Finite Group Databasebeta

← #403048

Subgroups — #403048

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1987C211yesmaximal, characteristic
2C2×#1927C211yesmaximal, characteristic
3#15191C211yesmaximal, characteristic
4#13821C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11255C211yesmaximal, characteristic
7#11220C212yesmaximal
8#11077C211yesmaximal, characteristic
9#11067C212yesmaximal
10#10932C212yesmaximal
11#10806C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1987—24no—
25#1927—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1446—22no—
36#1442—21no—
37#1442—21no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42—21no—
60C22×C42—22no—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63C2×#106—22no—
64C2×#106—24no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—42no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103Q814yescomplemented
75#391—24no—
76#391—22no—
77#391—22no—
78#391—42no—
79#389—24no—
80#389—24no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—42no—
84C4×C4⋊C4—44no—
85C4×C4⋊C4D414yes—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D411yescharacteristic
98C2×#95D412yes—
99C2×#95D411yescharacteristic
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#336—22no—
107#336—42no—
108#336—42no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—42no—
121C22×Q8—81no—
122C22×Q8—81no—
123C22×Q8C2×Q812yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8C2×Q812yes—
152C4⋊Q8—28no—
153C4⋊Q8—48no—
154C4⋊Q8—44no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×Q812yes—
182C2×C4⋊C4#4114yes—
183C2×C4⋊C4#4114yes—
184C2×C4⋊C4#4112yes—
185C2×C4⋊C4#4114yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—82no—
200C2×C42C2×D412yes—
201C2×C42#4112yes—
202#106—44no—
203C4⋊C8—44no—
204#103—28no—
205#103—48no—
206#103—44no—
207#95—28no—
208#95—24no—
209#95—22no—
210#95—42no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—82no—
218#95—81no—
219#95—82no—
220#95SD1614yes—
221C24C22×D411yescharacteristic
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—48no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—88no—
244C2×Q8—88no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
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—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4C2×SD1614yes—
303C22×C4C2×SD1612yes—
304C22×C4#13712yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4112yes—
309C22×C4C2×#4112yes—
310C22×C4C2×#4112yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4H(5,𝔽2)12yes—
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C42−1+411yescharacteristic
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—416no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4C22⋊Q814yes—
351C4⋊C4#12414yes—
352C42—22no—
353C42—22no—
354C42—24no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—48no—
360C42—88no—
361C23C2×#12111yescharacteristic
362C23#52512yes—
363C23#54611yescharacteristic
364C23#54612yes—
365C23D4×Q811yescharacteristic
366C23C22×SD1611yescharacteristic
367C23C2×#13711yescharacteristic
368C23#57512yes—
369C23#57611yescharacteristic
370C23#57711yescharacteristic
371C23#57812yes—
372Q8—42no—
373Q8—44no—
374Q8—42no—
375Q8—816no—
376Q8—81no—
377Q8—81no—
378Q8—82no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
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—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4#46114yes—
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4#52511yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4#52912yes—
443C2×C4#52912yes—
444C2×C4#53211yescharacteristic
445C2×C4#53312yes—
446C2×C4#53511yescharacteristic
447C2×C4#53711yescharacteristic
448C2×C4#53812yes—
449C2×C4#54212yes—
450C2×C4#55112yes—
451C2×C4#55211yescharacteristic
452C2×C4#55412yes—
453C2×C4#55411yescharacteristic
454C2×C4#55512yes—
455C2×C4#55612yes—
456C2×C4#55912yes—
457C8—42no—
458C8—81no—
459C8—82no—
460C22#241111yescharacteristic
461C22#241711yescharacteristic
462C22#245912yes—
463C22#246512yes—
464C22#285011yescharacteristic
465C22C2×#46111yescharacteristic
466C22#304011yescharacteristic
467C22#304411yescharacteristic
468C22#304511yescharacteristic
469C22#310112yes—
470C22#311612yes—
471C22#312112yes—
472C22#312412yes—
473C22#329511yescharacteristic
474C22#329812yes—
475C22#331112yes—
476C22#331212yes—
477C22#331312yes—
478C22#332212yes—
479C22#333111yescharacteristic
480C22Q8×SD1611yescharacteristic
481C22#338511yescharacteristic
482C22#339012yes—
483C4—24no—
484C4—44no—
485C4—44no—
486C4—42no—
487C4—44no—
488C4—82no—
489C4—84no—
490C4—82no—
491C4—84no—
492C4—161no—
493C4#239611yescharacteristic
494C4#253212yes—
495C4#256311yescharacteristic
496C4#258912yes—
497C4#259211yescharacteristic
498C4#266211yescharacteristic
499C2#1486611yescharacteristic
500C2#2448011yescharacteristic
501C2#2449311yescharacteristic
502C2#2481511yescharacteristic
503C2#2483211yescharacteristic
504C2#2672212yes—
505C2#2675012yes—
506C2#2700012yes—
507C2#2702812yes—
508C2#3302611yescharacteristic
509C2#3303511yescharacteristic
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
74. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19872
C2×#19272
#151912
#138212
#113842
#112552
#112202
#110772
#110672
#109322
#108062