Finite Group Databasebeta

← #118610

Subgroups — #118610

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#10851C211yesmaximal, characteristic
6#10849C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1992—22no—
16#1990—21no—
17#1990—21no—
18#1987—21no—
19#1981—21no—
20#1948—22no—
21#1892—22no—
22#1853—22no—
23#1844—21no—
24#1834—21no—
25#1803—21no—
26#1803—21no—
27#1736—22no—
28C2×#340—22no—
29C2×#339—22no—
30#1503—22no—
31#1503C414yescomplemented
32C2×#328—22no—
33#1283—22no—
34C22×M16D412yes—
35C23×C8D412yes—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431C2×C412yes—
62#431D412yes—
63C2×#106—24no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×#103—22no—
72C2×#103—22no—
73C2×C22⋊C8D412yes—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384D411yescharacteristic
85#384D411yescharacteristic
86#384D412yes—
87#382—21no—
88#380—21no—
89#380—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—
96C2×#95—21no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—44no—
103#339—44no—
104#328—28no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16SD1614yes—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—42no—
124C22×C8—42no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C4⋊C412yes—
132C4⋊Q8C4⋊C412yes—
133#125—22no—
134#125—22no—
135#125—41no—
136#125—41no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—22no—
154#117—22no—
155#117—41no—
156#117—41no—
157#117—42no—
158#117—42no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—82no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C42—22no—
191C2×C42—41no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×Q811yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208#106—84no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215#103—82no—
216#103—82no—
217C22⋊C8—22no—
218C22⋊C8C22⋊C414yes—
219#97—41no—
220#97—41no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—41no—
224C4×C8—41no—
225#95—22no—
226#95—41no—
227#95—41no—
228#95—41no—
229C24#12711yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—81no—
239C2×Q8—81no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4#12411yescharacteristic
280C22×C4C2×SD1612yes—
281C22×C4#13512yes—
282C22×C4#13712yes—
283C22×C4#13712yes—
284M16—24no—
285M16—24no—
286M16—42no—
287M16—42no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—44no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—82no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—44no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—82no—
327C4⋊C4—81no—
328C4⋊C4—81no—
329C4⋊C4—82no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C42—42no—
335C42—42no—
336C42—44no—
337C42—82no—
338C42—82no—
339C42C2×C4⋊C411yescharacteristic
340C42#11711yescharacteristic
341C42C4×D411yescharacteristic
342C42C4×D411yescharacteristic
343C42#12111yescharacteristic
344C42#12111yescharacteristic
345C42#12111yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42C22⋊Q811yescharacteristic
348C42C22⋊Q811yescharacteristic
349C42#12311yescharacteristic
350C42#12311yescharacteristic
351C42#12411yescharacteristic
352C42#12411yescharacteristic
353C42#12611yescharacteristic
354C42#12611yescharacteristic
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23#39011yescharacteristic
360C23#39211yescharacteristic
361C23#39911yescharacteristic
362C23#49211yescharacteristic
363C23#49511yescharacteristic
364C23#49711yescharacteristic
365C23#49711yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—82no—
369Q8—82no—
370Q8—81no—
371Q8—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—162no—
413C2×C4—162no—
414C2×C4#38011yescharacteristic
415C2×C4#38511yescharacteristic
416C2×C4#38511yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4#38811yescharacteristic
419C2×C4#38811yescharacteristic
420C2×C4#39311yescharacteristic
421C2×C4#39411yescharacteristic
422C2×C4#39411yescharacteristic
423C2×C4#39611yescharacteristic
424C2×C4#39611yescharacteristic
425C2×C4#39711yescharacteristic
426C2×C4C4×SD1612yes—
427C2×C4#44112yes—
428C2×C4#44812yes—
429C2×C4#45212yes—
430C2×C4#45311yescharacteristic
431C2×C4#45411yescharacteristic
432C2×C4#45511yescharacteristic
433C2×C4#45611yescharacteristic
434C2×C4#46112yes—
435C2×C4#46412yes—
436C2×C4#46512yes—
437C2×C4#47012yes—
438C2×C4#47512yes—
439C2×C4#47912yes—
440C2×C4#48312yes—
441C2×C4#48512yes—
442C8—44no—
443C8—44no—
444C8—82no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22#143911yescharacteristic
458C22#196811yescharacteristic
459C22#197111yescharacteristic
460C22#200611yescharacteristic
461C22#200711yescharacteristic
462C22#200811yescharacteristic
463C22#200911yescharacteristic
464C22#201311yescharacteristic
465C22#201511yescharacteristic
466C22#209611yescharacteristic
467C22#209711yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—84no—
475C4—82no—
476C4—82no—
477C4—162no—
478C4—161no—
479C4—161no—
480C4#185911yescharacteristic
481C4#186011yescharacteristic
482C4#186512yes—
483C4#192712yes—
484C4#203512yes—
485C4#203711yescharacteristic
486C4#203811yescharacteristic
487C4#206612yes—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2#1019511yescharacteristic
493C2#1019711yescharacteristic
494C2#1020111yescharacteristic
495C2#1020711yescharacteristic
496C2#1109311yescharacteristic
497C2#1111311yescharacteristic
498C2#1112211yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
31. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#218852
#213222
C2×#15032
#108512
#108492
C2×#12832