Finite Group Databasebeta

← #164952

Subgroups — #164952

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36468C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22835C211yesmaximal, characteristic, complemented
4#7718C212yesmaximal, complemented
5#7674C212yesmaximal, complemented
6#7663C211yesmaximal, characteristic, complemented
7#7660C211yesmaximal, characteristic, complemented
8#7651C211yesmaximal, characteristic, complemented
9#7620C212yesmaximal, complemented
10#7582C212yesmaximal, complemented
11#7569C211yesmaximal, characteristic, complemented
12C2×#553—21no—
13#2859C2211yescharacteristic
14#2854C2212yes—
15#2602—22no—
16#2562—21no—
17#2559—22no—
18#2537C411yescharacteristic
19#2537C411yescharacteristic
20#2535—21no—
21#2496C412yes—
22#2428C412yes—
23#2354C411yescharacteristic
24#2352—21no—
25D4×C4⋊C4C411yescharacteristic
26C4×#127C2211yescharacteristic
27C4×#121C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1924—22no—
32#1923—22no—
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1749C2212yes—
42C4×C22⋊C8C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1270C2211yescharacteristic
48#1270C2212yes—
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53#553—44no—
54#553—44no—
55C2×#127C2×C411yescharacteristic
56C2×#126—22no—
57C2×#125—21no—
58C2×#123—22no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—22no—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C42C2312yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88#400—22no—
89#397—22no—
90#394—22no—
91#394—22no—
92#394—22no—
93#392—22no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2311yescharacteristic
98#390C2311yescharacteristic
99#389—21no—
100#389—21no—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2312yes—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#387—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#385—22no—
113#384C2×C411yescharacteristic
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#382—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380C2×C412yes—
122#380C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22⋊C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161#127—22no—
162#127—41no—
163#126—48no—
164#125—42no—
165#125—42no—
166#123—44no—
167#123—44no—
168#121—24no—
169#121—48no—
170#121—42no—
171#121—44no—
172#121—44no—
173#121—82no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—44no—
182C4×D4—44no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—82no—
188#117—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C42—22no—
223C2×C42—82no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2411yescharacteristic
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240C22⋊C8—82no—
241C22⋊C8—82no—
242#97—82no—
243C4×C8—42no—
244C4×C8—42no—
245C4×C8—82no—
246#95—22no—
247#95—22no—
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—
257C24C2×C22⋊C412yes—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4#13112yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—42no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—42no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C42—21no—
373C42—21no—
374C42—22no—
375C42—42no—
376C42—42no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—42no—
381C42—44no—
382C42#13512yes—
383C42#13512yes—
384C42AGL(1,ℤ/8)12yes—
385C42AGL(1,ℤ/8)12yes—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C22×C22⋊C411yescharacteristic
408D4—42no—
409D4—42no—
410D4—44no—
411D4—84no—
412D4—84no—
413D4—81no—
414D4—81no—
415D4—88no—
416D4—164no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
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—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—164no—
460C2×C4#41312yes—
461C2×C4#41612yes—
462C2×C4#41612yes—
463C2×C4#41812yes—
464C2×C4#41812yes—
465C2×C4C2×#13111yescharacteristic
466C2×C4#56811yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
470C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
471C8—42no—
472C8—42no—
473C8—81no—
474C8—82no—
475C8—81no—
476C8—82no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22#288011yescharacteristic
499C22#288311yescharacteristic
500C22C2×#41311yescharacteristic
501C22C2×#41611yescharacteristic
502C22C2×#41811yescharacteristic
503C22#290111yescharacteristic
504C22#290111yescharacteristic
505C4—21no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—42no—
514C4—41no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—81no—
519C4—82no—
520C4—82no—
521C4#155314yes—
522C2—82no—
523C2—82no—
524C2—161no—
525C2#1742411yescharacteristic
526C2#1750811yescharacteristic
527C2#1751111yescharacteristic
528C2#1761711yescharacteristic
529C2#1762611yescharacteristic
530C2#1768711yescharacteristic
531C2C2×#155311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36468C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22839C2semidirect
3. #22835C2semidirect
4. #7718C2semidirect
5. #7674C2semidirect
6. #7663C2semidirect
7. #7660C2semidirect
8. #7651C2semidirect
9. #7620C2semidirect
10. #7582C2semidirect
11. #7569C2semidirect

Maximal subgroups

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

subgroupindex
#364682
#228392
#228352
#77182
#76742
#76632
#76602
#76512
#76202
#75822
#75692