Finite Group Databasebeta

← #163928

Subgroups — #163928

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic
2#22643C211yesmaximal, characteristic, complemented
3#22636C211yesmaximal, characteristic, complemented
4#7642C212yesmaximal, complemented
5#7628C212yesmaximal, complemented
6#7614C212yesmaximal, complemented
7#7608C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7572C211yesmaximal, characteristic
10#7572C211yesmaximal, characteristic
11#7564C211yesmaximal, characteristic, complemented
12C2×#518C411yescharacteristic
13#2357—21no—
14#2357—21no—
15#2354—21no—
16#2354—21no—
17#2342—22no—
18#2342—22no—
19#2319C412yes—
20#2319C412yes—
21#2317C411yescharacteristic
22#2309C411yescharacteristic
23#2296C2211yescharacteristic
24#2293C2212yes—
25#2293C2212yes—
26C2×#390C2211yescharacteristic
27#2274C411yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1924—22no—
33#1839C2212yescomplemented
34#1831C2212yescomplemented
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38#1770—21no—
39#1767C2212yescomplemented
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1745C2212yescomplemented
43#1271C2211yescharacteristic, complemented
44#1271C2211yescharacteristic, complemented
45#1271C2211yescharacteristic, complemented
46#1271C2211yescharacteristic, complemented
47#1271C2212yescomplemented
48#1271C2212yescomplemented
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C23×D4C2×C411yescharacteristic
54#518—24no—
55#518—24no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×#97—22no—
80C2×#97—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2312yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C411yescharacteristic
94#389C2×C411yescharacteristic
95#389C2311yescharacteristic
96#389C2311yescharacteristic
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#385—21no—
106#385—21no—
107#385—22no—
108#384—21no—
109#382—21no—
110#381—22no—
111#381—22no—
112#380—22no—
113#380—22no—
114#379—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
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⋊C4—22no—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130#376C2×C412yes—
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
142C25—21no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4M1614yes—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C4×D4—28no—
162C4×D4—28no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—42no—
171#117—42no—
172#117—82no—
173#117—82no—
174#117—82no—
175#117—82no—
176#117M1614yes—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—216no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—81no—
208C2×C22⋊C4—81no—
209C2×C22⋊C4—81no—
210C2×C22⋊C4—81no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—42no—
218C2×C42C22×C412yes—
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2411yescharacteristic
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—44no—
231C4⋊C8—82no—
232C22⋊C8—42no—
233C22⋊C8—42no—
234C22⋊C8—81no—
235C22⋊C8—81no—
236#97—44no—
237#97—82no—
238C4×C8—42no—
239C4×C8—42no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24—22no—
253C24—24no—
254C24—42no—
255C24—42no—
256C24—81no—
257C24—81no—
258C24—81no—
259C24—81no—
260C24C2×C22⋊C411yescharacteristic
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—48no—
266C2×D4—48no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4C2×M1614yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×M1612yes—
302C22×C4C2×M1612yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4C2×M1612yes—
343C4⋊C4C2×M1612yes—
344C22⋊C4—28no—
345C22⋊C4—28no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—24no—
367C42—22no—
368C42—22no—
369C42—42no—
370C42—42no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—42no—
375C42—82no—
376C42—82no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—24no—
384C23—48no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—84no—
392C23—84no—
393C23—162no—
394C23—162no—
395C23—162no—
396C23—162no—
397C23#40712yes—
398C23C22×C22⋊C411yescharacteristic
399D4—44no—
400D4—44no—
401D4—48no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—88no—
448C2×C4—88no—
449C2×C4—162no—
450C2×C4#40712yes—
451C2×C4#41012yes—
452C2×C4#41312yes—
453C2×C4C22×M1611yescharacteristic
454C2×C4C22×M1611yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57611yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57711yescharacteristic
459C8—42no—
460C8—42no—
461C8—82no—
462C8—81no—
463C8—81no—
464C8—82no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—168no—
480C22C2×#40711yescharacteristic
481C22C2×#41011yescharacteristic
482C22C2×#41311yescharacteristic
483C22#289711yescharacteristic
484C22#289711yescharacteristic
485C22#290111yescharacteristic
486C22#290111yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—41no—
500C4—41no—
501C4—82no—
502C4—81no—
503C4—81no—
504C4#146414yes—
505C2—42no—
506C2—84no—
507C2C2×#146411yescharacteristic
508C2#1745311yescharacteristic
509C2#1745311yescharacteristic
510C2#1760711yescharacteristic
511C2#1761011yescharacteristic
512C2#1769111yescharacteristic
513C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×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
2. #22643C2semidirect
3. #22636C2semidirect
4. #7642C2semidirect
5. #7628C2semidirect
6. #7614C2semidirect
7. #7608C2semidirect
8. #7598C2semidirect
11. #7564C2semidirect
33. #1839C22semidirect
34. #1831C22semidirect
39. #1767C22semidirect
42. #1745C22semidirect
43. #1271C22semidirect
44. #1271C22semidirect
45. #1271C22semidirect
46. #1271C22semidirect
47. #1271C22semidirect
48. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#226432
#226362
#76422
#76282
#76142
#76082
#75982
#75722
#75642