Finite Group Databasebeta

← #164342

Subgroups — #164342

Order 512.

901 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36290C211yesmaximal, characteristic
2#20558C211yesmaximal, characteristic
3#20558C211yesmaximal, characteristic
4#7691C212yesmaximal, complemented
5#7691C212yesmaximal, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7627C212yesmaximal, complemented
10#7568C211yesmaximal, characteristic, complemented
11C2×#1271C212yesmaximal, complemented
12C2×C8×D4—21no—
13C2×C8×D4—21no—
14#2859C2211yescharacteristic
15#2851C2212yes—
16#2571C411yescharacteristic
17#2552C411yescharacteristic
18#2509—21no—
19#2507—21no—
20#2505—22no—
21#2473—22no—
22#2442C412yes—
23#2442C412yes—
24#2349—21no—
25D4×C4⋊C4—21no—
26D4×C4⋊C4C411yescharacteristic
27D4×C4⋊C4C411yescharacteristic
28C4×#125C2211yescharacteristic
29C2×#389C2212yes—
30C42×D4C2211yescharacteristic
31#1770—21no—
32#1770—21no—
33#1767C2211yescharacteristic
34#1767C2211yescharacteristic
35#1752—22no—
36#1752—22no—
37#1749C2212yes—
38#1749C2212yes—
39#1744C2211yescharacteristic
40#1744C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C23×C8—22no—
54C23×C8—22no—
55C2×#125—21no—
56C2×#125C2×C411yescharacteristic
57C2×#125C2×C412yes—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C42C2312yes—
67C8×D4—28no—
68C8×D4—28no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×#97—22no—
75C2×#97—21no—
76C2×#97—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#400—22no—
80#400—22no—
81#400—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2311yescharacteristic
97#389C2311yescharacteristic
98#389C2312yes—
99#387—21no—
100#387—21no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382—21no—
113#382—21no—
114#382C2×C411yescharacteristic
115#382C2×C412yes—
116#380—22no—
117#380—22no—
118#380C2×C412yes—
119#380C2×C412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C2×#95—22no—
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×D4C22⋊C411yescharacteristic
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—28no—
155C22×C8—28no—
156C22×C8—82no—
157C22×C8—82no—
158C22×C8—82no—
159C22×C8—82no—
160#125—22no—
161#125—41no—
162#125—81no—
163#125—81no—
164#125—82no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—44no—
169C4×D4—44no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
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×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—82no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—24no—
219C4⋊C8—24no—
220C22⋊C8—28no—
221C22⋊C8—28no—
222#97—81no—
223#97—81no—
224#97—82no—
225#97—82no—
226C4×C8—24no—
227C4×C8—24no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C24C2×C22⋊C412yes—
246C2×D4#10114yes—
247C2×D4#10414yes—
248C2×D4#10414yes—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—88no—
302C2×C8—88no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4#13112yes—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—42no—
350C42—41no—
351C42—42no—
352C42—42no—
353C42—42no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23C2×#10114yes—
359C23C2×#10414yes—
360C23C2×#10414yes—
361C23C22×C22⋊C411yescharacteristic
362D4#330116yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—88no—
412C2×C4—164no—
413C2×C4C2×#10112yes—
414C2×C4C2×#10412yes—
415C2×C4C2×#10412yes—
416C2×C4C2×#13111yescharacteristic
417C2×C4#56811yescharacteristic
418C2×C4#57611yescharacteristic
419C2×C4#57711yescharacteristic
420C2×C4#57711yescharacteristic
421C2×C4#57811yescharacteristic
422C8—42no—
423C8—42no—
424C8—44no—
425C8—44no—
426C8—42no—
427C8—42no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—24no—
434C22C2×#33018yes—
435C22#288011yescharacteristic
436C22#288311yescharacteristic
437C22C22×#10111yescharacteristic
438C22#289711yescharacteristic
439C22#289811yescharacteristic
440C22C22×#10411yescharacteristic
441C22C22×#10411yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—41no—
450C4—42no—
451C4—41no—
452C4—41no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—81no—
457C4—82no—
458C4—81no—
459C4—81no—
460C4C2×#33014yes—
461C2—28no—
462C2#1742611yescharacteristic
463C2#1751711yescharacteristic
464C2#1751911yescharacteristic
465C2#1754711yescharacteristic
466C2#1754811yescharacteristic
467C2C22×#33011yescharacteristic
468C2#1777411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
4. #7691C2semidirect
5. #7691C2semidirect
6. #7683C2semidirect
7. #7680C2semidirect
8. #7636C2semidirect
9. #7627C2semidirect
10. #7568C2semidirect
11. C2×#1271C2semidirect

Maximal subgroups

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

subgroupindex
#362902
#205582
#76912
#76832
#76802
#76362
#76272
#75682
C2×#12712