Finite Group Databasebeta

← #238191

Subgroups — #238191

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21400C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5#10357C212yesmaximal, complemented
6#10355C212yesmaximal, complemented
7#10350C212yesmaximal, complemented
8#10349C212yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10#7897C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#533C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857—22no—
18#2659—22no—
19#2552—21no—
20#2552C2212yes—
21#2551—21no—
22#2551C2212yes—
23#2550—22no—
24#2550—22no—
25#2316—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1989—22no—
32#1988C2212yes—
33#1987C2212yes—
34#1987C2212yes—
35#1981C2212yes—
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1892—22no—
39#1892—22no—
40#1803—21no—
41#1803—21no—
42C4×#103—22no—
43#1736—21no—
44#1736—21no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×M16—21no—
52C23×C8—21no—
53C23×C8—21no—
54#544—44no—
55#544—44no—
56#533—22no—
57#533—22no—
58#533—22no—
59#533Q814yescomplemented
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—22no—
66C2×#125—22no—
67C2×#125—21no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#423—22no—
81#423—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#398—24no—
98#398—24no—
99#395—24no—
100#395—24no—
101#391—24no—
102#389—22no—
103#389—22no—
104#389—24no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#384—22no—
110#384D411yescharacteristic
111#384D411yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#382—21no—
115#382—21no—
116#380—24no—
117#380—21no—
118#380—21no—
119#380D412yes—
120#380D412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2312yes—
129C2×#95—21no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138C22×Q8—24no—
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—42no—
147C2×M16—42no—
148C2×M16—42no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—81no—
158C22×C8—81no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—44no—
166C4⋊Q8—48no—
167C4⋊Q8—44no—
168C4⋊Q8—82no—
169C4⋊Q8—82no—
170C4⋊Q8C2×Q812yes—
171C4⋊Q8C2×Q812yes—
172#125—24no—
173#125—44no—
174#125—48no—
175#125—44no—
176#125—42no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—416no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—42no—
185C4×Q8—42no—
186#117—24no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—42no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4SD1614yes—
216C2×C4⋊C4Q1614yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—41no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—41no—
231C2×C42—81no—
232C2×C42—81no—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×Q812yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42C2411yescharacteristic
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247#103—44no—
248#103—44no—
249#103—82no—
250#103—82no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—42no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264C24#12011yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—48no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—84no—
277C2×Q8—84no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4C22⋊Q812yes—
302C22×C4C22⋊Q812yes—
303C22×C4C22⋊Q812yes—
304C22×C4#12412yes—
305C22×C4#12412yes—
306C22×C4#12412yes—
307C22×C4C2×SD1612yes—
308C22×C4C2×Q1612yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—84no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—48no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—84no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4C2×SD1614yes—
367C4⋊C4C2×Q1614yes—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—44no—
376C42—44no—
377C42—82no—
378C42C22⋊Q812yes—
379C42C22⋊Q812yes—
380C42C22⋊Q812yes—
381C42#12412yes—
382C42#12412yes—
383C42#12412yes—
384C42C22×Q811yescharacteristic
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#39312yes—
393C23#45012yes—
394C23#45112yes—
395C23C2×#12011yescharacteristic
396Q8—44no—
397Q8—42no—
398Q8—42no—
399Q8—82no—
400Q8—81no—
401Q8—81no—
402Q8—168no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
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—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4#39312yes—
442C2×C4#39312yes—
443C2×C4#39312yes—
444C2×C4#45012yes—
445C2×C4#45112yes—
446C2×C4#47514yes—
447C2×C4#47714yes—
448C2×C4#48714yes—
449C2×C4#48914yes—
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×#12411yescharacteristic
454C2×C4C2×#12411yescharacteristic
455C2×C4C2×#12411yescharacteristic
456C2×C4C22×SD1611yescharacteristic
457C2×C4C22×Q1611yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57611yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57811yescharacteristic
464C8—42no—
465C8—42no—
466C8—82no—
467C8—82no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22#202414yes—
475C22C2×#39311yescharacteristic
476C22C2×#45011yescharacteristic
477C22C2×#45111yescharacteristic
478C22#301211yescharacteristic
479C22#301311yescharacteristic
480C22#301811yescharacteristic
481C22#301811yescharacteristic
482C4—22no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—44no—
487C4—44no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4#202414yes—
495C4C2×#47511yescharacteristic
496C4C2×#47711yescharacteristic
497C4#308311yescharacteristic
498C4#308411yescharacteristic
499C4C2×#48711yescharacteristic
500C4C2×#48911yescharacteristic
501C4#313611yescharacteristic
502C4#313711yescharacteristic
503C2—24no—
504C2#1866311yescharacteristic
505C2#1866511yescharacteristic
506C2#1867511yescharacteristic
507C2#1867711yescharacteristic
508C2C2×#202411yescharacteristic
509C2#2345811yescharacteristic
510C2#2346111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
5. #10357C2semidirect
6. #10355C2semidirect
7. #10350C2semidirect
8. #10349C2semidirect
59. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#218852
#214002
#213122
#103572
#103552
#103502
#103492
#80452
#78972
C2×#12792