Finite Group Databasebeta

← #398469

Subgroups — #398469

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#1949C212yesmaximal
3C2×#1865C211yesmaximal, characteristic
4#15230C211yesmaximal, characteristic
5#13817C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#11077C212yesmaximal
10#11052C212yesmaximal
11#10907C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#477—28no—
15C2×#477—24no—
16C22×#107C2212yes—
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C2×#391C2212yes—
24C2×#389C2211yescharacteristic
25C2×#382C2212yes—
26C2×C4×C4⋊C4C2212yes—
27#1949—28no—
28#1865—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1446—22no—
37#1441—21no—
38#1441—21no—
39#1436—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2312yes—
70#477—232no—
71C2×#107—24no—
72C2×#107—24no—
73C2×#107D418yes—
74C2×#106—22no—
75C2×#106—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—24no—
84C2×#103—22no—
85C2×#103—24no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—42no—
89#391—24no—
90#391Q818yescomplemented
91#389—22no—
92#389—22no—
93#382—24no—
94#382—42no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—42no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109C2×#95D412yes—
110#340—24no—
111#340—22no—
112#340—24no—
113#340—24no—
114#340—42no—
115#336—22no—
116#336—24no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8—42no—
128C22×Q8—81no—
129C22×Q8—81no—
130C22×Q8C2×D412yes—
131C22×Q8C2×Q814yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D414yes—
159C22×C8#4112yes—
160C4⋊Q8—28no—
161C4⋊Q8—48no—
162C4⋊Q8—44no—
163C4×Q8—28no—
164C4×Q8—44no—
165C4×Q8—48no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C4⋊C4#4112yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—44no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—82no—
210C2×C42C2×D412yes—
211C2×C42C2×Q814yes—
212C2×C42#4112yes—
213C2×C42#4114yes—
214#107—216no—
215#106—44no—
216C4⋊C8—216no—
217C4⋊C8—28no—
218#103—28no—
219#103—216no—
220#103—44no—
221#95—28no—
222#95—28no—
223#95—22no—
224#95—22no—
225#95—42no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—82no—
232#95—81no—
233#95—82no—
234C24C22×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—48no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—88no—
256C2×Q8—88no—
257C2×Q8C4⋊Q814yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4C22⋊Q814yes—
315C22×C4C22⋊Q814yes—
316C22×C4#13512yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D412yes—
320C22×C4C22×Q812yes—
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4H(5,𝔽2)12yes—
326C22×C42−1+411yescharacteristic
327C22×C42−1+411yescharacteristic
328C2×C8—24no—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—84no—
365C4⋊C4#12014yes—
366C42—22no—
367C42—24no—
368C42—22no—
369C42—24no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—48no—
374C42—88no—
375C23C2×#12111yescharacteristic
376C23C2×C22⋊Q812yes—
377C23D4212yes—
378C23#54711yescharacteristic
379C23#54811yescharacteristic
380C23C2×#13511yescharacteristic
381C23C2×#13711yescharacteristic
382C23#57512yes—
383C23#57611yescharacteristic
384C23#57711yescharacteristic
385C23#57812yes—
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—42no—
390Q8—44no—
391Q8—44no—
392Q8—816no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#46414yes—
453C2×C4C2×#12011yescharacteristic
454C2×C4#52511yescharacteristic
455C2×C4#52512yes—
456C2×C4C2×C4⋊Q811yescharacteristic
457C2×C4#53211yescharacteristic
458C2×C4#53212yes—
459C2×C4#53312yes—
460C2×C4#53411yescharacteristic
461C2×C4#53611yescharacteristic
462C2×C4#53912yes—
463C2×C4#54212yes—
464C2×C4#54612yes—
465C2×C4D4×Q812yes—
466C2×C4D4×Q812yes—
467C2×C4#55211yescharacteristic
468C2×C4#55412yes—
469C2×C4#55412yes—
470C2×C4#55611yescharacteristic
471C8—44no—
472C8—42no—
473C8—81no—
474C22#241011yescharacteristic
475C22#241511yescharacteristic
476C22#244912yes—
477C22#245212yes—
478C22#285711yescharacteristic
479C22C2×#46411yescharacteristic
480C22#304011yescharacteristic
481C22#304411yescharacteristic
482C22#304511yescharacteristic
483C22#307612yes—
484C22#307812yes—
485C22#308212yes—
486C22#308312yes—
487C22#326712yes—
488C22#327012yes—
489C22#327312yes—
490C22#328512yes—
491C22#334011yescharacteristic
492C22#334912yes—
493C22#335711yescharacteristic
494C22#336111yescharacteristic
495C22#337211yescharacteristic
496C22#337712yes—
497C4—24no—
498C4—42no—
499C4—44no—
500C4—44no—
501C4—44no—
502C4—44no—
503C4—84no—
504C4—84no—
505C4—82no—
506C4—161no—
507C4#238811yescharacteristic
508C4#243411yescharacteristic
509C4#246512yes—
510C4#255612yes—
511C4#258311yescharacteristic
512C4#266311yescharacteristic
513C2#1476311yescharacteristic
514C2#2436111yescharacteristic
515C2#2437311yescharacteristic
516C2#2467911yescharacteristic
517C2#2469911yescharacteristic
518C2#2633912yes—
519C2#2635812yes—
520C2#2646812yes—
521C2#2647412yes—
522C2#3320211yescharacteristic
523C2#3320611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
90. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#19492
C2×#18652
#152302
#138172
#113842
#113222
#112272
#110772
#110522
#109072