Finite Group Databasebeta

← #164793

Subgroups — #164793

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36504C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic, complemented
3#20684C211yesmaximal, characteristic, complemented
4#7730C212yesmaximal, complemented
5#7687C211yesmaximal, characteristic, complemented
6#7670C212yesmaximal, complemented
7#7667C211yesmaximal, characteristic, complemented
8#7648C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7583C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×Q8—21no—
15C2×#408—21no—
16#2857C2211yescharacteristic
17#2850C2212yes—
18#2542C411yescharacteristic
19#2540C412yes—
20#2537C411yescharacteristic
21#2536C412yes—
22#2479—22no—
23#2467—22no—
24#2423—21no—
25#2407—21no—
26#2363—21no—
27#2355C411yescharacteristic
28#2354—21no—
29C4×C22⋊Q8C2212yes—
30C4×#121C2211yescharacteristic
31#1954—21no—
32#1954—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1797C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39C4×C22⋊C8C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56#556—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#126—22no—
59C2×#124C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C412yes—
64C2×#121—21no—
65C2×#121C2×C411yescharacteristic
66C2×#121C2×C412yes—
67C4×#41—44no—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D412yes—
77C2×#117—21no—
78C2×#117D412yes—
79C2×#117D411yescharacteristic
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C8×Q8—44no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85#408—44no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94#399—22no—
95#399—22no—
96#398—22no—
97#394—22no—
98#394—22no—
99#392—22no—
100#391—21no—
101#391C2312yes—
102#390—21no—
103#390—22no—
104#390—22no—
105#390C2311yescharacteristic
106#389—21no—
107#389—21no—
108#389C2311yescharacteristic
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—21no—
114#387—22no—
115#387—22no—
116#386—22no—
117#386—22no—
118#386—21no—
119#385—22no—
120#384—21no—
121#384C2×C411yescharacteristic
122#382C2×C412yes—
123#380—21no—
124#380—21no—
125#380—21no—
126#380C2×C411yescharacteristic
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C412yes—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4C22⋊C411yescharacteristic
153C22×D4C22⋊C412yes—
154C22×D4C22×C411yescharacteristic
155C23×C4—21no—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C4⋊Q8—42no—
175#126—48no—
176#124—44no—
177C22⋊Q8—24no—
178C22⋊Q8—42no—
179C22⋊Q8—44no—
180C22⋊Q8—82no—
181C22⋊Q8—82no—
182#121—22no—
183#121—41no—
184#121—44no—
185#121—82no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—44no—
189C4×D4—42no—
190C4×D4—44no—
191C4×D4—42no—
192C4×D4—44no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196#117—42no—
197#117—44no—
198#117—44no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2411yescharacteristic
243C4⋊C8—44no—
244C4⋊C8—42no—
245C22⋊C8—44no—
246C22⋊C8—42no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—81no—
250C22⋊C8—82no—
251C4×C8—44no—
252C4×C8—42no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95—22no—
262C24—21no—
263C24C2×C22⋊C412yes—
264C24C2×C22⋊C411yescharacteristic
265#41—48no—
266#41—48no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8#9914yes—
274C2×D4—24no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—81no—
288C2×D4—82no—
289C2×D4—82no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4#13512yes—
327C22×C4#13512yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331M16—84no—
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
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—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—84no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—44no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—81no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4#13112yes—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—44no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—41no—
392C42—42no—
393C42—41no—
394C42—44no—
395C42—42no—
396C42—82no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23#41612yes—
418C23C22×C22⋊C411yescharacteristic
419Q8—28no—
420Q8—82no—
421Q8—82no—
422D4—48no—
423D4—44no—
424D4—81no—
425D4—81no—
426D4—84no—
427D4—164no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4C2×#9914yes—
477C2×C4C2×#9912yes—
478C2×C4#41612yes—
479C2×C4#42112yes—
480C2×C4C2×#13111yescharacteristic
481C2×C4#56811yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57811yescharacteristic
486C8—42no—
487C8—82no—
488C8—81no—
489C8—81no—
490C8—81no—
491C8—82no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22#288011yescharacteristic
514C22C22×#9911yescharacteristic
515C22#288811yescharacteristic
516C22C2×#41611yescharacteristic
517C22#289811yescharacteristic
518C22C2×#42111yescharacteristic
519C22#290111yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—24no—
525C4—21no—
526C4—21no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—82no—
533C4—82no—
534C4—81no—
535C4—81no—
536C4#151614yes—
537C2—42no—
538C2—82no—
539C2—161no—
540C2#1741911yescharacteristic
541C2#1750811yescharacteristic
542C2#1752511yescharacteristic
543C2C2×#151611yescharacteristic
544C2#1760611yescharacteristic
545C2#1768111yescharacteristic
546C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36504C2
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. #22530C2semidirect
3. #20684C2semidirect
4. #7730C2semidirect
5. #7687C2semidirect
6. #7670C2semidirect
7. #7667C2semidirect
8. #7648C2semidirect
9. #7616C2semidirect
10. #7583C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#365042
#225302
#206842
#77302
#76872
#76702
#76672
#76482
#76162
#75832
#75652