Finite Group Databasebeta

← #165060

Subgroups — #165060

Order 512.

909 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22835C211yesmaximal, characteristic, complemented
4#7718C212yesmaximal, complemented
5#7674C212yesmaximal, complemented
6#7659C211yesmaximal, characteristic, complemented
7#7659C211yesmaximal, characteristic, complemented
8#7652C211yesmaximal, characteristic, complemented
9#7620C212yesmaximal, complemented
10#7582C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12C2×#559—21no—
13#2858C2211yescharacteristic
14#2854C2212yes—
15#2602—22no—
16#2592—21no—
17#2557—22no—
18#2547C411yescharacteristic
19#2544—21no—
20#2541C411yescharacteristic
21#2491C412yes—
22#2422C412yes—
23#2367—21no—
24#2366C411yescharacteristic
25#2365C411yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
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#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53#559—44no—
54#559—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127—21no—
58C2×#127C2×C411yescharacteristic
59C2×#124—21no—
60C2×#124—22no—
61C2×#124—21no—
62C2×#124—21no—
63C2×#123—22no—
64C2×#123—22no—
65C2×#121C2×C412yes—
66C2×#121C2×C412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
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#397—22no—
89#397—22no—
90#394—22no—
91#394—22no—
92#393—22no—
93#392—22no—
94#391—21no—
95#391—21no—
96#391—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2×C412yes—
103#390C2312yes—
104#389—21no—
105#389C2×C412yes—
106#389C2311yescharacteristic
107#389C2311yescharacteristic
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#385—22no—
117#384C2×C411yescharacteristic
118#380C2×C412yes—
119#380C2×C412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
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×Q8—21no—
143C22×Q8—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22⋊C411yescharacteristic
149C22×D4C22×C412yes—
150C23×C4—22no—
151C23×C4C22×C412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—41no—
162C4⋊Q8—42no—
163#127—42no—
164#127—44no—
165#124—48no—
166#124—44no—
167#124—42no—
168#124—42no—
169#124—42no—
170#124—42no—
171#123—48no—
172#123—44no—
173#123—44no—
174#121—24no—
175#121—42no—
176#121—82no—
177C4×Q8—44no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—42no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—82no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—82no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2411yescharacteristic
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C22⋊C8—44no—
237C22⋊C8—44no—
238C22⋊C8—82no—
239C22⋊C8—82no—
240#97—82no—
241C4×C8—42no—
242C4×C8—42no—
243C4×C8—82no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255C24—22no—
256C24C2×C22⋊C412yes—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—81no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—82no—
280C2×D4—82no—
281C2×D4—82no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—81no—
348C4⋊C4#13112yes—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C42—21no—
366C42—21no—
367C42—22no—
368C42—42no—
369C42—42no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—44no—
375C42—42no—
376C42—42no—
377C42#13512yes—
378C42#13512yes—
379C42AGL(1,ℤ/8)12yes—
380C42AGL(1,ℤ/8)12yes—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23C22×C22⋊C411yescharacteristic
396Q8—84no—
397Q8—84no—
398D4—44no—
399D4—82no—
400D4—82no—
401D4—88no—
402D4—84no—
403D4—162no—
404D4—164no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—164no—
449C2×C4#41312yes—
450C2×C4#41612yes—
451C2×C4#41612yes—
452C2×C4#41812yes—
453C2×C4#41812yes—
454C2×C4C2×#13111yescharacteristic
455C2×C4#56811yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
460C8—42no—
461C8—42no—
462C8—81no—
463C8—82no—
464C8—81no—
465C8—82no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22#288011yescharacteristic
481C22#288311yescharacteristic
482C22C2×#41311yescharacteristic
483C22C2×#41611yescharacteristic
484C22C2×#41811yescharacteristic
485C22#290111yescharacteristic
486C22#290111yescharacteristic
487C4—21no—
488C4—21no—
489C4—22no—
490C4—22no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—41no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—82no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4#155314yes—
505C2—82no—
506C2—162no—
507C2#1742911yescharacteristic
508C2#1751011yescharacteristic
509C2#1751111yescharacteristic
510C2#1761711yescharacteristic
511C2#1762611yescharacteristic
512C2#1769111yescharacteristic
513C2C2×#155311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
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. #7659C2semidirect
7. #7659C2semidirect
8. #7652C2semidirect
9. #7620C2semidirect
10. #7582C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#364692
#228392
#228352
#77182
#76742
#76592
#76522
#76202
#75822
#75702