes—
398C23#44812yes—
399C23#44912yes—
400C23#44912yes—
401C23#46014yes—
402C23#46414yes—
403C23C2×C4×D411yescharacteristic
404C23C2×#12011yescharacteristic
405C23C2×#12111yescharacteristic
406C23C22×SD1611yescharacteristic
407C23C2×#13511yescharacteristic
408C23C2×AGL(1,ℤ/8)11yescharacteristic
409C23C2×#13711yescharacteristic
410Q8—44no—
411Q8—48no—
412Q8—82no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—88no—
452C2×C4—88no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—164no—
464C2×C4#38612yes—
465C2×C4#38612yes—
466C2×C4#45814yes—
467C2×C4C22×C22⋊C411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12311yescharacteristic
471C8—44no—
472C8—82no—
473C22—22no—
474C22—22no—
475C22—24no—
476C22—24no—
477C22—22no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—22no—
485C22—21no—
486C22—22no—
487C22—21no—
488C22—22no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22#188014yes—
494C22#188014yes—
495C22C2×#38611yescharacteristic
496C22C2×C4×SD1611yescharacteristic
497C22C2×#44011yescharacteristic
498C22C2×#44811yescharacteristic
499C22C2×#44911yescharacteristic
500C22#302212yes—
501C22C2×#45812yes—
502C22C2×#46011yescharacteristic
503C22C2×#46411yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—22no—
509C4—44no—
510C4—44no—
511C4—44no—
512C4—82no—
513C4—84no—
514C4—84no—
515C4—162no—
516C4#189014yes—
517C2—21no—
518C2—24no—
519C2—24no—
520C2—21no—
521C2—21no—
522C2—21no—
523C2#1820412yes—
524C2#1820612yes—
525C2C2×#188011yescharacteristic
526C2#2094411yescharacteristic
527C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9044C2semidirect
13. C2×#483C4semidirect
78. #483C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
#342122
C2×#18232
C2×#15202
#139042
#90442
#80232
#79072