Finite Group Databasebeta

← #163877

Subgroups — #163877

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34254C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#7598C211yesmaximal, characteristic
5#7596C212yesmaximal, complemented
6#7581C212yesmaximal, complemented
7#7577C212yesmaximal, complemented
8#7575C211yesmaximal, characteristic
9#7575C211yesmaximal, characteristic
10#7564C211yesmaximal, characteristic
11#7562C212yesmaximal, complemented
12C2×#519C411yescharacteristic
13#2356—21no—
14#2356—21no—
15#2355—21no—
16#2355—21no—
17#2343—22no—
18#2343—22no—
19#2320C412yes—
20#2320C412yes—
21#2317C411yescharacteristic
22#2310C411yescharacteristic
23#2296C2211yescharacteristic
24#2295C2212yes—
25#2295C2212yes—
26C2×#391C2211yescharacteristic
27#2274C411yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1923—22no—
32#1923—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37C8×C4⋊C4—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1744C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C23×Q8C2×C411yescharacteristic
54#519—24no—
55#519C818yescomplemented
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C412yes—
60C2×#117—22no—
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C4×C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—22no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2312yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C411yescharacteristic
90#389C2×C411yescharacteristic
91#389C2311yescharacteristic
92#389C2311yescharacteristic
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#385—21no—
102#385—21no—
103#385—22no—
104#384—21no—
105#383—22no—
106#383—22no—
107#382—21no—
108#382—22no—
109#382—22no—
110#381—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D412yes—
126#376—22no—
127#376—22no—
128#376C2×C412yes—
129C2×#95—21no—
130C2×#95—21no—
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8C2×C814yes—
147C22×Q8C22×C412yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C4×Q8—28no—
160C4×Q8—28no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8C2×C818yes—
164C4×Q8C2×C818yes—
165#117—22no—
166#117—216no—
167#117—42no—
168#117—42no—
169#117C2×C814yes—
170#117C2×C814yes—
171#117M1614yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—44no—
215C4⋊C8—82no—
216C22⋊C8—42no—
217C22⋊C8—42no—
218C22⋊C8—81no—
219C22⋊C8—81no—
220C4×C8—42no—
221C4×C8—42no—
222C4×C8—44no—
223C4×C8—82no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—48no—
248C2×Q8—48no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8C22×C814yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C22×C812yes—
284C22×C4C2×M1612yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4C22⋊C814yes—
325C4⋊C4C22⋊C814yes—
326C4⋊C4C22×C812yes—
327C4⋊C4C22×C812yes—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4C22⋊C818yes—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—28no—
337C42—28no—
338C42—24no—
339C42—22no—
340C42—22no—
341C42—42no—
342C42—42no—
343C42—41no—
344C42—41no—
345C42—42no—
346C42—42no—
347C42—82no—
348C42—82no—
349C42C22×C814yes—
350C42C2×M1614yes—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C2×C22⋊C812yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—44no—
360Q8—44no—
361Q8—48no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—162no—
410C2×C4C2×C22⋊C814yes—
411C2×C4C2×C22⋊C814yes—
412C2×C4C2×C22⋊C812yes—
413C2×C4#41012yes—
414C2×C4#41312yes—
415C2×C4C23×C811yescharacteristic
416C2×C4C22×M1611yescharacteristic
417C2×C4#57711yescharacteristic
418C2×C4#57711yescharacteristic
419C2×C4#57811yescharacteristic
420C2×C4#57811yescharacteristic
421C8—42no—
422C8—42no—
423C8—81no—
424C8—82no—
425C8—82no—
426C8—81no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22C22×C22⋊C811yescharacteristic
435C22C2×#41011yescharacteristic
436C22C2×#41311yescharacteristic
437C22#289811yescharacteristic
438C22#289811yescharacteristic
439C22#290111yescharacteristic
440C22#290111yescharacteristic
441C4—24no—
442C4—24no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—44no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—81no—
458C4—81no—
459C4#145814yes—
460C2—42no—
461C2C2×#145811yescharacteristic
462C2#1743911yescharacteristic
463C2#1743911yescharacteristic
464C2#1760511yescharacteristic
465C2#1761211yescharacteristic
466C2#1769111yescharacteristic
467C2#1777811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #7596C2semidirect
6. #7581C2semidirect
7. #7577C2semidirect
11. #7562C2semidirect
55. #519C8semidirect

Maximal subgroups

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

subgroupindex
#342542
#226392
#226322
#75982
#75962
#75812
#75772
#75752
#75642
#75622