Finite Group Databasebeta

← #162483

Subgroups — #162483

Order 512.

859 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35196C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic, complemented
3#20568C211yesmaximal, characteristic, complemented
4#7710C212yesmaximal, complemented
5#7701C212yesmaximal, complemented
6#7690C212yesmaximal, complemented
7#7681C212yesmaximal, complemented
8#7590C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7571C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14#2855C2212yes—
15#2850C2212yes—
16#2649—22no—
17#2571C411yescharacteristic
18#2547C411yescharacteristic
19#2542C411yescharacteristic
20#2537C411yescharacteristic
21#2516—22no—
22#2497—21no—
23#2426C412yes—
24#2421C412yes—
25#2420—21no—
26#2324—21no—
27#2314—21no—
28C4×C4⋊Q8C2211yescharacteristic
29#2293C2211yescharacteristic
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#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#124C2×C412yes—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×#121—21no—
60C2×#121C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×#117—21no—
67C22×C42C2312yes—
68#436—28no—
69C8×D4—44no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×#97—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#400—22no—
81#400—22no—
82#399—22no—
83#397—22no—
84#397—22no—
85#396—22no—
86#394—22no—
87#394—22no—
88#391C2×C412yes—
89#391C2312yes—
90#390—21no—
91#390—22no—
92#390C2×C411yescharacteristic
93#390C2311yescharacteristic
94#389—21no—
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2×C411yescharacteristic
99#389C2311yescharacteristic
100#388—22no—
101#388—22no—
102#388—22no—
103#388—21no—
104#388—21no—
105#388—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#386—22no—
110#386—22no—
111#385—22no—
112#384—21no—
113#384C2×C411yescharacteristic
114#382—21no—
115#382C2×C412yes—
116#381—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380C2×C412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
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×C412yes—
143C22×D4—22no—
144C22×D4C22⋊C411yescharacteristic
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C2×M16—28no—
151C2×M16—82no—
152C2×M16—82no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—44no—
160C22×C8—44no—
161C22×C8—82no—
162C4⋊Q8—24no—
163#124—82no—
164C22⋊Q8—82no—
165C22⋊Q8—82no—
166#121—44no—
167#121—82no—
168C4×Q8—82no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—22no—
174#117—42no—
175#117—42no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—82no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—24no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—82no—
224C22⋊C8—28no—
225C22⋊C8—44no—
226#97—82no—
227C4×C8—24no—
228C4×C8—42no—
229C4×C8—81no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C24—21no—
245C24C2×C22⋊C412yes—
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—44no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4#10414yes—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—88no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—88no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—44no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—81no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—21no—
342C42—21no—
343C42—22no—
344C42—42no—
345C42—41no—
346C42—42no—
347C42—41no—
348C42—42no—
349C42—41no—
350C42—42no—
351C42—82no—
352C42#13114yes—
353C42AGL(1,ℤ/8)12yes—
354C42#13712yes—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23C2×#10414yes—
369C23C22×C22⋊C411yescharacteristic
370Q8—84no—
371D4—28no—
372D4—82no—
373D4—164no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4#40812yes—
418C2×C4#40812yes—
419C2×C4#41712yes—
420C2×C4#41712yes—
421C2×C4C2×#10412yes—
422C2×C4C2×#13112yes—
423C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
424C2×C4C2×#13711yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57811yescharacteristic
427C8—42no—
428C8—42no—
429C8—44no—
430C8—42no—
431C8—81no—
432C8—82no—
433C22—24no—
434C22—24no—
435C22—24no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22C2×#40811yescharacteristic
448C22#288311yescharacteristic
449C22#288811yescharacteristic
450C22C2×#41711yescharacteristic
451C22#289811yescharacteristic
452C22C22×#10411yescharacteristic
453C22#290111yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—41no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—82no—
469C4—82no—
470C4#148314yes—
471C2—44no—
472C2—161no—
473C2#1738911yescharacteristic
474C2C2×#148311yescharacteristic
475C2#1748411yescharacteristic
476C2#1754811yescharacteristic
477C2#1763011yescharacteristic
478C2#1765211yescharacteristic
479C2#1776411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
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. #7701C2semidirect
6. #7690C2semidirect
7. #7681C2semidirect
8. #7590C2semidirect
9. #7574C2semidirect
10. #7571C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#351962
#226962
#205682
#77102
#77012
#76902
#76812
#75902
#75742
#75712
#75602