Finite Group Databasebeta

← #195191

Subgroups — #195191

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2#34379C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic, complemented
4C2×#1507C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic, complemented
6#9045C218yesmaximal, complemented
7#8012C211yesmaximal, characteristic, complemented
8#7915C211yesmaximal, characteristic, complemented
9C22×#121C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#481—24no—
12C2×#481—22no—
13C2×#481C418yescomplemented
14C2×#481C2214yes—
15C22×#106—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×#381—22no—
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1824C2214yescomplemented
33#1823—21no—
34#1823—21no—
35#1736—22no—
36#1507—24no—
37C2×#327C2214yes—
38#1445C2214yescomplemented
39C4⋊#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#481—216no—
77#481—28no—
78#481C2×C4116yescomplemented
79C2×#106—28no—
80C2×#106—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#381—48no—
104#380—22no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380D411yescharacteristic
110#380D411yescharacteristic
111#380C2312yes—
112#379—22no—
113#379—21no—
114C4×C4⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—28no—
121C2×#95C2312yes—
122C2×#95C2312yes—
123#340—24no—
124#340—24no—
125#327—28no—
126C25—21no—
127C25C2×D411yescharacteristic
128C22×D4—22no—
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—44no—
135C22×D4—84no—
136C22×D4C22⋊C414yes—
137C22×D4C2×D412yes—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—42no—
149C23×C4C22×C412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D414yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C2×M16—44no—
161C2×M16—42no—
162C22×C8—24no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—44no—
166#121—24no—
167#121—24no—
168#121—24no—
169#121—24no—
170#121—24no—
171#121—22no—
172#121—22no—
173#121C22⋊C418yes—
174#121C22×C414yes—
175#121C2×D414yes—
176C4×D4—84no—
177C4×D4—88no—
178#117—44no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C22⋊C414yes—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—84no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—44no—
228C2×C42—42no—
229#106—216no—
230#102—216no—
231C22⋊C8—24no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—42no—
236C22⋊C8C22×C414yes—
237#95—24no—
238#95—24no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—416no—
247C24—24no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—24no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—82no—
256C24—82no—
257C24C4×D412yes—
258C24#12011yescharacteristic
259C24#12011yescharacteristic
260C24#12112yes—
261C24C22×D411yescharacteristic
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—28no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—48no—
268C2×D4—44no—
269C2×D4—48no—
270C2×D4—44no—
271C2×D4—84no—
272C2×D4—84no—
273C2×D4—88no—
274C2×D4—88no—
275C2×D4C2×C22⋊C414yes—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—416no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—82no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—88no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C4×D412yes—
319C22×C4C4×D412yes—
320C22×C4C4×D412yes—
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12312yes—
327C22×C4#12312yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334M16—84no—
335C2×C8—24no—
336C2×C8—44no—
337C2×C8—48no—
338C2×C8—44no—
339C2×C8—42no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—82no—
352C4⋊C4—84no—
353C4⋊C4C2×C22⋊C414yes—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—28no—
357C22⋊C4—48no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C42—44no—
368C42—84no—
369C42—84no—
370C23—22no—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—24no—
383C23—24no—
384C23—42no—
385C23—42no—
386C23—48no—
387C23—48no—
388C23—44no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—48no—
393C23—84no—
394C23—84no—
395C23—164no—
396C23—164no—
397C23#38612yes—
398C23#38612yes—
399C23#45312yes—
400C23#45412yes—
401C23C2×C4×D411yescharacteristic
402C23C2×#12011yescharacteristic
403C23C2×#12111yescharacteristic
404C23#57714yes—
405D4—44no—
406D4—48no—
407D4—48no—
408D4—48no—
409D4—82no—
410D4—88no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—84no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—88no—
448C2×C4—88no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—88no—
453C2×C4—168no—
454C2×C4#38612yes—
455C2×C4#38612yes—
456C2×C4#45814yes—
457C2×C4C22×C22⋊C411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×#12111yescharacteristic
460C2×C4C2×#12311yescharacteristic
461C8—44no—
462C8—82no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—48no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—84no—
484C22—84no—
485C22—168no—
486C22#188314yes—
487C22C2×#38611yescharacteristic
488C22#294912yes—
489C22#300912yes—
490C22C2×#45311yescharacteristic
491C22C2×#45411yescharacteristic
492C22C2×#45812yes—
493C22#304212yes—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—22no—
499C4—44no—
500C4—84no—
501C4—84no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4#189014yes—
506C2—22no—
507C2—24no—
508C2—22no—
509C2—42no—
510C2—84no—
511C2#1826512yes—
512C2#1827112yes—
513C2#2093311yescharacteristic
514C2C2×#188311yescharacteristic
515C2C2×#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. #13904C2semidirect
6. #9045C2semidirect
7. #8012C2semidirect
8. #7915C2semidirect
13. C2×#481C4semidirect
32. #1824C22semidirect
38. #1445C22semidirect
40. #1294C22semidirect
41. #1280C22semidirect
67. C22×C4⋊C4C2×C4semidirect
78. #481C2×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×#4812
#343792
#207002
C2×#15072
#139042
#90452
#80122
#79152