Finite Group Databasebeta

← #391822

Subgroups — #391822

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic, complemented
5#11385C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11371C211yesmaximal, characteristic, complemented
8#11208C212yesmaximal, complemented
9#11207C212yesmaximal, complemented
10#11200C212yesmaximal
11#11199C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2629—24no—
16#2629C418yescomplemented
17C2×#398—22no—
18C2×#396—22no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1818—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1441—22no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—22no—
46C2×#125—42no—
47C2×#125—42no—
48C2×#125C2×C414yes—
49C2×#117—42no—
50C2×#117—42no—
51C2×#117C2×C414yes—
52C2×#117D414yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×C22⋊C8—24no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—42no—
73#398—216no—
74#396—216no—
75#389—24no—
76#389C2×C418yes—
77#384—22no—
78#384—42no—
79#384C2×C414yes—
80#382—42no—
81#380—24no—
82#380—42no—
83#380—42no—
84#380C2×C418yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336—22no—
102#336—24no—
103C25C2×D411yescharacteristic
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C22×C412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138#125—48no—
139#125—48no—
140#117—48no—
141#117—48no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C22⋊C418yes—
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4C22×C414yes—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4C22⋊C418yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C22⋊C8—48no—
204#95—28no—
205#95—28no—
206#95—42no—
207#95—42no—
208#95—44no—
209#95—82no—
210#95—82no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24C2×C22⋊C412yes—
222C24C22×D411yescharacteristic
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C414yes—
293C22×C4C2×C22⋊C414yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4H(5,𝔽2)12yes—
297C22×C4H(5,𝔽2)12yes—
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+411yescharacteristic
300C22×C42−1+411yescharacteristic
301C22×C42−1+411yescharacteristic
302C22×C42−1+411yescharacteristic
303C22×C42−1+412yes—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—48no—
323C4⋊C4—416no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C4⋊C4—88no—
339C22⋊C4—48no—
340C22⋊C4—88no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23C22×C22⋊C411yescharacteristic
371C23#53411yescharacteristic
372C23#53411yescharacteristic
373C23#53511yescharacteristic
374C23#53611yescharacteristic
375C23#53611yescharacteristic
376C23#53611yescharacteristic
377C23#57612yes—
378C23#57712yes—
379C23#57712yes—
380C23#57812yes—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4#51812yes—
441C2×C4#51911yescharacteristic
442C2×C4#51911yescharacteristic
443C2×C4#52012yes—
444C2×C4#53412yes—
445C2×C4#53611yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#53712yes—
448C2×C4#56212yes—
449C2×C4#56212yes—
450C2×C4#56212yes—
451C2×C4#56212yes—
452C2×C4#56312yes—
453C2×C4#56312yes—
454C2×C4#56312yes—
455C2×C4#56312yes—
456C8—82no—
457C8—82no—
458C22—44no—
459C22—42no—
460C22—44no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—82no—
468C22—82no—
469C22#230811yescharacteristic
470C22#231011yescharacteristic
471C22#231111yescharacteristic
472C22#281611yescharacteristic
473C22#281611yescharacteristic
474C22#281711yescharacteristic
475C22#281711yescharacteristic
476C22#289712yes—
477C22#289812yes—
478C22#319812yes—
479C22#320112yes—
480C22#320312yes—
481C22#320412yes—
482C22#322011yescharacteristic
483C22#322111yescharacteristic
484C22#322211yescharacteristic
485C22#322311yescharacteristic
486C22#324112yes—
487C22#324112yes—
488C22#324212yes—
489C22#324212yes—
490C22#324312yes—
491C22#324412yes—
492C4—44no—
493C4—44no—
494C4—42no—
495C4—84no—
496C4—84no—
497C4—161no—
498C4—161no—
499C4—161no—
500C4—161no—
501C4#237812yes—
502C4#237812yes—
503C4#281412yes—
504C4#281412yes—
505C2—82no—
506C2#1466011yescharacteristic
507C2#2177212yes—
508C2#2184312yes—
509C2#2187211yescharacteristic
510C2#2187311yescharacteristic
511C2#3278211yescharacteristic
512C2#3278311yescharacteristic
513C2#3278411yescharacteristic
514C2#3278511yescharacteristic
515C2#3279312yes—
516C2#3279512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11385C2semidirect
5. #11385C2semidirect
6. #11373C2semidirect
7. #11371C2semidirect
8. #11208C2semidirect
9. #11207C2semidirect
16. #2629C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#18182
#143402
#113852
#113732
#113712
#112082
#112072
#112002
#111992