Finite Group Databasebeta

← #239661

Subgroups — #239661

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7#10419C214yesmaximal, complemented
8#10418C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#428—21no—
16C2×#423C2211yescharacteristic
17#2546—22no—
18#2546—22no—
19#2546—22no—
20#2546C2214yes—
21C2×#398—22no—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389—21no—
25#2285C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2212yes—
28#1990C2212yes—
29#1987—22no—
30#1949—22no—
31#1948C2212yes—
32#1947C2212yes—
33#1824—22no—
34#1824—21no—
35#1803C2212yes—
36#1803C2212yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1549—24no—
41C2×#332C2214yes—
42#1487—24no—
43C2×Q8⋊C8C2214yes—
44#1279C2214yes—
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C22×M16—21no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—48no—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8C2312yes—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8D412yes—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#428—44no—
84C2×#107—22no—
85C2×#106—22no—
86#423—22no—
87#423D414yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—22no—
93C2×#103—22no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98#398—28no—
99#398—28no—
100#391—24no—
101#391—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—24no—
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#382—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2312yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#340—22no—
125#340—22no—
126#332—28no—
127Q8⋊C8—28no—
128C22×Q8—24no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—42no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8C2×D412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C2×M16—42no—
150C2×M16—42no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—416no—
161C4⋊Q8—48no—
162C4⋊Q8—48no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177#117—42no—
178#117—42no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4SD1614yes—
208C2×C4⋊C4Q1614yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—41no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—41no—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#107—44no—
231#106—44no—
232C4⋊C8—24no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8C2×D414yes—
238#103—44no—
239#103—44no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—42no—
245#95—28no—
246#95—28no—
247#95—22no—
248#95—22no—
249#95—41no—
250#95—41no—
251C24#12011yescharacteristic
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
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—42no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8AGL(1,ℤ/8)14yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12312yes—
305C22×C4#12312yes—
306C22×C4#12412yes—
307C22×C4C2×SD1612yes—
308C22×C4C2×Q1612yes—
309C22×C4#13712yes—
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—48no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—416no—
333C4⋊C4—416no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4C2×SD1614yes—
351C4⋊C4C2×Q1614yes—
352C22⋊C4—82no—
353C42—44no—
354C42—42no—
355C42—42no—
356C42—44no—
357C42—44no—
358C42—82no—
359C42#12112yes—
360C42#12112yes—
361C42#12112yes—
362C42#12312yes—
363C42#12312yes—
364C42#12412yes—
365C42C22×D411yescharacteristic
366C42C2×#4111yescharacteristic
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23#39412yes—
374C23#44812yes—
375C23#45112yes—
376C23C2×#12011yescharacteristic
377Q8—24no—
378Q8—24no—
379Q8—48no—
380Q8—42no—
381Q8—42no—
382Q8—88no—
383Q8—84no—
384Q8—84no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—48no—
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—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—84no—
414C2×C4—88no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#39412yes—
422C2×C4#39412yes—
423C2×C4#39412yes—
424C2×C4#44812yes—
425C2×C4#45112yes—
426C2×C4#46114yes—
427C2×C4#46214yes—
428C2×C4#48714yes—
429C2×C4#49114yes—
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#12411yescharacteristic
436C2×C4C22×SD1611yescharacteristic
437C2×C4C22×Q1611yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)12yes—
439C2×C4C2×#13711yescharacteristic
440C2×C4C2×#13711yescharacteristic
441C2×C4#57512yes—
442C8—44no—
443C8—82no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22#204314yes—
452C22C2×#39411yescharacteristic
453C22C2×#44811yescharacteristic
454C22C2×#45111yescharacteristic
455C22#300311yescharacteristic
456C22#300511yescharacteristic
457C22#302011yescharacteristic
458C22#302011yescharacteristic
459C4—22no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—44no—
464C4—44no—
465C4—44no—
466C4—84no—
467C4—82no—
468C4—161no—
469C4—161no—
470C4#204314yes—
471C4C2×#46111yescharacteristic
472C4C2×#46211yescharacteristic
473C4#305512yes—
474C4#309412yes—
475C4C2×#48711yescharacteristic
476C4C2×#49111yescharacteristic
477C2—24no—
478C2#1875811yescharacteristic
479C2#1875911yescharacteristic
480C2#1877811yescharacteristic
481C2#1877911yescharacteristic
482C2#2348011yescharacteristic
483C2#2349411yescharacteristic
484C2C2×#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. #10419C2semidirect
8. #10418C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
#218342
#211702
#207342
C2×#15492
C2×#14872
#104192
#104182
#78922