Finite Group Databasebeta

← #162477

Subgroups — #162477

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35197C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic, complemented
3#20568C211yesmaximal, characteristic, complemented
4#7710C212yesmaximal, complemented
5#7698C212yesmaximal, complemented
6#7690C212yesmaximal, complemented
7#7678C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14#2854C2212yes—
15#2850C2212yes—
16#2648—22no—
17#2571C411yescharacteristic
18#2548C411yescharacteristic
19#2541C411yescharacteristic
20#2538C411yescharacteristic
21#2513—21no—
22#2457—22no—
23#2445—21no—
24#2427C412yes—
25#2420C412yes—
26#2328—21no—
27#2319—21no—
28C4×#127C2211yescharacteristic
29#2295C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1939—21no—
32#1850—22no—
33#1831C2211yescharacteristic
34#1797C2212yes—
35#1770—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1753—22no—
39#1749C2212yes—
40#1744C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×M16—22no—
54C23×C8—22no—
55C2×#127—21no—
56C2×#127C2×C411yescharacteristic
57C2×#124C2×C412yes—
58C2×#123—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×#121C2×C412yes—
61C2×#121C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C22×C42C2312yes—
70#436—28no—
71C8×D4—44no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#400—22no—
83#397—22no—
84#397—22no—
85#396—22no—
86#394—22no—
87#394—22no—
88#394—22no—
89#392—22no—
90#391C2×C411yescharacteristic
91#391C2311yescharacteristic
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390C2×C412yes—
99#390C2312yes—
100#389—21no—
101#389—21no—
102#389C2×C412yes—
103#389C2×C411yescharacteristic
104#389C2311yescharacteristic
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#387—21no—
110#387—21no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#384C2×C411yescharacteristic
117#382—21no—
118#382C2×C412yes—
119#381—22no—
120#380—22no—
121#380C2×C412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130#376—21no—
131C43C2311yescharacteristic
132#336—21no—
133#336—22no—
134#336—22no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22⋊C411yescharacteristic
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4C22×C412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C2×M16—28no—
154C2×M16—82no—
155C2×M16—82no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—44no—
163C22×C8—44no—
164C22×C8—82no—
165#127—24no—
166#127—44no—
167#124—82no—
168#123—82no—
169#123—82no—
170C22⋊Q8—82no—
171#121—82no—
172#121—82no—
173C4×Q8—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—81no—
181C4×D4—81no—
182C4×D4—82no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—82no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—24no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—82no—
228C22⋊C8—28no—
229C22⋊C8—44no—
230#97—82no—
231C4×C8—24no—
232C4×C8—42no—
233C4×C8—81no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C24—22no—
249C24C2×C22⋊C412yes—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4#10414yes—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300M16—88no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—24no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—88no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—21no—
352C42—21no—
353C42—22no—
354C42—42no—
355C42—41no—
356C42—42no—
357C42—41no—
358C42—42no—
359C42—41no—
360C42—42no—
361C42—42no—
362C42#13114yes—
363C42AGL(1,ℤ/8)12yes—
364C42#13712yes—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23C2×#10414yes—
379C23C22×C22⋊C411yescharacteristic
380Q8—82no—
381D4—28no—
382D4—84no—
383D4—84no—
384D4—84no—
385D4—164no—
386D4—164no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—84no—
429C2×C4—164no—
430C2×C4#40812yes—
431C2×C4#40812yes—
432C2×C4#41712yes—
433C2×C4#41712yes—
434C2×C4C2×#10412yes—
435C2×C4C2×#13112yes—
436C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
437C2×C4C2×#13711yescharacteristic
438C2×C4#57611yescharacteristic
439C2×C4#57711yescharacteristic
440C8—42no—
441C8—42no—
442C8—44no—
443C8—42no—
444C8—81no—
445C8—82no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22C2×#40811yescharacteristic
461C22#288311yescharacteristic
462C22#288811yescharacteristic
463C22C2×#41711yescharacteristic
464C22#289711yescharacteristic
465C22C22×#10411yescharacteristic
466C22#290111yescharacteristic
467C4—22no—
468C4—22no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—41no—
478C4—81no—
479C4—82no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4#148314yes—
484C2—44no—
485C2—162no—
486C2#1738911yescharacteristic
487C2C2×#148311yescharacteristic
488C2#1748311yescharacteristic
489C2#1754711yescharacteristic
490C2#1763011yescharacteristic
491C2#1765211yescharacteristic
492C2#1776211yescharacteristic
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. #22696C2semidirect
3. #20568C2semidirect
4. #7710C2semidirect
5. #7698C2semidirect
6. #7690C2semidirect
7. #7678C2semidirect
8. #7587C2semidirect
9. #7577C2semidirect
10. #7569C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#351972
#226962
#205682
#77102
#76982
#76902
#76782
#75872
#75772
#75692
#75622