Finite Group Databasebeta

← #239879

Subgroups — #239879

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#21884C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4#21169C211yesmaximal, characteristic, complemented
5C22×#332C211yesmaximal, characteristic, complemented
6C2×#1480C211yesmaximal, characteristic
7#10417C214yesmaximal, complemented
8#10416C214yesmaximal, complemented
9#7891C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#427—21no—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2862—21no—
17#2858—21no—
18#2640—22no—
19#2563—22no—
20#2547C2214yes—
21#2504—22no—
22#2436—22no—
23#2311—21no—
24#2311—21no—
25#2285C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1990C2214yes—
28#1988—21no—
29#1987—22no—
30#1982—21no—
31#1949—22no—
32#1948C2212yes—
33#1947C2212yes—
34#1893—22no—
35#1824—22no—
36#1802C2214yes—
37#1735—21no—
38C2×#332—24no—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1480—24no—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1279C2214yes—
45C22×#41—21no—
46C23×Q8—21no—
47C23×C8—21no—
48C23×C8—21no—
49#532—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8C2312yes—
62C2×#127D411yescharacteristic
63C2×#124—24no—
64C2×#124—21no—
65C2×#121—21no—
66C2×#121D412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2312yes—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42C2311yescharacteristic
75#431—22no—
76#431—22no—
77#427—44no—
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D414yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102—24no—
91C2×C22⋊C8—21no—
92C2×C4×C8C2312yes—
93#397—28no—
94#393—28no—
95#391—22no—
96#391—22no—
97#391—24no—
98#390—24no—
99#390—22no—
100#390—22no—
101#386—24no—
102#386—24no—
103#384C2312yes—
104#382—24no—
105#382—21no—
106#380—24no—
107#380—24no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116#376—21no—
117C43—21no—
118#340—24no—
119#340—28no—
120#340—22no—
121#340—22no—
122#332—24no—
123#332—24no—
124D4⋊C8—24no—
125D4⋊C8—24no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—82no—
138C22×Q8—82no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4C2×D412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—24no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—82no—
157C4⋊Q8—24no—
158C4⋊Q8—22no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169#127—22no—
170#127—22no—
171#124—42no—
172#124—42no—
173#124—84no—
174#124—84no—
175#121—24no—
176#121—24no—
177#121—42no—
178#121—42no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—42no—
184C4×D4—42no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—41no—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—44no—
227#106—44no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—22no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8C2×D412yes—
236C4⋊C8C2×D412yes—
237#102—84no—
238C22⋊C8—42no—
239C4×C8—22no—
240C4×C8—22no—
241#95—28no—
242#95—28no—
243#95—22no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24—22no—
248C24#12011yescharacteristic
249#41—84no—
250#41—84no—
251#41—84no—
252#41—84no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—84no—
262C2×Q8—88no—
263C2×Q8—84no—
264C2×Q8—84no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—84no—
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12312yes—
300C22×C4#12312yes—
301C22×C4#12412yes—
302C22×C4#13512yes—
303C22×C4#13512yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4#13712yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—48no—
316C2×C8—44no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—44no—
320C2×C8—84no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C22⋊C4—82no—
347C42—44no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—82no—
353C42#12112yes—
354C42#12112yes—
355C42#12112yes—
356C42#12312yes—
357C42#12312yes—
358C42#12412yes—
359C42C22×D411yescharacteristic
360C42C2×#4111yescharacteristic
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C23—22no—
364C23—22no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—44no—
370C23#39412yes—
371C23#44912yes—
372C23#45212yes—
373C23C2×#12011yescharacteristic
374Q8—88no—
375Q8—84no—
376Q8—82no—
377Q8—82no—
378Q8—84no—
379D4—44no—
380D4—42no—
381D4—42no—
382D4—84no—
383D4—88no—
384D4—84no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—88no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#39412yes—
413C2×C4#39412yes—
414C2×C4#39412yes—
415C2×C4#44912yes—
416C2×C4#45212yes—
417C2×C4#46314yes—
418C2×C4#46414yes—
419C2×C4#48814yes—
420C2×C4#48914yes—
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12311yescharacteristic
425C2×C4C2×#12311yescharacteristic
426C2×C4C2×#12411yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
430C2×C4C2×#13711yescharacteristic
431C2×C4#57612yes—
432C2×C4#57712yes—
433C8—42no—
434C8—42no—
435C8—82no—
436C8—82no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—24no—
441C22—82no—
442C22—82no—
443C22—84no—
444C22—84no—
445C22—82no—
446C22#204214yes—
447C22C2×#39411yescharacteristic
448C22C2×#44911yescharacteristic
449C22C2×#45211yescharacteristic
450C22#301212yes—
451C22#302012yes—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—84no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—162no—
461C4#204214yes—
462C4C2×#46311yescharacteristic
463C4C2×#46411yescharacteristic
464C4#305711yescharacteristic
465C4#305811yescharacteristic
466C4#309612yes—
467C4C2×#48811yescharacteristic
468C4C2×#48911yescharacteristic
469C2—24no—
470C2—162no—
471C2#1888812yes—
472C2#1888912yes—
473C2#2348611yescharacteristic
474C2#2349411yescharacteristic
475C2C2×#204211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×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
3. #21834C2semidirect
4. #21169C2semidirect
5. C22×#332C2semidirect
7. #10417C2semidirect
8. #10416C2semidirect
9. #7891C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#218842
#218342
#211692
C22×#3322
C2×#14802
#104172
#104162
#78912