Finite Group Databasebeta

← #116283

Subgroups — #116283

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#389—22no—
13C2×#384—21no—
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1990—21no—
18#1989—22no—
19#1988—21no—
20#1987—21no—
21#1956—21no—
22#1872—22no—
23#1803—22no—
24#1736—21no—
25#1736—21no—
26#1503—22no—
27#1503C414yescomplemented
28C2×#328—22no—
29#1444—22no—
30C4⋊#95—21no—
31#1303—22no—
32#1296—22no—
33#1279—22no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431C2×C412yes—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8D412yes—
77C2×#103—24no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#389—48no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—42no—
88#384—42no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4D412yes—
97C2×#95—24no—
98C2×#95—21no—
99C2×#95D412yes—
100#340—24no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#328—28no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8C4⋊C412yescomplemented
131C4⋊Q8C4⋊C412yescomplemented
132#125—22no—
133#125—22no—
134#125—41no—
135#125—41no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152#117—22no—
153#117—22no—
154#117—41no—
155#117—41no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4SD1612yes—
192C2×C4⋊C4Q1612yes—
193C2×C22⋊C4—41no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—44no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224#103—84no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C4×C8—22no—
230C4×C8—41no—
231C4×C8—41no—
232C4×C8—41no—
233#95—22no—
234#95—48no—
235#95—41no—
236#95—41no—
237#95C22⋊C414yes—
238C24#12011yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—81no—
246C2×Q8—81no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4C22⋊Q811yescharacteristic
286C22×C4C22⋊Q811yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4#12411yescharacteristic
290C22×C4#12411yescharacteristic
291C22×C4#12411yescharacteristic
292C22×C4#12711yescharacteristic
293C22×C4C2×SD1611yescharacteristic
294C22×C4C2×Q1611yescharacteristic
295C22×C4#13511yescharacteristic
296C22×C4#13511yescharacteristic
297C22×C4AGL(1,ℤ/8)11yescharacteristic
298C22×C4#13711yescharacteristic
299C22×C4#13711yescharacteristic
300C22×C4#13711yescharacteristic
301M16—81no—
302M16—81no—
303M16—81no—
304M16—81no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—81no—
313C2×C8—81no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—816no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—82no—
354C42—82no—
355C42C2×C4⋊C411yescharacteristic
356C42#11711yescharacteristic
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42#12111yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42C22⋊Q811yescharacteristic
363C42C22⋊Q811yescharacteristic
364C42C22⋊Q811yescharacteristic
365C42#12311yescharacteristic
366C42#12311yescharacteristic
367C42#12311yescharacteristic
368C42#12311yescharacteristic
369C42#12611yescharacteristic
370C42#12611yescharacteristic
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23#38611yescharacteristic
376C23#39211yescharacteristic
377C23#39311yescharacteristic
378C23#45011yescharacteristic
379C23#45111yescharacteristic
380C23#45211yescharacteristic
381C23#45211yescharacteristic
382Q8—44no—
383Q8—82no—
384Q8—82no—
385Q8—82no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#38011yescharacteristic
429C2×C4#38511yescharacteristic
430C2×C4#38511yescharacteristic
431C2×C4#38811yescharacteristic
432C2×C4#38811yescharacteristic
433C2×C4#39011yescharacteristic
434C2×C4#39411yescharacteristic
435C2×C4#39411yescharacteristic
436C2×C4#39611yescharacteristic
437C2×C4#39611yescharacteristic
438C2×C4#39711yescharacteristic
439C2×C4#39911yescharacteristic
440C2×C4C4×SD1611yescharacteristic
441C2×C4C4×Q1611yescharacteristic
442C2×C4#44011yescharacteristic
443C2×C4#44111yescharacteristic
444C2×C4#44811yescharacteristic
445C2×C4#44911yescharacteristic
446C2×C4#45111yescharacteristic
447C2×C4#45211yescharacteristic
448C2×C4#45312yes—
449C2×C4#45612yes—
450C2×C4#46111yescharacteristic
451C2×C4#46211yescharacteristic
452C2×C4#46311yescharacteristic
453C2×C4#46411yescharacteristic
454C2×C4#46511yescharacteristic
455C2×C4#46711yescharacteristic
456C2×C4#46811yescharacteristic
457C2×C4#47011yescharacteristic
458C2×C4#47511yescharacteristic
459C2×C4#47711yescharacteristic
460C2×C4#47911yescharacteristic
461C2×C4#47911yescharacteristic
462C2×C4#48711yescharacteristic
463C2×C4#48811yescharacteristic
464C2×C4#48911yescharacteristic
465C2×C4#49111yescharacteristic
466C8—82no—
467C8—82no—
468C8—82no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22#143911yescharacteristic
482C22#187711yescharacteristic
483C22#187811yescharacteristic
484C22#200011yescharacteristic
485C22#200111yescharacteristic
486C22#200211yescharacteristic
487C22#200511yescharacteristic
488C22#200811yescharacteristic
489C22#200911yescharacteristic
490C22#202411yescharacteristic
491C22#202611yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—48no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—162no—
504C4#185912yes—
505C4#186311yescharacteristic
506C4#186511yescharacteristic
507C4#193311yescharacteristic
508C4#193511yescharacteristic
509C4#203912yes—
510C4#204211yescharacteristic
511C4#204311yescharacteristic
512C4#206611yescharacteristic
513C4#206811yescharacteristic
514C2—21no—
515C2—21no—
516C2—21no—
517C2—21no—
518C2#1024611yescharacteristic
519C2#1024811yescharacteristic
520C2#1025111yescharacteristic
521C2#1025311yescharacteristic
522C2#1106711yescharacteristic
523C2#1107811yescharacteristic
524C2#1109311yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
27. #1503C4semidirect
130. C4⋊Q8C4⋊C4semidirect
131. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#218602
#213122
C2×#15032
#139102
C2×#13032
#80432
#78922