Finite Group Databasebeta

← #195160

Subgroups — #195160

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2#34358C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic, complemented
4C2×#1510C211yesmaximal, characteristic
5#14342C211yesmaximal, characteristic, complemented
6#9045C218yesmaximal, complemented
7#8024C211yesmaximal, characteristic, complemented
8#7913C211yesmaximal, 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#2364—22no—
21#2342—22no—
22D4×C22⋊C4—22no—
23#2333—22no—
24#2325C2214yes—
25#2313—21no—
26#2308—21no—
27#2280C2211yescharacteristic
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30#1824C2214yescomplemented
31#1823—22no—
32#1736—22no—
33#1510—24no—
34C2×#327C2214yes—
35#1445—22no—
36#1445C2214yescomplemented
37C22⋊#95—22no—
38#1294C2214yescomplemented
39#1280C2214yescomplemented
40C23×D4—21no—
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C22×M16—21no—
45C23×C8—21no—
46C2×#121—21no—
47C2×#121—21no—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121C2×C414yes—
51C2×#121D414yes—
52C2×#121D412yes—
53C2×#121C2312yes—
54C2×C4×D4—24no—
55C2×#117D412yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yescomplemented
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4D412yes—
71#481—216no—
72#481—28no—
73#481C2×C4116yescomplemented
74C2×#106—28no—
75C2×#106—24no—
76C2×#102—28no—
77C2×#102—24no—
78#407—22no—
79#407—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8C2×C414yescomplemented
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86#390—24no—
87#388—24no—
88#388—24no—
89#387—24no—
90#387—24no—
91#386—24no—
92#386—24no—
93#386—24no—
94#386—24no—
95#385—24no—
96#381—21no—
97#381—21no—
98#381—21no—
99#381—21no—
100#381—21no—
101#381—22no—
102#381—22no—
103#380—22no—
104#380D412yes—
105#380C2312yes—
106#379—22no—
107#379—21no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C2×#95—21no—
113C2×#95—28no—
114C2×#95D412yes—
115C2×#95C2312yes—
116C2×#95C2312yes—
117#340—28no—
118#327—28no—
119C25—21no—
120C25C2×D411yescharacteristic
121C22×D4—22no—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—24no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—44no—
128C22×D4—84no—
129C22×D4C22⋊C414yes—
130C22×D4C2×D412yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—44no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—42no—
140C23×C4—84no—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C2×M16—44no—
153C2×M16—42no—
154C22×C8—24no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—44no—
158#121—24no—
159#121—24no—
160#121—24no—
161#121—24no—
162#121—24no—
163#121—22no—
164#121—22no—
165#121C22⋊C418yes—
166#121C22×C414yes—
167#121C2×D414yes—
168C4×D4—88no—
169#117—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C22⋊C414yes—
191C2×C4⋊C4C22×C414yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—84no—
210C2×C42—24no—
211C2×C42—22no—
212#106—216no—
213#102—216no—
214C22⋊C8—24no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—42no—
219C22⋊C8C22×C414yes—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—42no—
231#95—88no—
232C24—24no—
233C24—24no—
234C24—22no—
235C24—24no—
236C24—42no—
237C24—44no—
238C24—82no—
239C24—82no—
240C24C4×D412yes—
241C24#12011yescharacteristic
242C24#12011yescharacteristic
243C24#12112yes—
244C24C22×D411yescharacteristic
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—28no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—48no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—48no—
254C2×D4—84no—
255C2×D4—82no—
256C2×D4—84no—
257C2×D4—88no—
258C2×D4—88no—
259C2×D4C2×C22⋊C414yes—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—82no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4—88no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313M16—84no—
314C2×C8—24no—
315C2×C8—48no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—42no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—84no—
329C4⋊C4C2×C22⋊C414yes—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C42—82no—
343C42—84no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—416no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—48no—
364C23—84no—
365C23—84no—
366C23—168no—
367C23#38612yes—
368C23#38612yes—
369C23#45814yes—
370C23C2×C4×D411yescharacteristic
371C23C2×#12011yescharacteristic
372C23C2×#12111yescharacteristic
373C23#57714yes—
374D4—44no—
375D4—48no—
376D4—48no—
377D4—82no—
378D4—84no—
379D4—88no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—88no—
411C2×C4—88no—
412C2×C4—84no—
413C2×C4—88no—
414C2×C4—82no—
415C2×C4—88no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—168no—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4C22×C22⋊C411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12311yescharacteristic
425C8—44no—
426C8—82no—
427C22—28no—
428C22—28no—
429C22—22no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—48no—
439C22—48no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22—84no—
445C22—84no—
446C22—168no—
447C22#189014yes—
448C22C2×#38611yescharacteristic
449C22#294912yes—
450C22#300912yes—
451C22#302112yes—
452C22C2×#45812yes—
453C22#304212yes—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—22no—
459C4—22no—
460C4—44no—
461C4—84no—
462C4—84no—
463C4—84no—
464C4—162no—
465C2—22no—
466C2—24no—
467C2—22no—
468C2—42no—
469C2—84no—
470C2#1823314yes—
471C2#2093311yescharacteristic
472C2#2094511yescharacteristic
473C2C2×#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. #14342C2semidirect
6. #9045C2semidirect
7. #8024C2semidirect
8. #7913C2semidirect
13. C2×#481C4semidirect
30. #1824C22semidirect
36. #1445C22semidirect
38. #1294C22semidirect
39. #1280C22semidirect
64. C22×C4⋊C4C2×C4semidirect
73. #481C2×C4semidirect
83. 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
#343582
#207002
C2×#15102
#143422
#90452
#80242
#79132