Finite Group Databasebeta

← #117547

Subgroups — #117547

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21804C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3C2×#1548C211yesmaximal, characteristic
4#14155C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13C2×#391—22no—
14#2285C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1981—22no—
18#1956—21no—
19#1956—21no—
20#1949—22no—
21#1948—21no—
22#1947—21no—
23#1787—22no—
24#1736—21no—
25#1548—22no—
26#1548C414yescomplemented
27C2×#333—22no—
28#1445—22no—
29C4⋊#95—21no—
30C4⋊#95—21no—
31#1303—22no—
32#1294—22no—
33#1279—22no—
34C23×Q8D412yes—
35C22×M16—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#428—42no—
66#428—42no—
67C2×#107—22no—
68C2×#106—22no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8D412yes—
77C2×#103—24no—
78C2×#103—24no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#391—48no—
84#384D411yescharacteristic
85#384D411yescharacteristic
86#382—21no—
87#380—21no—
88#380—21no—
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⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C2×#95—24no—
100C2×#95—24no—
101C2×#95—21no—
102C2×#95D412yes—
103#340—22no—
104#340—22no—
105#333—28no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—44no—
112C22×Q8—44no—
113C22×Q8C4⋊C414yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—41no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146#117—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
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—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4SD1612yes—
183C2×C4⋊C4Q1612yes—
184C2×C22⋊C4—41no—
185C2×C42—44no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q811yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207#107—44no—
208#106—44no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—41no—
216C4⋊C8C22⋊C414yes—
217#103—84no—
218#103—84no—
219C22⋊C8—22no—
220C22⋊C8—41no—
221C22⋊C8—41no—
222C22⋊C8—41no—
223C4×C8—22no—
224C4×C8—41no—
225C4×C8—41no—
226C4×C8—41no—
227#95—22no—
228#95—22no—
229#95—48no—
230#95—48no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235C24#12011yescharacteristic
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—48no—
239C2×Q8—48no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—82no—
247C2×Q8—82no—
248C22×C4—22no—
249C22×C4—22no—
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—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4C2×C4⋊C411yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12411yescharacteristic
293C22×C4#12711yescharacteristic
294C22×C4C2×SD1611yescharacteristic
295C22×C4C2×Q1611yescharacteristic
296C22×C4#13511yescharacteristic
297C22×C4#13511yescharacteristic
298C22×C4AGL(1,ℤ/8)11yescharacteristic
299C22×C4#13711yescharacteristic
300C22×C4#13711yescharacteristic
301C22×C4#13711yescharacteristic
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—81no—
314C2×C8—81no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C22⋊C4—81no—
342C22⋊C4—81no—
343C42—82no—
344C42—82no—
345C42—81no—
346C42—81no—
347C42—81no—
348C42—81no—
349C42—81no—
350C42—81no—
351C42C2×C22⋊C411yescharacteristic
352C42#11711yescharacteristic
353C42C4×D411yescharacteristic
354C42C4×D411yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42C22⋊Q811yescharacteristic
358C42C22⋊Q811yescharacteristic
359C42C22⋊Q811yescharacteristic
360C42C22⋊Q811yescharacteristic
361C42#12311yescharacteristic
362C42#12311yescharacteristic
363C42#12411yescharacteristic
364C42#12411yescharacteristic
365C42#12611yescharacteristic
366C42#12611yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#38511yescharacteristic
372C23#39211yescharacteristic
373C23#39411yescharacteristic
374C23#44811yescharacteristic
375C23#44911yescharacteristic
376C23#45111yescharacteristic
377C23#45211yescharacteristic
378Q8—48no—
379Q8—84no—
380Q8—84no—
381Q8—84no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4#38011yescharacteristic
425C2×C4#38511yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38811yescharacteristic
428C2×C4#38811yescharacteristic
429C2×C4#39011yescharacteristic
430C2×C4#39311yescharacteristic
431C2×C4#39411yescharacteristic
432C2×C4#39611yescharacteristic
433C2×C4#39611yescharacteristic
434C2×C4#39711yescharacteristic
435C2×C4#39911yescharacteristic
436C2×C4C4×SD1611yescharacteristic
437C2×C4C4×Q1611yescharacteristic
438C2×C4#44011yescharacteristic
439C2×C4#44111yescharacteristic
440C2×C4#45011yescharacteristic
441C2×C4#45111yescharacteristic
442C2×C4#45211yescharacteristic
443C2×C4#45211yescharacteristic
444C2×C4#45312yes—
445C2×C4#45512yes—
446C2×C4#46111yescharacteristic
447C2×C4#46211yescharacteristic
448C2×C4#46311yescharacteristic
449C2×C4#46411yescharacteristic
450C2×C4#46511yescharacteristic
451C2×C4#46711yescharacteristic
452C2×C4#46811yescharacteristic
453C2×C4#47011yescharacteristic
454C2×C4#47511yescharacteristic
455C2×C4#47711yescharacteristic
456C2×C4#47911yescharacteristic
457C2×C4#47911yescharacteristic
458C2×C4#48711yescharacteristic
459C2×C4#48811yescharacteristic
460C2×C4#48911yescharacteristic
461C2×C4#49111yescharacteristic
462C8—44no—
463C8—82no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22#143911yescharacteristic
478C22#186311yescharacteristic
479C22#186511yescharacteristic
480C22#200011yescharacteristic
481C22#200111yescharacteristic
482C22#200211yescharacteristic
483C22#200411yescharacteristic
484C22#200911yescharacteristic
485C22#200911yescharacteristic
486C22#204211yescharacteristic
487C22#204311yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—84no—
495C4—82no—
496C4—82no—
497C4—84no—
498C4—161no—
499C4—161no—
500C4#185612yes—
501C4#187711yescharacteristic
502C4#187811yescharacteristic
503C4#193311yescharacteristic
504C4#193511yescharacteristic
505C4#202411yescharacteristic
506C4#202611yescharacteristic
507C4#203712yes—
508C4#206611yescharacteristic
509C4#206811yescharacteristic
510C2—21no—
511C2—21no—
512C2—21no—
513C2—21no—
514C2#1024011yescharacteristic
515C2#1024411yescharacteristic
516C2#1024811yescharacteristic
517C2#1025211yescharacteristic
518C2#1106711yescharacteristic
519C2#1113011yescharacteristic
520C2#1113511yescharacteristic
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
26. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#218042
#211702
C2×#15482
#141552
C2×#13032
#80142
#78922