Finite Group Databasebeta

← #164823

Subgroups — #164823

Order 512.

891 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36496C211yesmaximal, characteristic
2#22020C212yesmaximal, complemented
3#7719C212yesmaximal, complemented
4#7716C212yesmaximal, complemented
5#7676C212yesmaximal, complemented
6#7651C211yesmaximal, characteristic, complemented
7#7649C211yesmaximal, characteristic, complemented
8#7637C211yesmaximal, characteristic, complemented
9#7603C211yesmaximal, characteristic, complemented
10#7601C211yesmaximal, characteristic, complemented
11#7569C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13C2×#554C411yescharacteristic
14#2861C2211yescharacteristic
15#2861C2211yescharacteristic
16#2858C2211yescharacteristic
17#2567C411yescharacteristic
18#2566C411yescharacteristic
19#2566C411yescharacteristic
20#2544C411yescharacteristic
21#2544C411yescharacteristic
22#2541C411yescharacteristic
23#2461—22no—
24#2461—22no—
25#2426—22no—
26#2361—22no—
27C4×#127C2211yescharacteristic
28C4×#127C2211yescharacteristic
29C4×#124C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1985—22no—
32#1940—22no—
33#1916—22no—
34#1916—22no—
35#1839C2212yes—
36#1839C2212yes—
37#1831C2212yes—
38#1831C2212yes—
39C4×C4⋊C8C2212yes—
40#1745C2212yes—
41#1272C2212yes—
42#1272C2212yes—
43#1270C2212yes—
44#1270C2212yes—
45#1270C2212yes—
46#1270C2212yes—
47#1270C2211yescharacteristic
48#1270C2211yescharacteristic
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51#559—44no—
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#127C2×C411yescharacteristic
57C2×#125C2×C411yescharacteristic
58C2×#125C2×C411yescharacteristic
59C2×#124—21no—
60C2×#124—22no—
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#123—21no—
65C2×#123—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#121—22no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D412yes—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C411yescharacteristic
81C22×C4⋊C4—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×#97—22no—
86C2×C4×C8—22no—
87#396—22no—
88#396—22no—
89#395—22no—
90#394—22no—
91#394—22no—
92#392—22no—
93#391C2×C411yescharacteristic
94#391C2×C411yescharacteristic
95#391C2311yescharacteristic
96#390—22no—
97#390—22no—
98#390—22no—
99#390C2×C411yescharacteristic
100#390C2×C411yescharacteristic
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2311yescharacteristic
104#390C2311yescharacteristic
105#390C2311yescharacteristic
106#389—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—22no—
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128C43C2311yescharacteristic
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140C22×Q8—21no—
141C22×Q8C22⋊C411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—42no—
160#127—22no—
161#127—22no—
162#127—41no—
163#127—41no—
164#127—42no—
165#125—42no—
166#125—42no—
167#124—22no—
168#124—41no—
169#124—42no—
170#124—42no—
171#124—44no—
172#124—82no—
173#124—82no—
174#123—44no—
175#123—44no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179#121—48no—
180C4×Q8—24no—
181C4×Q8—44no—
182C4×Q8—42no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—44no—
189C4×D4—82no—
190C4×D4—82no—
191C4×D4—82no—
192C4×D4—82no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—21no—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2411yescharacteristic
243C4⋊C8—44no—
244C4⋊C8—82no—
245C4⋊C8—82no—
246#97—44no—
247C4×C8—82no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255C24—21no—
256C24—21no—
257C24—21no—
258C2×Q8—22no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—82no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4#13112yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—44no—
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×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D412yes—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4#10014yes—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—24no—
380C42—42no—
381C42—42no—
382C42—41no—
383C42—42no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23C22×C22⋊C411yescharacteristic
400Q8—44no—
401Q8—81no—
402Q8—81no—
403Q8—162no—
404D4—42no—
405D4—42no—
406D4—88no—
407D4—81no—
408D4—81no—
409D4—81no—
410D4—81no—
411D4—84no—
412D4—88no—
413D4—162no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—84no—
447C2×C4—162no—
448C2×C4—162no—
449C2×C4—162no—
450C2×C4—162no—
451C2×C4C2×#10012yes—
452C2×C4C2×#10012yes—
453C2×C4C2×#10012yes—
454C2×C4#42114yes—
455C2×C4C2×#13111yescharacteristic
456C2×C4#56811yescharacteristic
457C2×C4#57612yes—
458C2×C4#57712yes—
459C8—82no—
460C8—82no—
461C8—82no—
462C8—82no—
463C22—42no—
464C22—44no—
465C22—81no—
466C22—82no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—81no—
477C22—81no—
478C22#288011yescharacteristic
479C22#288411yescharacteristic
480C22C22×#10011yescharacteristic
481C22#289712yes—
482C22C2×#42112yes—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—24no—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—41no—
492C4—42no—
493C4—42no—
494C4—41no—
495C4—41no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C2—82no—
502C2—161no—
503C2—161no—
504C2#1742511yescharacteristic
505C2#1752112yes—
506C2#1760212yes—
507C2#1766311yescharacteristic
508C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22020C2semidirect
3. #7719C2semidirect
4. #7716C2semidirect
5. #7676C2semidirect
6. #7651C2semidirect
7. #7649C2semidirect
8. #7637C2semidirect
9. #7603C2semidirect
10. #7601C2semidirect
11. #7569C2semidirect

Maximal subgroups

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

subgroupindex
#364962
#220202
#77192
#77162
#76762
#76512
#76492
#76372
#76032
#76012
#75692