Finite Group Databasebeta

← #164154

Subgroups — #164154

Order 512.

1105 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36493C211yesmaximal, characteristic
2#22469C211yesmaximal, characteristic, complemented
3#20378C211yesmaximal, characteristic, complemented
4#7690C212yesmaximal, complemented
5#7684C211yesmaximal, characteristic, complemented
6#7670C212yesmaximal, complemented
7#7664C211yesmaximal, characteristic, complemented
8#7620C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7566C211yesmaximal, characteristic, complemented
11#7565C212yesmaximal, complemented
12C2×#559C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×D4—21no—
15C2×#408—21no—
16C2×#405—21no—
17#2856C2211yescharacteristic
18#2850C2212yes—
19#2541C411yescharacteristic
20#2540C412yes—
21#2538C411yescharacteristic
22#2535C412yes—
23#2471—22no—
24#2456—22no—
25#2448—21no—
26#2431—21no—
27#2366—21no—
28#2363—21no—
29#2355C411yescharacteristic
30C4×C22⋊Q8C2211yescharacteristic
31C4×#121C2212yes—
32#1984—21no—
33#1869—22no—
34#1817C2212yes—
35#1797C2211yescharacteristic
36#1750—22no—
37#1749C2212yes—
38C4×C22⋊C8C2211yescharacteristic
39C4×C22⋊C8C2212yes—
40#1728C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56C23×C8—22no—
57#559—44no—
58C2×#127—21no—
59C2×#127C2×C411yescharacteristic
60C2×#124—21no—
61C2×#124C2×C412yes—
62C2×#123—22no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C412yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121C2×C412yes—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C4×#41—44no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×D4—22no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D412yes—
81C2×#117D411yescharacteristic
82C2×#117D412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85C8×D4—44no—
86C2×C4⋊C8—21no—
87#408—28no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93#405—44no—
94C2×#97—22no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97#399—22no—
98#397—22no—
99#396—22no—
100#392—22no—
101#392—22no—
102#392—22no—
103#391—21no—
104#391C2311yescharacteristic
105#390—21no—
106#390—21no—
107#390—22no—
108#390—22no—
109#390—21no—
110#390—21no—
111#390C2312yes—
112#389—21no—
113#389C2311yescharacteristic
114#388—22no—
115#388—22no—
116#388—22no—
117#387—22no—
118#387—21no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#385—21no—
125#384C2×C411yescharacteristic
126#382C2×C412yes—
127#380C2×C411yescharacteristic
128#380C2×C412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336—21no—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144#336C2312yes—
145C2×#41—24no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—42no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22×C411yescharacteristic
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22⋊C412yes—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22×C412yes—
159C23×C4—22no—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—24no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—44no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—44no—
180C22×C8—82no—
181#127—42no—
182#127—44no—
183#124—44no—
184#124—44no—
185#123—48no—
186C22⋊Q8—22no—
187C22⋊Q8—41no—
188C22⋊Q8—82no—
189#121—24no—
190#121—42no—
191#121—42no—
192#121—42no—
193#121—81no—
194#121—81no—
195#121—82no—
196#121—82no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×D4—24no—
201C4×D4—44no—
202C4×D4—42no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—82no—
207C4×D4—82no—
208#117—24no—
209#117—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42C22×C412yes—
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2411yescharacteristic
251C4⋊C8—42no—
252C22⋊C8—28no—
253C22⋊C8—28no—
254C22⋊C8—42no—
255C22⋊C8—44no—
256C22⋊C8—81no—
257#97—44no—
258C4×C8—44no—
259C4×C8—42no—
260#95—22no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24—22no—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C412yes—
272#41—48no—
273#41—48no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4#10414yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4#13512yes—
343C22×C4#13512yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347M16—48no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—48no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—88no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C4⋊C4#9914yes—
383C4⋊C4#13112yes—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—81no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C42—24no—
402C42—22no—
403C42—24no—
404C42—22no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—82no—
411C23—21no—
412C23—22no—
413C23—21no—
414C23—22no—
415C23—22no—
416C23—21no—
417C23—22no—
418C23—22no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23#41612yes—
431C23C2×#10414yes—
432C23C22×C22⋊C411yescharacteristic
433Q8—44no—
434Q8—81no—
435Q8—81no—
436D4—28no—
437D4—48no—
438D4—82no—
439D4—82no—
440D4—88no—
441D4—84no—
442D4—162no—
443D4—164no—
444D4—164no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—164no—
493C2×C4C2×#9914yes—
494C2×C4C2×#9912yes—
495C2×C4#41612yes—
496C2×C4C2×#10412yes—
497C2×C4C2×#13111yescharacteristic
498C2×C4#56811yescharacteristic
499C2×C4C2×#13511yescharacteristic
500C2×C4C2×#13511yescharacteristic
501C2×C4#57611yescharacteristic
502C2×C4#57711yescharacteristic
503C8—42no—
504C8—42no—
505C8—42no—
506C8—44no—
507C8—81no—
508C8—82no—
509C22—22no—
510C22—22no—
511C22—24no—
512C22—22no—
513C22—24no—
514C22—24no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—42no—
521C22—41no—
522C22—41no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22#288011yescharacteristic
531C22C22×#9911yescharacteristic
532C22#288811yescharacteristic
533C22C2×#41611yescharacteristic
534C22#289711yescharacteristic
535C22C22×#10411yescharacteristic
536C22#290111yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—24no—
544C4—42no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—41no—
549C4—81no—
550C4—82no—
551C4—81no—
552C4—81no—
553C4#150114yes—
554C2—42no—
555C2—44no—
556C2—162no—
557C2#1741911yescharacteristic
558C2#1751011yescharacteristic
559C2#1751711yescharacteristic
560C2C2×#150111yescharacteristic
561C2#1760311yescharacteristic
562C2#1765211yescharacteristic
563C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36493C2
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
2. #22469C2semidirect
3. #20378C2semidirect
4. #7690C2semidirect
5. #7684C2semidirect
6. #7670C2semidirect
7. #7664C2semidirect
8. #7620C2semidirect
9. #7617C2semidirect
10. #7566C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#364932
#224692
#203782
#76902
#76842
#76702
#76642
#76202
#76172
#75662
#75652