Finite Group Databasebeta

← #197964

Subgroups — #197964

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#2974C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic, complemented
6#9227C214yesmaximal, complemented
7#9219C214yesmaximal, complemented
8#8044C211yesmaximal, characteristic, complemented
9#7879C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#474—24no—
13C2×#474—21no—
14C2×#474—21no—
15C2×#474C418yescomplemented
16C2×#474C2212yes—
17C2×#474C2212yes—
18#2974—24no—
19#2974—24no—
20C2×#436C2212yes—
21C2×#435—22no—
22C2×C8×D4C2212yes—
23C2×#432C2211yescharacteristic
24C2×#432C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C22×#107—21no—
27C22×C4⋊C8C2211yescharacteristic
28C22×#102—21no—
29C2×#407—21no—
30C22×C22⋊C8—21no—
31#2285C2211yescharacteristic
32#1981—22no—
33#1956C2214yescomplemented
34#1824—21no—
35#1802C2212yes—
36#1802C2212yes—
37#1787—21no—
38#1787—21no—
39#1786—21no—
40#1762—21no—
41#1762—21no—
42#1556—24no—
43C2×#332C2214yescomplemented
44#1296C2214yescomplemented
45#1277C2214yescomplemented
46C23×Q8—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C22×M16—22no—
50C23×C8—21no—
51C23×C8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×D4—21no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×C4×D4D414yes—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C42C2311yescharacteristic
76#474—28no—
77#474—28no—
78#474—24no—
79#474—24no—
80#474C2×C418yes—
81#474C2×C418yes—
82#436—24no—
83#436—24no—
84#435—216no—
85C8×D4—24no—
86C8×D4—24no—
87#432—22no—
88#432—22no—
89#432—22no—
90#432D414yes—
91#431—22no—
92#431—22no—
93C2×#107—28no—
94C2×#107—22no—
95C2×#107—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8C2×C414yes—
99C2×C4⋊C8D412yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2312yes—
103C2×#103—24no—
104C2×#103—24no—
105C2×#102—28no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110#407—28no—
111C2×C22⋊C8—28no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8C2312yes—
121#384C2312yes—
122#382—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127#340—22no—
128#340—22no—
129#332—28no—
130C25—21no—
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—41no—
135C22×Q8—84no—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—42no—
140C22×D4C22⋊C414yes—
141C22×D4C2×D411yescharacteristic
142C22×D4C2×D411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—41no—
151C23×C4—84no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C2×M16—28no—
156C2×M16—84no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—28no—
161C22×C8—22no—
162C22×C8—44no—
163C22×C8—84no—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8C22×C414yes—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—24no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4C22⋊C414yes—
187C4×D4C22⋊C414yes—
188C4×D4C22×C412yes—
189C4×D4C22×C412yes—
190C4×D4C2×D412yes—
191C4×D4C2×D412yes—
192#117—82no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C22⋊C414yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—24no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C42—41no—
217C2×C42C22×C412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42#4111yescharacteristic
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225#107—28no—
226#107—28no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8C22×C414yes—
234C4⋊C8C2×D414yes—
235#103—84no—
236#103—84no—
237#102—28no—
238#102—28no—
239#102—44no—
240#102—44no—
241C22⋊C8—28no—
242C22⋊C8—28no—
243C22⋊C8—22no—
244C22⋊C8—24no—
245C22⋊C8—22no—
246C22⋊C8—42no—
247#97—24no—
248#97—42no—
249C4×C8—22no—
250C4×C8—22no—
251#95—41no—
252#95—41no—
253C24—21no—
254C24—21no—
255C24—24no—
256C24—44no—
257C24—42no—
258C24—42no—
259C24—82no—
260C24#12111yescharacteristic
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—88no—
267C2×Q8—84no—
268C2×Q8—88no—
269C2×D4—28no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4C2×C22⋊C412yes—
280C2×D4C2×C22⋊C412yes—
281C2×D4#13714yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—88no—
302C22×C4—88no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—84no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C4×D412yes—
309C22×C4C4×D412yes—
310C22×C4#12012yes—
311C22×C4#12112yes—
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12312yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319M16—88no—
320C2×C8—24no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—48no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—88no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—82no—
346C4⋊C4C2×C22⋊C412yes—
347C4⋊C4C2×C22⋊C412yes—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—82no—
355C42—82no—
356C42C2×C22⋊C412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12012yes—
360C42#12112yes—
361C42#12312yes—
362C42C23×C411yescharacteristic
363C42C22×D411yescharacteristic
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C23—22no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—22no—
371C23—48no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—48no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—84no—
381C23—88no—
382C23—88no—
383C23#38612yes—
384C23#47112yes—
385C23#47312yes—
386C23C2×#12111yescharacteristic
387Q8—44no—
388Q8—44no—
389Q8—88no—
390Q8—84no—
391D4—22no—
392D4—22no—
393D4—22no—
394D4—24no—
395D4—48no—
396D4—42no—
397D4#41914yes—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#38612yes—
429C2×C4#38612yes—
430C2×C4#38612yes—
431C2×C4C22×C22⋊C411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C2×#12011yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#13712yes—
438C2×C4#57512yes—
439C2×C4#57614yes—
440C8—44no—
441C8—42no—
442C8—42no—
443C8—44no—
444C8—82no—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—24no—
450C22—44no—
451C22—44no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—816no—
458C22—82no—
459C22—82no—
460C22#190814yes—
461C22C2×#38611yescharacteristic
462C22#304112yes—
463C22#305612yes—
464C22C2×#47111yescharacteristic
465C22C2×#47311yescharacteristic
466C4—21no—
467C4—21no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—42no—
472C4—42no—
473C4—84no—
474C4—162no—
475C4C2×#41911yescharacteristic
476C4#289611yescharacteristic
477C4#294812yes—
478C4#300812yes—
479C4#305612yes—
480C2—24no—
481C2—44no—
482C2—82no—
483C2#1945712yes—
484C2#1945912yes—
485C2#2093411yescharacteristic
486C2#2097211yescharacteristic
487C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ 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
3. #21804C2semidirect
5. C2×#1556C2semidirect
6. #9227C2semidirect
7. #9219C2semidirect
8. #8044C2semidirect
9. #7879C2semidirect
15. C2×#474C4semidirect
33. #1956C22semidirect
43. C2×#332C22semidirect
44. #1296C22semidirect
45. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#29742
#218042
#207382
C2×#15562
#92272
#92192
#80442
#78792