Finite Group Databasebeta

← #393392

Subgroups — #393392

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1989C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic
4#14283C211yesmaximal, characteristic
5#13750C211yesmaximal, characteristic
6#10908C212yesmaximal
7#10908C212yesmaximal
8#10849C211yesmaximal, characteristic
9#10833C211yesmaximal, characteristic
10#10807C212yesmaximal
11#10806C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×#376C2211yescharacteristic
22#1989—24no—
23#1989C418yescomplemented
24#1948—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1441—21no—
37#1441—21no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4×#95—21no—
42C4×#95—22no—
43C4×#95—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C42—22no—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#106—24no—
70C2×#106—24no—
71C2×#106—24no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2×C414yes—
78#384—22no—
79#384—22no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4C2×C414yes—
86C4×C4⋊C4D414yes—
87C4×C4⋊C4Q818yes—
88#376—22no—
89#376—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D412yes—
100C2×#95D411yescharacteristic
101#340—24no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340C2×C418yes—
109#340C2×C418yes—
110#336—24no—
111#336—24no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8C22×C414yes—
140#125—28no—
141#125—28no—
142#125—44no—
143#125—44no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—81no—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C4⋊C4C2×Q814yes—
185C2×C4⋊C4#4114yes—
186C2×C4⋊C4#4114yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—28no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—28no—
195C2×C42—81no—
196C2×C42—81no—
197C2×C42—82no—
198C2×C42—82no—
199C2×C42—82no—
200C2×C42—82no—
201C2×C42—82no—
202C2×C42C22×C412yes—
203C2×C42C2×D412yes—
204C2×C42C2×Q814yes—
205C2×C42#4114yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210#106—48no—
211C4⋊C8—28no—
212C4⋊C8—44no—
213#95—28no—
214#95—24no—
215#95—24no—
216#95—28no—
217#95—22no—
218#95—24no—
219#95—22no—
220#95—82no—
221#95—82no—
222#95—82no—
223#95—82no—
224C24C22×D411yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C4×D412yes—
280C22×C4C4×D412yes—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4C22⋊Q814yes—
284C22×C4C22⋊Q814yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×Q812yes—
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C4C4×Q818yes—
338C4⋊C4#12414yes—
339C4⋊C4C4⋊Q814yes—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—84no—
345C42—84no—
346C42—88no—
347C42—88no—
348C42#11714yes—
349C42C4×D414yes—
350C42#12414yes—
351C42#12614yes—
352C23C2×C4×D411yescharacteristic
353C23C2×#12111yescharacteristic
354C23C2×C22⋊Q812yes—
355C23C2×#12311yescharacteristic
356C23#52512yes—
357C23#57611yescharacteristic
358C23#57712yes—
359C23#57712yes—
360C23#57711yescharacteristic
361C23#57711yescharacteristic
362C23#57811yescharacteristic
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—28no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#38814yes—
424C2×C4#38814yes—
425C2×C4C2×#11711yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×C4×Q812yes—
428C2×C4C4×#4112yes—
429C2×C4C2×#12411yescharacteristic
430C2×C4C2×#12411yescharacteristic
431C2×C4C2×#12611yescharacteristic
432C2×C4#52912yes—
433C2×C4#52912yes—
434C2×C4#52912yes—
435C2×C4#52912yes—
436C2×C4C2×C4⋊Q811yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4#53211yescharacteristic
439C2×C4#53312yes—
440C2×C4#53312yes—
441C2×C4#53312yes—
442C2×C4#53312yes—
443C8—44no—
444C8—81no—
445C8—81no—
446C22C2×#38811yescharacteristic
447C22C4×C22⋊Q812yes—
448C22#285011yescharacteristic
449C22#285512yes—
450C22#285711yescharacteristic
451C22#294611yescharacteristic
452C22#294711yescharacteristic
453C22#295212yes—
454C22#304211yescharacteristic
455C22#304311yescharacteristic
456C22#304511yescharacteristic
457C22#304611yescharacteristic
458C22#308112yes—
459C22#308212yes—
460C22#308412yes—
461C22#308412yes—
462C22#309811yescharacteristic
463C22#309911yescharacteristic
464C22#309912yes—
465C22#311912yes—
466C22#312012yes—
467C22#312312yes—
468C22#312612yes—
469C4—24no—
470C4—24no—
471C4—44no—
472C4—48no—
473C4—44no—
474C4—44no—
475C4—44no—
476C4—82no—
477C4—82no—
478C4—162no—
479C4#229512yes—
480C4C4×#12411yescharacteristic
481C4C4×C4⋊Q811yescharacteristic
482C4#286111yescharacteristic
483C4#286211yescharacteristic
484C4#286312yes—
485C2#1470311yescharacteristic
486C2#2106211yescharacteristic
487C2#2106811yescharacteristic
488C2#2161912yes—
489C2#2162212yes—
490C2#3303911yescharacteristic
491C2#3304011yescharacteristic
492C2#3315712yes—
493C2#3315812yes—
494C2#3320411yescharacteristic
495C2#3320511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1989C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19892
C2×#19482
#142832
#137502
#109082
#108492
#108332
#108072
#108062