Finite Group Databasebeta

← #193220

Subgroups — #193220

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21961C211yesmaximal, characteristic
3#21400C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#8929C212yesmaximal, complemented
7#8927C212yesmaximal, complemented
8#8926C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7897C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2370—22no—
18#2369C2212yes—
19#2365C2212yes—
20#2355—21no—
21#2355—22no—
22Q8×C4⋊C4—21no—
23#2329—22no—
24#2317—21no—
25C4×C22⋊Q8—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1949—22no—
35#1948C2212yes—
36#1947—21no—
37#1947—21no—
38#1947C2212yes—
39#1947C2212yes—
40#1947C2212yes—
41#1892—22no—
42#1808—22no—
43C4×#103—22no—
44#1736—21no—
45#1550C418yescomplemented
46C2×#333C2212yes—
47C2×#332C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×M16—21no—
53C23×C8—21no—
54C23×C8—21no—
55#533—22no—
56#533—22no—
57#533C2×C414yes—
58#533Q814yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8D412yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8D412yes—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#428—44no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#106—22no—
79#423—22no—
80#423C2×C414yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2×C414yes—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—24no—
95#391—22no—
96#389—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#387—24no—
101#387—24no—
102#387—24no—
103#385—24no—
104#384—22no—
105#384D411yescharacteristic
106#384D411yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—22no—
110#382—21no—
111#382—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380D412yes—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C4⋊C4C2312yes—
130C4×C22⋊C4—24no—
131C2×#95—21no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#333C2×C418yescomplemented
139#332C2×C418yescomplemented
140C22×Q8—21no—
141C22×Q8C2×D412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—24no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—81no—
159C22×C8—81no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C22×C412yes—
165C4⋊Q8C22×C412yes—
166C4⋊Q8C2×Q812yes—
167C4⋊Q8C2×Q812yes—
168#125—24no—
169#125—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8C4⋊C418yes—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—82no—
178C4×Q8—82no—
179C4×Q8C4⋊C418yes—
180#117—24no—
181#117—24no—
182#117—42no—
183#117—42no—
184#117—42no—
185#117—42no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4Q1614yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—41no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—41no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×Q812yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234#107—44no—
235#107—82no—
236#107—82no—
237#106—44no—
238C4⋊C8—24no—
239C4⋊C8—24no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8C22×C414yes—
243C4⋊C8C22×C414yes—
244#103—44no—
245#103—44no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—41no—
260#95—41no—
261C24#12011yescharacteristic
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8C2×C4⋊C414yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4C2×C4⋊C412yes—
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4C22⋊Q812yes—
299C22×C4C22⋊Q812yes—
300C22×C4#12312yes—
301C22×C4C2×Q1612yes—
302C22×C4#13512yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4#13712yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308M16—48no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—48no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—84no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4C2×C4⋊C414yes—
358C4⋊C4C2×Q1614yes—
359C22⋊C4—28no—
360C22⋊C4—44no—
361C22⋊C4—82no—
362C42—28no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—82no—
368C42—84no—
369C42—84no—
370C42—82no—
371C42C2×C4⋊C412yes—
372C42C4×D412yes—
373C42C4×D412yes—
374C42C22⋊Q812yes—
375C42C22⋊Q812yes—
376C42#12312yes—
377C42C23×C411yescharacteristic
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23#38512yes—
386C23#44912yes—
387C23#45112yes—
388C23C2×#12011yescharacteristic
389Q8—44no—
390Q8—42no—
391Q8—42no—
392Q8—42no—
393Q8—42no—
394Q8—82no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4#38512yes—
434C2×C4#38512yes—
435C2×C4#38512yes—
436C2×C4C4×Q1614yes—
437C2×C4#44014yes—
438C2×C4#44912yes—
439C2×C4#45112yes—
440C2×C4#47714yes—
441C2×C4#47914yes—
442C2×C4C22×C4⋊C411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C2×C4C22×Q1611yescharacteristic
449C2×C4C2×#13511yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57611yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57811yescharacteristic
456C8—42no—
457C8—42no—
458C8—44no—
459C8—44no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22#186314yes—
467C22C2×#38511yescharacteristic
468C22C2×#45111yescharacteristic
469C22C2×#44911yescharacteristic
470C22#301211yescharacteristic
471C22#301311yescharacteristic
472C22#301811yescharacteristic
473C22#301811yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—44no—
479C4—44no—
480C4—44no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4—161no—
485C4—161no—
486C4#186314yes—
487C4C2×C4×Q1611yescharacteristic
488C4C2×#44011yescharacteristic
489C4#294611yescharacteristic
490C4#294711yescharacteristic
491C4C2×#47711yescharacteristic
492C4C2×#47911yescharacteristic
493C4#308311yescharacteristic
494C4#308411yescharacteristic
495C2—24no—
496C2#1866511yescharacteristic
497C2#1866811yescharacteristic
498C2#1867711yescharacteristic
499C2#1868011yescharacteristic
500C2#2085511yescharacteristic
501C2C2×#186311yescharacteristic
502C2#2087811yescharacteristic
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. #8929C2semidirect
7. #8927C2semidirect
8. #8926C2semidirect
9. #8922C2semidirect
45. #1550C4semidirect
138. #333C2×C4semidirect
139. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219612
#214002
#211702
C2×#15502
#89292
#89272
#89262
#89222
#78972
C2×#12792