Finite Group Databasebeta

← #164600

Subgroups — #164600

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36274C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic
3#20684C211yesmaximal, characteristic
4#7721C212yesmaximal, complemented
5#7695C212yesmaximal, complemented
6#7687C211yesmaximal, characteristic, complemented
7#7667C211yesmaximal, characteristic, complemented
8#7617C211yesmaximal, characteristic, complemented
9#7609C212yesmaximal, complemented
10#7595C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×D4×Q8C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#436—21no—
15C2×#408—21no—
16#2857C2211yescharacteristic
17#2855C2212yes—
18#2582C411yescharacteristic
19#2568C411yescharacteristic
20#2451—21no—
21#2450—22no—
22#2439—21no—
23#2438—22no—
24#2417C412yes—
25#2409C412yes—
26#2349—21no—
27D4×C4⋊C4C411yescharacteristic
28#2339—21no—
29C4×C22⋊Q8C2211yescharacteristic
30C2×#391C2212yes—
31#1984—21no—
32#1870—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1728C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2211yescharacteristic
49#1272C2212yes—
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C22×#41D411yescharacteristic
54C23×Q8C2×C412yes—
55C22×M16—21no—
56C22×M16—22no—
57C23×C8—21no—
58D4×Q8—44no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C4×#41—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D412yes—
75C2×#117D412yes—
76C2×#117D411yescharacteristic
77C22×C4⋊C4C2×C412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2312yes—
80#436—44no—
81C2×C4⋊C8—21no—
82#408—44no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89#400—22no—
90#398—22no—
91#398—22no—
92#396—22no—
93#395—22no—
94#393—22no—
95#391—21no—
96#391—21no—
97#391—22no—
98#391—22no—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2312yes—
103#391C2311yescharacteristic
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389C2×C412yes—
109#389C2311yescharacteristic
110#387—22no—
111#387—22no—
112#387—21no—
113#386—22no—
114#386—22no—
115#385—22no—
116#385—21no—
117#385—22no—
118#385—22no—
119#384C2×C411yescharacteristic
120#382—21no—
121#381—22no—
122#380—21no—
123#380—21no—
124#380—22no—
125#380C2×C412yes—
126#380C2×C411yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C22⋊C4—21no—
130C2×#95—22no—
131C2×#95—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C22×Q8—22no—
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8—44no—
152C22×Q8—82no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22×C412yes—
155C22×Q8C22×C411yescharacteristic
156C22×D4C22⋊C412yes—
157C22×D4C22⋊C411yescharacteristic
158C23×C4—22no—
159C23×C4—22no—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—44no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—44no—
170C2×M16—42no—
171C2×M16—82no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—44no—
183C22⋊Q8—22no—
184C22⋊Q8—48no—
185C22⋊Q8—41no—
186C22⋊Q8—44no—
187C22⋊Q8—82no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197#117—44no—
198#117—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—42no—
230C2×C42—44no—
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2411yescharacteristic
238C4⋊C8—42no—
239C22⋊C8—42no—
240C22⋊C8—44no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—81no—
244C4×C8—42no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—21no—
260C24C2×C22⋊C412yes—
261C24C2×C22⋊C411yescharacteristic
262#41—48no—
263#41—48no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—82no—
277C2×Q8—88no—
278C2×Q8—88no—
279C2×Q8#13112yes—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—42no—
285C2×D4#9914yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4#13712yes—
327C22×C4#13712yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331M16—88no—
332M16—84no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—84no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—82no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—81no—
380C22⋊C4—82no—
381C42—22no—
382C42—22no—
383C42—41no—
384C42—42no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C23—21no—
391C23—22no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—22no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23C2×#9914yes—
403C23#41912yes—
404C23C22×C22⋊C411yescharacteristic
405Q8—42no—
406Q8—42no—
407Q8—44no—
408Q8—82no—
409Q8—82no—
410Q8—88no—
411D4—28no—
412D4—48no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—88no—
465C2×C4—164no—
466C2×C4C2×#9912yes—
467C2×C4#41912yes—
468C2×C4#42112yes—
469C2×C4C2×#13111yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57811yescharacteristic
475C8—42no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8—82no—
480C8—82no—
481C22—24no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—24no—
486C22—24no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—42no—
491C22—42no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22#288011yescharacteristic
496C22C22×#9911yescharacteristic
497C22#288811yescharacteristic
498C22C2×#41911yescharacteristic
499C22#289811yescharacteristic
500C22C2×#42111yescharacteristic
501C22#290111yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—41no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—82no—
515C4—82no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4#151114yes—
520C2—44no—
521C2—42no—
522C2#1741111yescharacteristic
523C2#1751411yescharacteristic
524C2#1752511yescharacteristic
525C2C2×#151111yescharacteristic
526C2#1760811yescharacteristic
527C2#1768111yescharacteristic
528C2#1776011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7721C2semidirect
5. #7695C2semidirect
6. #7687C2semidirect
7. #7667C2semidirect
8. #7617C2semidirect
9. #7609C2semidirect
10. #7595C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#362742
#224732
#206842
#77212
#76952
#76872
#76672
#76172
#76092
#75952
#75662