Finite Group Databasebeta

← #192433

Subgroups — #192433

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic
3#21759C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#13908C211yesmaximal, characteristic
6#8868C214yesmaximal, complemented
7#8867C214yesmaximal, complemented
8C2×#1294C211yesmaximal, characteristic
9#7907C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2368—22no—
14#2360—22no—
15#2345—22no—
16#2345—21no—
17Q8×C22⋊C4—21no—
18#2329C2214yes—
19#2315—21no—
20C4×C22⋊Q8—22no—
21C2×#380—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#2273—21no—
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1892—22no—
28#1872C2214yes—
29#1852—22no—
30#1852C2214yes—
31#1824—21no—
32#1808—22no—
33#1787—21no—
34C2×#328C418yescomplemented
35C2×#328C2212yes—
36C2×#328C2212yes—
37#1444C2212yes—
38#1442C2212yes—
39#1436—22no—
40C22⋊#95—21no—
41#1294C2212yes—
42#1294C2212yes—
43#1280C2214yes—
44C23×Q8D411yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—21no—
48C23×C8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C414yes—
54C2×C22⋊Q8D412yes—
55C2×C22⋊Q8Q814yes—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C42D412yes—
69C2×#107—24no—
70C2×#103—24no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8C2×C414yes—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79#391—22no—
80#391—24no—
81#389—22no—
82#389—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#385—24no—
87#385—24no—
88#380—22no—
89#380—22no—
90#380—21no—
91#380—21no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—44no—
97#380D412yes—
98#380C2312yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95C2×C414yes—
117C2×#95C2312yes—
118C2×#95C2311yescharacteristic
119C2×#95C2311yescharacteristic
120#340—24no—
121#340—24no—
122#328C2×C418yescomplemented
123#328C2×C418yescomplemented
124C25C2×D411yescharacteristic
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—82no—
128C22×Q8C4⋊C414yescomplemented
129C22×Q8C2×D412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—44no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C22×C412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×Q812yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—82no—
159C22×C8—82no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8C4⋊C418yescomplemented
168C22⋊Q8C22×C414yes—
169C22⋊Q8C2×Q814yes—
170C4×Q8—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—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C4⋊C414yescomplemented
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D414yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4Q1614yes—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—82no—
215#107—84no—
216#103—84no—
217C22⋊C8—24no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—42no—
222C22⋊C8C22×C414yes—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—44no—
235#95—44no—
236#95—41no—
237#95—41no—
238#95—82no—
239#95C22×C412yes—
240#95C22×C412yes—
241C24—42no—
242C24—42no—
243C24C4×D412yes—
244C24#12011yescharacteristic
245C24#12011yescharacteristic
246C24C22⋊Q812yes—
247C24C2×Q1612yes—
248C24#13512yes—
249C24AGL(1,ℤ/8)12yes—
250C24#13712yes—
251C24C22×D411yescharacteristic
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8C2×C4⋊C414yes—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—48no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—48no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—84no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—88no—
309C22×C4—88no—
310C22×C4C2×C4⋊C412yes—
311C22×C4C2×C4⋊C412yes—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4C22⋊Q812yes—
318C22×C4C22⋊Q812yes—
319C22×C4C22⋊Q812yes—
320C22×C4#12312yes—
321C22×C4#12312yes—
322C22×C4C2×Q1614yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q811yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C2×C8—24no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—42no—
333C2×C8—84no—
334C2×C8—84no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—88no—
347C4⋊C4C2×C4⋊C414yes—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—28no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C42—44no—
361C42—82no—
362C42—84no—
363C42—84no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—44no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23#38512yes—
379C23#38512yes—
380C23C4×Q1614yes—
381C23#44014yes—
382C23#44912yes—
383C23#44912yes—
384C23#45112yes—
385C23#45112yes—
386C23#47714yes—
387C23#47914yes—
388C23C2×C4×D411yescharacteristic
389C23C2×#12011yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23C22×Q1611yescharacteristic
392C23C2×#13511yescharacteristic
393C23C2×AGL(1,ℤ/8)11yescharacteristic
394C23C2×#13711yescharacteristic
395Q8—44no—
396Q8—48no—
397Q8—82no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
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—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—816no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—164no—
454C2×C4#38512yes—
455C2×C4#38512yes—
456C2×C4C22×C4⋊C411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C22⋊Q811yescharacteristic
459C2×C4C2×#12311yescharacteristic
460C8—44no—
461C8—82no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—22no—
466C22—21no—
467C22—22no—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—24no—
477C22—24no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22#186314yes—
483C22#186314yes—
484C22C2×#38511yescharacteristic
485C22C2×C4×Q1611yescharacteristic
486C22C2×#44011yescharacteristic
487C22C2×#45111yescharacteristic
488C22C2×#44911yescharacteristic
489C22#301812yes—
490C22#302612yes—
491C22C2×#47711yescharacteristic
492C22C2×#47911yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—44no—
500C4—44no—
501C4—44no—
502C4—44no—
503C4—82no—
504C4—82no—
505C4—84no—
506C4—162no—
507C2—21no—
508C2—21no—
509C2—21no—
510C2—24no—
511C2—24no—
512C2—21no—
513C2#1838312yes—
514C2#1838412yes—
515C2#2086311yescharacteristic
516C2C2×#186311yescharacteristic
517C2#2088911yescharacteristic
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. #8868C2semidirect
7. #8867C2semidirect
34. C2×#328C4semidirect
122. #328C2×C4semidirect
123. #328C2×C4semidirect
128. C22×Q8C4⋊C4semidirect
167. C22⋊Q8C4⋊C4semidirect
187. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342122
#219402
#217592
C22×#3282
#139082
#88682
#88672
C2×#12942
#79072