Finite Group Databasebeta

← #195190

Subgroups — #195190

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2#34379C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic, complemented
4C2×#1507C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic, complemented
6#9043C218yesmaximal, complemented
7#8012C211yesmaximal, characteristic, complemented
8#7915C211yesmaximal, characteristic, complemented
9C22×#121C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#480—24no—
12C2×#480—22no—
13C2×#480C418yescomplemented
14C2×#480C2214yes—
15C22×#107—21no—
16C22×#102—21no—
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2377—22no—
20#2361—22no—
21#2345—21no—
22D4×C22⋊C4—21no—
23D4×C22⋊C4—22no—
24#2325C2214yes—
25#2324—22no—
26#2315—21no—
27#2312—21no—
28#2280C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1824—21no—
32#1824—21no—
33#1823C2214yescomplemented
34#1736—22no—
35#1507—24no—
36C2×#327C2214yes—
37#1445C2214yescomplemented
38C4⋊#95—22no—
39C22⋊#95—22no—
40#1294C2214yescomplemented
41#1280C2214yescomplemented
42C23×D4—21no—
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C22×M16—21no—
47C23×C8—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121C2×C414yes—
53C2×#121D414yes—
54C2×#121D412yes—
55C2×#121C2312yes—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×#117D412yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C414yescomplemented
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76#480—216no—
77#480—28no—
78#480C2×C4116yescomplemented
79C2×#107—28no—
80C2×#107—24no—
81C2×#102—28no—
82C2×#102—24no—
83#407—22no—
84#407—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8C2×C414yescomplemented
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91#390—22no—
92#390—24no—
93#389—22no—
94#388—24no—
95#388—24no—
96#387—24no—
97#386—24no—
98#386—24no—
99#386—24no—
100#385—24no—
101#385—24no—
102#381—21no—
103#380—22no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380D411yescharacteristic
109#380D411yescharacteristic
110#380C2312yes—
111#379—22no—
112#379—21no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C2×#95—28no—
120C2×#95C2312yes—
121C2×#95C2312yes—
122#340—24no—
123#340—24no—
124#327—28no—
125C25—21no—
126C25C2×D411yescharacteristic
127C22×D4—22no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—44no—
134C22×D4—84no—
135C22×D4C22⋊C414yes—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—44no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—42no—
146C23×C4—84no—
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D414yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C2×M16—44no—
159C2×M16—42no—
160C22×C8—24no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—44no—
164#121—24no—
165#121—24no—
166#121—24no—
167#121—24no—
168#121—24no—
169#121—22no—
170#121—22no—
171#121C22⋊C418yes—
172#121C22×C414yes—
173#121C2×D414yes—
174C4×D4—84no—
175C4×D4—88no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22⋊C414yes—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—84no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—44no—
226C2×C42—42no—
227#107—216no—
228#102—216no—
229C22⋊C8—24no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—42no—
234C22⋊C8C22×C414yes—
235#95—24no—
236#95—24no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—88no—
245C24—24no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—24no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—82no—
254C24—82no—
255C24C4×D412yes—
256C24#12011yescharacteristic
257C24#12011yescharacteristic
258C24#12112yes—
259C24C22×D411yescharacteristic
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—28no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—48no—
266C2×D4—44no—
267C2×D4—48no—
268C2×D4—44no—
269C2×D4—84no—
270C2×D4—84no—
271C2×D4—88no—
272C2×D4—88no—
273C2×D4C2×C22⋊C414yes—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—88no—
304C22×C4—82no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—88no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12312yes—
322C22×C4#12312yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329M16—84no—
330C2×C8—24no—
331C2×C8—44no—
332C2×C8—48no—
333C2×C8—44no—
334C2×C8—42no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—82no—
347C4⋊C4—84no—
348C4⋊C4C2×C22⋊C414yes—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—28no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—84no—
360C22⋊C4—84no—
361C22⋊C4—84no—
362C42—44no—
363C42—84no—
364C42—84no—
365C23—22no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—22no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—24no—
378C23—24no—
379C23—42no—
380C23—42no—
381C23—48no—
382C23—48no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23—48no—
388C23—84no—
389C23—84no—
390C23—164no—
391C23—164no—
392C23#38612yes—
393C23#38612yes—
394C23#45512yes—
395C23#45612yes—
396C23C2×C4×D411yescharacteristic
397C23C2×#12011yescharacteristic
398C23C2×#12111yescharacteristic
399C23#57614yes—
400D4—44no—
401D4—48no—
402D4—48no—
403D4—48no—
404D4—82no—
405D4—88no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—84no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—88no—
442C2×C4—88no—
443C2×C4—84no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—88no—
447C2×C4—168no—
448C2×C4#38612yes—
449C2×C4#38612yes—
450C2×C4#45814yes—
451C2×C4C22×C22⋊C411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C8—44no—
456C8—82no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—22no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—48no—
471C22—48no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—84no—
478C22—84no—
479C22—168no—
480C22#188214yes—
481C22C2×#38611yescharacteristic
482C22#294812yes—
483C22#300812yes—
484C22C2×#45511yescharacteristic
485C22C2×#45611yescharacteristic
486C22C2×#45812yes—
487C22#304112yes—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—22no—
493C4—44no—
494C4—84no—
495C4—84no—
496C4—82no—
497C4—82no—
498C4—84no—
499C4#189014yes—
500C2—22no—
501C2—24no—
502C2—22no—
503C2—42no—
504C2—84no—
505C2#1826412yes—
506C2#1827012yes—
507C2#2093411yescharacteristic
508C2C2×#188211yescharacteristic
509C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
3. #20700C2semidirect
5. #13924C2semidirect
6. #9043C2semidirect
7. #8012C2semidirect
8. #7915C2semidirect
13. C2×#480C4semidirect
33. #1823C22semidirect
37. #1445C22semidirect
40. #1294C22semidirect
41. #1280C22semidirect
67. C22×C4⋊C4C2×C4semidirect
78. #480C2×C4semidirect
88. C2×C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
#343792
#207002
C2×#15072
#139242
#90432
#80122
#79152