Finite Group Databasebeta

← #239656

Subgroups — #239656

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7#10415C214yesmaximal, complemented
8#10414C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16#2567—22no—
17#2567—22no—
18#2567—22no—
19#2567C2214yes—
20C2×#395—22no—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389—21no—
24#2285C2211yescharacteristic
25C2×#384—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1981—22no—
28#1956C2212yes—
29#1956C2212yes—
30#1949—22no—
31#1948C2212yes—
32#1947C2212yes—
33#1824—21no—
34#1803C2212yes—
35#1803C2212yes—
36#1787—22no—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1548—24no—
41C2×#333C2214yes—
42#1487—24no—
43C2×Q8⋊C8C2214yes—
44#1279C2214yes—
45C23×Q8D411yescharacteristic
46C23×Q8D412yes—
47C22×M16—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8C2312yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8D412yes—
63C2×C4×Q8C2312yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#428—44no—
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423D414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#395—28no—
98#395—28no—
99#391—24no—
100#391—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#384—24no—
106#384—24no—
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2312yes—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#333—28no—
126Q8⋊C8—28no—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—42no—
135C22×Q8—82no—
136C22×Q8—82no—
137C22×Q8C2×D412yes—
138C22×Q8C2×D414yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173#117—42no—
174#117—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4—84no—
204C2×C4⋊C4SD1614yes—
205C2×C4⋊C4Q1614yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C22⋊C4—41no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—44no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—41no—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—44no—
227#106—44no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—42no—
233C4⋊C8C2×D414yes—
234#103—44no—
235#103—44no—
236#103—84no—
237#103—84no—
238C22⋊C8—42no—
239C22⋊C8—42no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—42no—
243#95—28no—
244#95—28no—
245#95—22no—
246#95—22no—
247#95—41no—
248#95—41no—
249C24#12011yescharacteristic
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8—88no—
268C2×Q8—88no—
269C2×Q8#13714yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#12412yes—
305C22×C4C2×SD1612yes—
306C22×C4C2×Q1612yes—
307C22×C4#13712yes—
308C22×C4#13712yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312M16—48no—
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—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
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—42no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—88no—
347C4⋊C4—88no—
348C4⋊C4C2×SD1614yes—
349C4⋊C4C2×Q1614yes—
350C22⋊C4—82no—
351C42—44no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—44no—
356C42—82no—
357C42#12112yes—
358C42#12112yes—
359C42#12112yes—
360C42#12312yes—
361C42#12312yes—
362C42#12412yes—
363C42C22×D411yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#39412yes—
372C23#44812yes—
373C23#45112yes—
374C23C2×#12011yescharacteristic
375Q8—24no—
376Q8—24no—
377Q8—48no—
378Q8—48no—
379Q8—42no—
380Q8—42no—
381Q8—84no—
382Q8—88no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—42no—
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—44no—
410C2×C4—44no—
411C2×C4—84no—
412C2×C4—88no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4#39412yes—
420C2×C4#39412yes—
421C2×C4#39412yes—
422C2×C4#44812yes—
423C2×C4#45112yes—
424C2×C4#46114yes—
425C2×C4#46214yes—
426C2×C4#48714yes—
427C2×C4#49114yes—
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×#12111yescharacteristic
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×#12311yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C2×C4C2×#12411yescharacteristic
434C2×C4C22×SD1611yescharacteristic
435C2×C4C22×Q1611yescharacteristic
436C2×C4C2×#13712yes—
437C2×C4C2×#13711yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4#57512yes—
440C8—44no—
441C8—82no—
442C8—82no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22#204314yes—
450C22C2×#39411yescharacteristic
451C22C2×#44811yescharacteristic
452C22C2×#45111yescharacteristic
453C22#300411yescharacteristic
454C22#300511yescharacteristic
455C22#301911yescharacteristic
456C22#301911yescharacteristic
457C4—22no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—44no—
462C4—44no—
463C4—44no—
464C4—82no—
465C4—84no—
466C4—161no—
467C4—161no—
468C4#204314yes—
469C4C2×#46111yescharacteristic
470C4C2×#46211yescharacteristic
471C4#305612yes—
472C4#309512yes—
473C4C2×#48711yescharacteristic
474C4C2×#49111yescharacteristic
475C2—24no—
476C2#1874811yescharacteristic
477C2#1874911yescharacteristic
478C2#1876811yescharacteristic
479C2#1876911yescharacteristic
480C2#2348111yescharacteristic
481C2#2349311yescharacteristic
482C2C2×#204311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10415C2semidirect
8. #10414C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#218042
#211702
#207342
C2×#15482
C2×#14872
#104152
#104142
#78922