Finite Group Databasebeta

← #403520

Subgroups — #403520

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2C2×#2066C212yesmaximal
3C2×#2058C211yesmaximal, characteristic
4C2×#2024C211yesmaximal, characteristic
5#14261C211yesmaximal, characteristic
6#11228C212yesmaximal
7#11208C211yesmaximal, characteristic
8#11052C211yesmaximal, characteristic
9#10932C212yesmaximal
10#10908C212yesmaximal
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17#2331—28no—
18C2×#391—22no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—22no—
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×#376C2211yescharacteristic
28#2066—28no—
29#2058—24no—
30#2024—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1443—22no—
40#1442—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8D414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—42no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—42no—
75#391—216no—
76#391—22no—
77#391—24no—
78#391—22no—
79#391—22no—
80#391Q814yescomplemented
81#389—24no—
82#389—22no—
83#389—22no—
84#389D418yes—
85#387—216no—
86#382—24no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—24no—
89#376—24no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D412yes—
102C2×#95D412yes—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#336—22no—
109#336—22no—
110#336—24no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—82no—
124C22×Q8—82no—
125C22×Q8C2×Q812yes—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8#4114yes—
151C4⋊Q8—216no—
152C4⋊Q8—216no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×Q812yes—
186C2×C4⋊C4#4114yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—28no—
195C2×C42—82no—
196C2×C42—82no—
197C2×C42C2×D414yes—
198C2×C42C2×D412yes—
199C2×C42C2×Q812yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202#106—48no—
203C4⋊C8—48no—
204#103—48no—
205#103—48no—
206#95—28no—
207#95—24no—
208#95—24no—
209#95—28no—
210#95—24no—
211#95—24no—
212#95—28no—
213#95—44no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—82no—
218#95—82no—
219C24C22×D411yescharacteristic
220C2×Q8—216no—
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—84no—
238C2×Q8—84no—
239C2×Q8—88no—
240C2×Q8—88no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
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—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4#12014yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4C22⋊Q814yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×Q811yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C42−1+411yescharacteristic
310C22×C42−1+412yes—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—216no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—88no—
342C4⋊C4—84no—
343C4⋊C4#12118yes—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—24no—
350C42—88no—
351C23C2×#12011yescharacteristic
352C23C2×#12111yescharacteristic
353C23C2×C22⋊Q811yescharacteristic
354C23#52511yescharacteristic
355C23#53211yescharacteristic
356C23#53611yescharacteristic
357C23#53711yescharacteristic
358C23#57712yes—
359C23#57712yes—
360C23#57712yes—
361C23#57812yes—
362Q8—44no—
363Q8—44no—
364Q8—816no—
365Q8—82no—
366Q8—82no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4C2×#12112yes—
428C2×C4#52512yes—
429C2×C4#52912yes—
430C2×C4#53312yes—
431C2×C4#54112yes—
432C2×C4#54212yes—
433C2×C4#54612yes—
434C2×C4#54712yes—
435C2×C4#54812yes—
436C2×C4#54812yes—
437C2×C4D4×Q811yescharacteristic
438C2×C4D4×Q812yes—
439C2×C4#55011yescharacteristic
440C2×C4#55411yescharacteristic
441C2×C4#55612yes—
442C2×C4#55611yescharacteristic
443C8—44no—
444C8—82no—
445C22#239611yescharacteristic
446C22#240911yescharacteristic
447C22#243011yescharacteristic
448C22#253811yescharacteristic
449C22#254511yescharacteristic
450C22#263711yescharacteristic
451C22#284711yescharacteristic
452C22#300912yes—
453C22#301312yes—
454C22#304211yescharacteristic
455C22#304211yescharacteristic
456C22#304311yescharacteristic
457C22#304411yescharacteristic
458C22#308212yes—
459C22#308412yes—
460C22#312112yes—
461C22#312312yes—
462C22#317312yes—
463C22#317812yes—
464C22#324212yes—
465C22#324412yes—
466C22#325812yes—
467C22#326312yes—
468C4—42no—
469C4—48no—
470C4—42no—
471C4—44no—
472C4—44no—
473C4—82no—
474C4—82no—
475C4—84no—
476C4—161no—
477C4—161no—
478C4#242612yes—
479C4#242912yes—
480C4#246512yes—
481C4#246912yes—
482C2#1487711yescharacteristic
483C2#2378912yes—
484C2#2431012yes—
485C2#2550912yes—
486C2#2828911yescharacteristic
487C2#2829111yescharacteristic
488C2#2866211yescharacteristic
489C2#2866611yescharacteristic
490C2#3016012yes—
491C2#3291411yescharacteristic
492C2#3291811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
80. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#20662
C2×#20582
C2×#20242
#142612
#112282
#112082
#110522
#109322
#109082
#108332