Finite Group Databasebeta

← #162476

Subgroups — #162476

Order 512.

859 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35207C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic, complemented
3#20568C211yesmaximal, characteristic, complemented
4#7711C212yesmaximal, complemented
5#7698C212yesmaximal, complemented
6#7691C212yesmaximal, complemented
7#7678C212yesmaximal, complemented
8#7591C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14#2854C2212yes—
15#2851C2212yes—
16#2653—22no—
17#2563C411yescharacteristic
18#2563C411yescharacteristic
19#2552C411yescharacteristic
20#2537C411yescharacteristic
21#2494—21no—
22#2494C412yes—
23#2490C412yes—
24#2458—22no—
25#2422—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#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×#123—22no—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×#117—21no—
66C22×C42C2312yes—
67#436—28no—
68C8×D4—44no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×#97—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#401—22no—
80#398—22no—
81#397—22no—
82#397—22no—
83#394—22no—
84#394—22no—
85#393—22no—
86#393—22no—
87#391—21no—
88#390—21no—
89#390—22no—
90#390C2×C412yes—
91#390C2×C411yescharacteristic
92#390C2311yescharacteristic
93#389—21no—
94#389—21no—
95#389C2×C412yes—
96#389C2×C411yescharacteristic
97#389C2312yes—
98#389C2311yescharacteristic
99#388—22no—
100#388—22no—
101#388—22no—
102#388—21no—
103#388—21no—
104#388—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#384—21no—
112#384C2×C411yescharacteristic
113#382—21no—
114#382C2×C412yes—
115#380—21no—
116#380—21no—
117#380—21no—
118#380C2×C412yes—
119#380C2×C412yes—
120#380C2×C412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C412yes—
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×Q8—21no—
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#125—82no—
164#123—82no—
165#123—82no—
166#121—44no—
167#121—82no—
168C4×Q8—81no—
169C4×Q8—81no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175#117—42no—
176#117—42no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
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×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—82no—
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221C4⋊C8—24no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—82no—
225C22⋊C8—28no—
226C22⋊C8—44no—
227#97—82no—
228C4×C8—24no—
229C4×C8—42no—
230C4×C8—81no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C24—21no—
246C24C2×C22⋊C412yes—
247C2×Q8—42no—
248C2×Q8—81no—
249C2×Q8—81no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—44no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4#10414yes—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—88no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—88no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—81no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—44no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—81no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—21no—
344C42—21no—
345C42—22no—
346C42—42no—
347C42—41no—
348C42—42no—
349C42—41no—
350C42—42no—
351C42—41no—
352C42—42no—
353C42—82no—
354C42#13114yes—
355C42AGL(1,ℤ/8)12yes—
356C42#13712yes—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C2×#10414yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—84no—
373D4—28no—
374D4—82no—
375D4—164no—
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—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4#40812yes—
420C2×C4#40812yes—
421C2×C4#41712yes—
422C2×C4#41712yes—
423C2×C4C2×#10412yes—
424C2×C4C2×#13112yes—
425C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
426C2×C4C2×#13711yescharacteristic
427C2×C4#57611yescharacteristic
428C2×C4#57711yescharacteristic
429C8—42no—
430C8—42no—
431C8—44no—
432C8—42no—
433C8—81no—
434C8—82no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22C2×#40811yescharacteristic
450C22#288311yescharacteristic
451C22#288911yescharacteristic
452C22C2×#41711yescharacteristic
453C22#289711yescharacteristic
454C22C22×#10411yescharacteristic
455C22#290111yescharacteristic
456C4—22no—
457C4—22no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—82no—
471C4—81no—
472C4#148314yes—
473C2—44no—
474C2—161no—
475C2#1738811yescharacteristic
476C2C2×#148311yescharacteristic
477C2#1748311yescharacteristic
478C2#1754711yescharacteristic
479C2#1763011yescharacteristic
480C2#1765311yescharacteristic
481C2#1776511yescharacteristic
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. #7711C2semidirect
5. #7698C2semidirect
6. #7691C2semidirect
7. #7678C2semidirect
8. #7591C2semidirect
9. #7574C2semidirect
10. #7570C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#352072
#226962
#205682
#77112
#76982
#76912
#76782
#75912
#75742
#75702
#75602