Finite Group Databasebeta

← #195675

Subgroups — #195675

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34235C211yesmaximal, characteristic
3#21811C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic
6#9059C214yesmaximal, complemented
7#9058C214yesmaximal, complemented
8#7904C211yesmaximal, characteristic
9#7895C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165C418yescomplemented
14C2×#501—22no—
15C2×#500C2212yes—
16C2×#498C2212yes—
17C2×#432C2211yescharacteristic
18C2×#428—22no—
19C22×C4⋊C8C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2365C2212yes—
22#2365C2212yes—
23#2360—22no—
24#2356—21no—
25#2351—21no—
26#2343—22no—
27#2336—22no—
28#2320—21no—
29C4×C22⋊Q8—22no—
30#2283C2211yescharacteristic
31#2276—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1988—21no—
34#1988—21no—
35#1988C2214yes—
36#1981C2214yes—
37#1960—21no—
38#1960—21no—
39#1787—22no—
40C2×#340—22no—
41#1552—24no—
42C2×#332C2212yes—
43C2×#332C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×M16—21no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125D412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#501—216no—
77#500—24no—
78#500C2×C418yescomplemented
79#498—24no—
80#498C2×C418yescomplemented
81#432—22no—
82#432—22no—
83#428—216no—
84C2×#107—24no—
85C2×#106—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103—24no—
92C2×#103—24no—
93C2×C22⋊C8—21no—
94C4×M16—22no—
95C4×M16C2×C414yes—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8C2312yes—
99#391—22no—
100#391—22no—
101#389—24no—
102#389—22no—
103#389—22no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#385—24no—
109#383—22no—
110#383—21no—
111#382—22no—
112#382D411yescharacteristic
113#382D411yescharacteristic
114#381—21no—
115#380D412yes—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—21no—
128#376—22no—
129#376—21no—
130C2×#95—21no—
131#340—24no—
132#340—24no—
133#332—24no—
134#332—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—24no—
138C22×Q8—22no—
139C22×Q8—41no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×Q8—82no—
143C23×C4—22no—
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—28no—
149C2×M16—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—44no—
154C22×C8—42no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8C22×C414yes—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C2×D412yes—
172C4⋊Q8C2×D412yes—
173#125—24no—
174#125C22⋊C418yes—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—84no—
186#117—24no—
187#117C22⋊C418yes—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—82no—
221C2×C42C22×C412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229#107—216no—
230#106—216no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—42no—
236#103—84no—
237#103—84no—
238C22⋊C8—42no—
239#97—24no—
240#97—42no—
241C4×C8—22no—
242C4×C8C22×C414yes—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—22no—
255#95—22no—
256C24#12011yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—88no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—88no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4C2×C22⋊C412yes—
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12312yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304M16—216no—
305C2×C8—28no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—84no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—84no—
335C4⋊C4C2×C22⋊C414yes—
336C4⋊C4C2×C22⋊C414yes—
337C4⋊C4#13714yes—
338C22⋊C4—28no—
339C22⋊C4—44no—
340C42—28no—
341C42—48no—
342C42—42no—
343C42—42no—
344C42—44no—
345C42—81no—
346C42—81no—
347C42—84no—
348C42C2×C22⋊C412yes—
349C42C4×D412yes—
350C42C4×D412yes—
351C42#12112yes—
352C42#12112yes—
353C42#12312yes—
354C42C23×C411yescharacteristic
355C42C22×D411yescharacteristic
356C42C2×#4111yescharacteristic
357C42C2×#4111yescharacteristic
358C23—22no—
359C23—21no—
360C23—21no—
361C23#38612yes—
362C23#45614yes—
363C23C2×#12011yescharacteristic
364Q8—44no—
365Q8—44no—
366Q8—42no—
367Q8—42no—
368Q8—44no—
369Q8—48no—
370Q8—88no—
371Q8—84no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—88no—
403C2×C4—88no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#38612yes—
407C2×C4#38612yes—
408C2×C4#38612yes—
409C2×C4C22×C22⋊C411yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C2×C4×D411yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12311yescharacteristic
415C2×C4C2×#13712yes—
416C2×C4#57512yes—
417C2×C4#57612yes—
418C2×C4#57712yes—
419C8—28no—
420C8—82no—
421C8—82no—
422C22—24no—
423C22—24no—
424C22—22no—
425C22—22no—
426C22#188614yes—
427C22C2×#38611yescharacteristic
428C22#300712yes—
429C22#300811yescharacteristic
430C22#301011yescharacteristic
431C22C2×#45612yes—
432C4—22no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—44no—
438C4—44no—
439C4—44no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—84no—
444C4C4×#13712yes—
445C4#294612yes—
446C4#303911yescharacteristic
447C4#304011yescharacteristic
448C4#304512yes—
449C2—24no—
450C2#1858712yes—
451C2#1858912yes—
452C2#2092711yescharacteristic
453C2#2093211yescharacteristic
454C2C2×#188611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #9059C2semidirect
7. #9058C2semidirect
13. #3165C4semidirect
78. #500C2×C4semidirect
80. #498C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342352
#218112
C2×#19882
C2×#15522
#90592
#90582
#79042
#78952