Finite Group Databasebeta

← #163941

Subgroups — #163941

Order 512.

824 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#34272C211yesmaximal, characteristic
2#22643C211yesmaximal, characteristic
3#22643C211yesmaximal, characteristic
4#7650C212yesmaximal, complemented
5#7636C212yesmaximal, complemented
6#7625C212yesmaximal, complemented
7#7622C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic
9#7576C211yesmaximal, characteristic
10#7573C211yesmaximal, characteristic
11#7564C211yesmaximal, characteristic
12C2×#544C411yescharacteristic
13#2370—21no—
14#2370—21no—
15#2369—21no—
16#2368—22no—
17#2365—21no—
18#2360—22no—
19#2331C412yes—
20#2330C412yes—
21#2317C411yescharacteristic
22C4×C4⋊Q8C2212yes—
23C4×#125C2212yes—
24#2296C2211yescharacteristic
25C2×#389C2211yescharacteristic
26#2274C411yescharacteristic
27#2274C411yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1924—22no—
33#1839C2212yes—
34#1839C2212yes—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38#1770—21no—
39#1767C2212yes—
40#1767C2212yes—
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53#544—24no—
54#544—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C412yes—
57C2×#125—21no—
58C2×#125C2×C412yes—
59C2×C22⋊Q8—22no—
60C2×C4×Q8—22no—
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×#97—22no—
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#391—22no—
82#391—22no—
83#391—22no—
84#389—21no—
85#389—21no—
86#389—22no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389C2×C411yescharacteristic
92#389C2×C412yes—
93#389C2×C411yescharacteristic
94#389C2311yescharacteristic
95#389C2312yes—
96#389C2311yescharacteristic
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—21no—
104#385—21no—
105#385—22no—
106#384C2×C412yes—
107#382C2×C412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
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⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129#376C2×C412yes—
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—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—28no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157#125—28no—
158#125—24no—
159#125—24no—
160C22⋊Q8—216no—
161C4×Q8—82no—
162C4×Q8—82no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—24no—
167#117—42no—
168#117M1614yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4M1614yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—42no—
219C4⋊C8—42no—
220C4⋊C8—44no—
221C4⋊C8—44no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C22⋊C8—81no—
225C22⋊C8—81no—
226#97—82no—
227#97—82no—
228C4×C8—42no—
229C4×C8—42no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246C24C2×C22⋊C411yescharacteristic
247C2×Q8—28no—
248C2×Q8—82no—
249C2×Q8—82no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×M1612yes—
280C22×C4C2×M1612yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—41no—
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—82no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4#10114yes—
328C4⋊C4C2×M1614yes—
329C4⋊C4C2×M1612yes—
330C4⋊C4C2×M1612yes—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—42no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—24no—
345C42—42no—
346C42—42no—
347C42—41no—
348C42—41no—
349C42—42no—
350C42—82no—
351C42—82no—
352C42—82no—
353C42—82no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23#40712yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—88no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—84no—
411C2×C4—162no—
412C2×C4#40712yes—
413C2×C4#41012yes—
414C2×C4C2×#10114yes—
415C2×C4C2×#10112yes—
416C2×C4C22×M1611yescharacteristic
417C2×C4C22×M1611yescharacteristic
418C2×C4#57611yescharacteristic
419C2×C4#57711yescharacteristic
420C2×C4#57711yescharacteristic
421C2×C4#57811yescharacteristic
422C8—42no—
423C8—42no—
424C8—82no—
425C8—81no—
426C8—82no—
427C8—81no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22C2×#40711yescharacteristic
436C22C2×#41011yescharacteristic
437C22C22×#10111yescharacteristic
438C22#289711yescharacteristic
439C22#289811yescharacteristic
440C22#290111yescharacteristic
441C22#290111yescharacteristic
442C4—24no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—44no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—41no—
456C4—41no—
457C4—81no—
458C4—82no—
459C4—81no—
460C4#146714yes—
461C2—42no—
462C2C2×#146711yescharacteristic
463C2#1745311yescharacteristic
464C2#1745411yescharacteristic
465C2#1760711yescharacteristic
466C2#1761211yescharacteristic
467C2#1768911yescharacteristic
468C2#1777711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×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
4. #7650C2semidirect
5. #7636C2semidirect
6. #7625C2semidirect
7. #7622C2semidirect

Maximal subgroups

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

subgroupindex
#342722
#226432
#76502
#76362
#76252
#76222
#75982
#75762
#75732
#75642