Finite Group Databasebeta

← #195192

Subgroups — #195192

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2#34380C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic
6#9044C218yesmaximal, complemented
7#8012C211yesmaximal, characteristic
8#7915C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#483—24no—
12C2×#483—22no—
13C2×#483C418yescomplemented
14C2×#483C2214yes—
15C22×#106—21no—
16C22×#103—21no—
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2378—22no—
20#2360—22no—
21#2344—21no—
22#2342—22no—
23Q8×C22⋊C4—21no—
24#2329C2214yes—
25#2323—22no—
26#2316—21no—
27#2313—21no—
28#2280C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1824—21no—
32#1824—21no—
33#1823C2214yes—
34#1736—22no—
35#1508—24no—
36C2×#328C2214yes—
37#1445C2214yes—
38C4⋊#95—22no—
39C22⋊#95—22no—
40#1294C2214yes—
41#1280C2214yes—
42C23×Q8D411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—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—
56C2×#117—22no—
57C2×#117D412yes—
58C22×C4⋊C4—21no—
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×C414yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D412yes—
74C22×C42—21no—
75#483—216no—
76#483—28no—
77#483C2×C4116yes—
78C2×#106—28no—
79C2×#106—24no—
80C2×#103—28no—
81C2×#103—24no—
82#407—22no—
83#407—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8C2×C414yes—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90#391—22no—
91#391—24no—
92#389—24no—
93#389—22no—
94#389—24no—
95#388—24no—
96#388—24no—
97#387—24no—
98#387—24no—
99#387—24no—
100#385—24no—
101#381—22no—
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—21no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—24no—
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#328—28no—
125C25C2×D411yescharacteristic
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—44no—
129C22×Q8C22⋊C414yes—
130C22×Q8C2×D412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
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×D411yescharacteristic
145C23×C4C2×D414yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
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—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8C22⋊C418yes—
166C22⋊Q8C22×C414yes—
167C22⋊Q8C2×D414yes—
168C4×Q8—84no—
169#117—44no—
170#117—84no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C22⋊C414yes—
200C2×C4⋊C4C22×C414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—44no—
222C2×C42—42no—
223#106—216no—
224#103—216no—
225C22⋊C8—24no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—42no—
230C22⋊C8C22×C414yes—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—88no—
243C24—24no—
244C24—22no—
245C24—22no—
246C24—42no—
247C24C4×D412yes—
248C24#12011yescharacteristic
249C24#12011yescharacteristic
250C24#12112yes—
251C24C22×D411yescharacteristic
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—44no—
255C2×Q8—48no—
256C2×Q8—44no—
257C2×Q8—84no—
258C2×Q8C2×C22⋊C414yes—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—28no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—88no—
293C22×C4—82no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—88no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315M16—84no—
316C2×C8—24no—
317C2×C8—44no—
318C2×C8—48no—
319C2×C8—44no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—28no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—82no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4C2×C22⋊C414yes—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—28no—
342C22⋊C4—48no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—44no—
350C42—84no—
351C42—84no—
352C42—84no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—22no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—22no—
365C23—24no—
366C23—42no—
367C23—42no—
368C23—48no—
369C23—48no—
370C23#38612yes—
371C23#38612yes—
372C23#45512yes—
373C23#45612yes—
374C23C2×C4×D411yescharacteristic
375C23C2×#12011yescharacteristic
376C23C2×#12111yescharacteristic
377C23#57714yes—
378Q8—44no—
379Q8—48no—
380Q8—82no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—84no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—88no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—84no—
425C2×C4—88no—
426C2×C4#38612yes—
427C2×C4#38612yes—
428C2×C4#45814yes—
429C2×C4C22×C22⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C8—44no—
434C8—82no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—48no—
449C22—48no—
450C22—42no—
451C22—42no—
452C22#188214yes—
453C22C2×#38611yescharacteristic
454C22#294912yes—
455C22#301012yes—
456C22C2×#45511yescharacteristic
457C22C2×#45611yescharacteristic
458C22C2×#45812yes—
459C22#304312yes—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—22no—
465C4—44no—
466C4—44no—
467C4—84no—
468C4—82no—
469C4—84no—
470C4—82no—
471C4—84no—
472C4#189014yes—
473C2—22no—
474C2—24no—
475C2—22no—
476C2—42no—
477C2#1826612yes—
478C2#1826812yes—
479C2#2093611yescharacteristic
480C2C2×#188211yescharacteristic
481C2C2×#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

Maximal subgroups

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

subgroupindex
C22×#4832
#343802
#207002
C2×#15082
#139242
#90442
#80122
#79152