Finite Group Databasebeta

← #302385

Subgroups — #302385

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 452 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#33195C211yesmaximal, characteristic
4#28687C211yesmaximal, characteristic, complemented
5#10575C214yesmaximal, complemented
6#9898C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8894C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×D4×Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#498—21no—
13C2×#474—21no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2857—21no—
17#2551—21no—
18#2551C2212yes—
19#2546—21no—
20#2440—22no—
21#2440—22no—
22#2434—22no—
23D4×C4⋊C4—21no—
24D4×C4⋊C4C2212yes—
25C4×C4⋊Q8C2211yescharacteristic
26C42×D4C2211yescharacteristic
27#2084—22no—
28#1990—22no—
29#1990—21no—
30#1990—22no—
31#1988—21no—
32#1948C2214yes—
33#1947—21no—
34#1947C2212yes—
35#1842C2211yescharacteristic
36#1834C2214yes—
37#1802—21no—
38#1802C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42#1296C2214yes—
43#1279C2212yes—
44#1279C2214yes—
45C23×Q8—22no—
46D4×Q8—44no—
47D4×Q8—44no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
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×C22⋊Q8—22no—
62C2×C22⋊Q8—24no—
63C2×C4×Q8—21no—
64C2×C4×D4—22no—
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C42—21no—
71#498—44no—
72#474—44no—
73C2×#107—22no—
74C2×#106—24no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#102—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2312yes—
84#398—24no—
85#395—24no—
86#395—24no—
87#391—22no—
88#391—21no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—21no—
94#389D412yes—
95#385—24no—
96#385—24no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D412yes—
104#384C2312yes—
105#384C2312yes—
106#380—22no—
107#380—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2311yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C43C2311yescharacteristic
117#340—22no—
118#340—22no—
119#340—24no—
120#340—28no—
121#336—21no—
122D4⋊C8Q818yescomplemented
123C22×Q8—24no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—48no—
127C22×Q8—48no—
128C22×Q8—41no—
129C22×Q8—84no—
130C22×D4C2×D411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—24no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—44no—
140C4⋊Q8—44no—
141C4⋊Q8—44no—
142C4⋊Q8—48no—
143C4⋊Q8—44no—
144C4⋊Q8—48no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8D814yes—
156C22⋊Q8—416no—
157C22⋊Q8—48no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4SD1614yes—
166C4×D4SD1614yes—
167C4×D4C2×Q814yes—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
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—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—24no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2411yescharacteristic
212#107—44no—
213#106—48no—
214C4⋊C8—22no—
215C4⋊C8—24no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8C2×Q814yescomplemented
219#102—44no—
220C4×C8—22no—
221C4×C8—42no—
222C4×C8—42no—
223C4×C8—42no—
224C4×C8C2×Q814yescomplemented
225#95—24no—
226#95—24no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95—22no—
232C24—21no—
233C2×Q8—24no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—84no—
242C2×Q8—832no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4C4⋊Q812yes—
250C2×D4C2×SD1612yes—
251C2×D4C2×SD1612yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—48no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—84no—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—44no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4C4⋊Q812yes—
317C4⋊C4C2×D814yes—
318C4⋊C4C2×SD1612yes—
319C4⋊C4C2×SD1612yes—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—48no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—42no—
329C42—41no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42C2×D812yes—
334C42C2×SD1612yes—
335C42C2×SD1612yes—
336C42AGL(1,ℤ/8)12yes—
337C42AGL(1,ℤ/8)12yes—
338C42#13712yes—
339C42C22×Q812yes—
340C422−1+412yes—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23C2×#12011yescharacteristic
349Q8—48no—
350Q8—816no—
351Q8—88no—
352Q8—164no—
353D4—22no—
354D4—22no—
355D4—22no—
356D4#49814yescomplemented
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—816no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4#44712yes—
390C2×C4#44712yes—
391C2×C4#44812yes—
392C2×C4#44812yes—
393C2×C4#45012yes—
394C2×C4#45012yes—
395C2×C4#45412yes—
396C2×C4#45512yes—
397C2×C4C2×C4⋊Q811yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4D4×Q812yes—
400C2×C4D4×Q812yes—
401C2×C4C22×D811yescharacteristic
402C2×C4C22×SD1611yescharacteristic
403C2×C4C22×SD1611yescharacteristic
404C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C4C2×#13711yescharacteristic
407C2×C4#57812yes—
408C8—44no—
409C8—44no—
410C8—82no—
411C22—24no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22#243411yescharacteristic
419C22C2×#44711yescharacteristic
420C22C2×#45011yescharacteristic
421C22C2×#44811yescharacteristic
422C22#301111yescharacteristic
423C22C2×#45511yescharacteristic
424C22C2×#45411yescharacteristic
425C4—22no—
426C4—22no—
427C4—22no—
428C4—42no—
429C4—44no—
430C4—42no—
431C4—42no—
432C4—84no—
433C4—82no—
434C4—162no—
435C4#162714yes—
436C4#163914yes—
437C4C2×#49811yescharacteristic
438C4#316511yescharacteristic
439C4#322711yescharacteristic
440C4#322911yescharacteristic
441C4Q8×D812yes—
442C4#338612yes—
443C2—44no—
444C2C2×#163911yescharacteristic
445C2C2×#162711yescharacteristic
446C2#1863211yescharacteristic
447C2#1864111yescharacteristic
448C2#2581111yescharacteristic
449C2#2582911yescharacteristic
450C2#2583811yescharacteristic
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. #28687C2semidirect
5. #10575C2semidirect
6. #9898C2semidirect
7. #9870C2semidirect
122. D4⋊C8Q8semidirect
218. C4⋊C8C2×Q8semidirect
224. C4×C8C2×Q8semidirect
356. D4#498semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#332892
#331952
#286872
#105752
#98982
#98702
#88942
C4×D4⋊C82