Finite Group Databasebeta

← #304520

Subgroups — #304520

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic, complemented
3#33162C211yesmaximal, characteristic
4#29219C211yesmaximal, characteristic, complemented
5#10350C214yesmaximal, complemented
6#9898C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8985C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×D4×Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#461—21no—
13C2×#461—21no—
14C2×#461C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2857—21no—
18#2551—21no—
19#2551C2212yes—
20#2546—21no—
21#2440—22no—
22#2440—22no—
23#2434—22no—
24D4×C4⋊C4—21no—
25D4×C4⋊C4C2212yes—
26C4×C4⋊Q8C2211yescharacteristic
27C42×D4C2211yescharacteristic
28#2099—22no—
29#1990—22no—
30#1990—21no—
31#1990C2214yes—
32#1988C2214yes—
33#1981—21no—
34#1981—21no—
35#1981C2212yes—
36#1842C2211yescharacteristic
37#1834—22no—
38#1834—21no—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42#1296C2214yes—
43#1279C2214yes—
44#1279C2212yes—
45C23×Q8—22no—
46C22×SD16—22no—
47D4×Q8—44no—
48D4×Q8—44no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—44no—
58C2×C4⋊Q8—44no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8C2312yes—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—24no—
64C2×C4×Q8—21no—
65C2×C4×D4—22no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C42—21no—
72#461—28no—
73#461—44no—
74#461—44no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2312yes—
86#398—24no—
87#395—24no—
88#395—24no—
89#391—22no—
90#391—21no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—21no—
96#389D412yes—
97#385—24no—
98#385—24no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384—21no—
105#384D411yescharacteristic
106#384D412yes—
107#384C2312yes—
108#380—22no—
109#380—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2311yescharacteristic
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C43C2311yescharacteristic
119#340—24no—
120#340—28no—
121#340—24no—
122#336—21no—
123D4⋊C8—24no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—48no—
129C22×Q8—48no—
130C22×Q8—84no—
131C22×D4C2×D411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C2×SD16—44no—
137C2×SD16—44no—
138C2×SD16—44no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—48no—
148C4⋊Q8—44no—
149C4⋊Q8—48no—
150C4⋊Q8—44no—
151C4⋊Q8—44no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8SD1614yescomplemented
161C22⋊Q8—416no—
162C22⋊Q8—48no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4D814yes—
171C4×D4SD1614yes—
172C4×D4#4114yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D414yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—24no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—24no—
219C4⋊C8—22no—
220C4⋊C8—42no—
221C4⋊C8—42no—
222#103—44no—
223#103—44no—
224#103—44no—
225C4×C8—24no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—42no—
229C4×C8—42no—
230#95—24no—
231#95—24no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—22no—
237C24—21no—
238C2×Q8—24no—
239C2×Q8—48no—
240C2×Q8—48no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—84no—
253C2×Q8—832no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4#12412yes—
258C2×D4C2×D812yes—
259C2×D4C2×SD1612yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—48no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—84no—
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287SD16—88no—
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4#12412yes—
324C4⋊C4C2×D812yes—
325C4⋊C4C2×SD1612yes—
326C4⋊C4C2×SD1614yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—48no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—42no—
335C42—41no—
336C42—41no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42C2×D812yes—
341C42C2×SD1612yes—
342C42C2×SD1612yes—
343C42AGL(1,ℤ/8)12yes—
344C42AGL(1,ℤ/8)12yes—
345C42#13712yes—
346C42C2×#4112yes—
347C422−1+412yes—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23C2×#12011yescharacteristic
356Q8—48no—
357Q8—816no—
358Q8—88no—
359Q8—82no—
360Q8—82no—
361Q8—82no—
362Q8—82no—
363D4—22no—
364D4—22no—
365D4—22no—
366D4#48614yes—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—816no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4#44712yes—
400C2×C4#44712yes—
401C2×C4#44812yes—
402C2×C4#44812yes—
403C2×C4#45012yes—
404C2×C4#45012yes—
405C2×C4#45512yes—
406C2×C4#45612yes—
407C2×C4C2×#12411yescharacteristic
408C2×C4#53611yescharacteristic
409C2×C4#54812yes—
410C2×C4#54812yes—
411C2×C4C22×D811yescharacteristic
412C2×C4C22×SD1611yescharacteristic
413C2×C4C22×SD1611yescharacteristic
414C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C2×C4#57812yes—
418C8—44no—
419C8—82no—
420C8—82no—
421C22—24no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22#243611yescharacteristic
429C22C2×#44711yescharacteristic
430C22C2×#45011yescharacteristic
431C22C2×#44811yescharacteristic
432C22#301111yescharacteristic
433C22C2×#45511yescharacteristic
434C22C2×#45611yescharacteristic
435C4—22no—
436C4—22no—
437C4—22no—
438C4—42no—
439C4—44no—
440C4—42no—
441C4—42no—
442C4—84no—
443C4—82no—
444C4—162no—
445C4#163914yes—
446C4#165114yes—
447C4C2×#48611yescharacteristic
448C4#314111yescharacteristic
449C4#322611yescharacteristic
450C4#322711yescharacteristic
451C4#335912yes—
452C4#336712yes—
453C2—44no—
454C2C2×#163911yescharacteristic
455C2C2×#165111yescharacteristic
456C2#1863211yescharacteristic
457C2#1865011yescharacteristic
458C2#2590311yescharacteristic
459C2#2592311yescharacteristic
460C2#2592911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33289C2semidirect
4. #29219C2semidirect
5. #10350C2semidirect
6. #9898C2semidirect
7. #9870C2semidirect
160. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#332892
#331622
#292192
#103502
#98982
#98702
#89852
C4×D4⋊C82