Finite Group Databasebeta

← #236132

Subgroups — #236132

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7#10067C214yesmaximal, complemented
8#10059C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8—21no—
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3230—24no—
15#3230—24no—
16C2×#500C2212yes—
17C2×#487C2212yes—
18#3104—24no—
19#3104—24no—
20C2×#484—21no—
21C2×#484—22no—
22C2×#477—22no—
23C2×#477C2212yes—
24C2×#462C2212yes—
25C2×#451—21no—
26C2×C4×Q16—22no—
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30#2546—22no—
31#2546—22no—
32#2546—22no—
33#2546C2214yes—
34C2×#398—22no—
35C2×#391—21no—
36C2×#389—22no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1989—22no—
39#1988C2212yes—
40#1987C2212yes—
41#1736—22no—
42#1549—24no—
43C2×#332C2214yes—
44#1487—24no—
45C2×Q8⋊C8C2214yes—
46#1279C2214yes—
47C23×Q8—21no—
48C23×Q8D411yescharacteristic
49C22×Q16—22no—
50C22×M16—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533D414yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—48no—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2312yes—
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—21no—
78C2×C4×Q8—22no—
79C2×C4×Q8D412yes—
80C2×C4×Q8C2312yes—
81C2×#117—21no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D412yes—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#500—24no—
92#500—24no—
93#487—24no—
94#487—24no—
95#484—28no—
96#484—216no—
97#477—216no—
98#477—24no—
99#477—24no—
100#462—24no—
101#462—24no—
102#451—28no—
103C4×Q16—216no—
104#431—22no—
105#431—22no—
106#431D414yes—
107#431D414yes—
108C2×#107—24no—
109C2×#107—22no—
110#423—22no—
111#423—22no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#103—22no—
117C2×#103—24no—
118C2×#103—22no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2312yes—
123#398—28no—
124#398—28no—
125#391—24no—
126#391—24no—
127#389—28no—
128#389—28no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2312yes—
133C2×#95—24no—
134C2×#95—21no—
135#340—24no—
136#340—24no—
137#332—28no—
138Q8⋊C8—28no—
139C22×Q8—24no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—42no—
148C22×Q8—44no—
149C22×Q8—44no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8C2×D412yes—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×Q16—28no—
161C2×Q16—44no—
162C2×Q16—44no—
163C2×M16—42no—
164C2×M16—44no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—22no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—416no—
181C4⋊Q8—48no—
182C4⋊Q8—48no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8C2×D412yes—
188C4⋊Q8C2×D412yes—
189#125—24no—
190#125—42no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—42no—
194C22⋊Q8—42no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209#117—24no—
210#117—42no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—416no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4Q1618yescomplemented
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—41no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—41no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42#4112yes—
255C2×C42C2411yescharacteristic
256#107—28no—
257#107—216no—
258C4⋊C8—24no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—42no—
263C4⋊C8C2×D414yes—
264#103—28no—
265#103—28no—
266#103—216no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C4×C8—24no—
270C4×C8—22no—
271C4×C8C2×D414yes—
272#95—216no—
273#95—22no—
274#95—22no—
275C24#12011yescharacteristic
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—42no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—42no—
297C2×Q8—44no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—88no—
301C2×Q8—88no—
302C2×Q8AGL(1,ℤ/8)14yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—28no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4#12012yes—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12012yes—
327C22×C4#12011yescharacteristic
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12712yes—
332C22×C4C2×Q1614yes—
333C22×C4#13714yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337Q16—416no—
338M16—84no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—44no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—44no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—432no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4C2×Q1618yes—
372C22⋊C4—44no—
373C42—44no—
374C42—42no—
375C42—42no—
376C42—44no—
377C42—44no—
378C42—44no—
379C42#12012yes—
380C42#12012yes—
381C42#12112yes—
382C42#12112yes—
383C42#12112yes—
384C42#12712yes—
385C42C22×D411yescharacteristic
386C42C22×D411yescharacteristic
387C42C22×D411yescharacteristic
388C42C2×#4111yescharacteristic
389C23—21no—
390C23—21no—
391C23—22no—
392C23#39212yes—
393C23#45114yes—
394C23C2×#12011yescharacteristic
395Q8—24no—
396Q8—24no—
397Q8—48no—
398Q8—44no—
399Q8—42no—
400Q8—42no—
401Q8—44no—
402Q8—82no—
403Q8—88no—
404Q8—84no—
405Q8—82no—
406Q8—84no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—416no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—84no—
431C2×C4—88no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4#39212yes—
436C2×C4#39212yes—
437C2×C4#39212yes—
438C2×C4#45114yes—
439C2×C4#46218yes—
440C2×C4#49414yes—
441C2×C4#49714yes—
442C2×C4C2×#12011yescharacteristic
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12711yescharacteristic
448C2×C4C22×Q1612yes—
449C2×C4C2×AGL(1,ℤ/8)12yes—
450C2×C4C2×#13712yes—
451C2×C4#57512yes—
452C8—44no—
453C8—44no—
454C8—82no—
455C22—24no—
456C22—24no—
457C22—22no—
458C22—22no—
459C22#201614yes—
460C22C2×#39211yescharacteristic
461C22C2×#45112yes—
462C22#300311yescharacteristic
463C22#300511yescharacteristic
464C22#302012yes—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—22no—
469C4—48no—
470C4—44no—
471C4—84no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4#201614yes—
476C4#300612yes—
477C4C2×#46212yes—
478C4#305512yes—
479C4C2×#49411yescharacteristic
480C4C2×#49711yescharacteristic
481C2—24no—
482C2#1875912yes—
483C2#1877912yes—
484C2#2339111yescharacteristic
485C2#2340311yescharacteristic
486C2C2×#201611yescharacteristic
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. #10067C2semidirect
8. #10059C2semidirect
235. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#31042
C2×#25462
#213122
C2×#15492
C2×#14872
#100672
#100592
#78922