Finite Group Databasebeta

← #304524

Subgroups — #304524

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 454 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#28748C211yesmaximal, characteristic, complemented
5#10350C214yesmaximal, complemented
6#9898C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8981C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×D4×Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#487—21no—
13C2×#474C2212yes—
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#2043—22no—
28#1990—22no—
29#1990C2214yes—
30#1988C2212yes—
31#1981C2214yes—
32#1956—21no—
33#1947—21no—
34#1947—21no—
35#1842C2211yescharacteristic
36#1834—22no—
37#1802—21no—
38#1802—21no—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42#1296C2214yes—
43#1279C2214yes—
44#1279C2212yes—
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—44no—
57C2×C4⋊Q8—44no—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8C2312yes—
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#487—44no—
72#474—28no—
73C2×#107—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#103—24no—
79C2×#103—24no—
80C2×#102—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2312yes—
85#398—24no—
86#395—24no—
87#395—24no—
88#391—22no—
89#391—21no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—21no—
95#389D412yes—
96#385—24no—
97#385—24no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384D411yescharacteristic
105#384D412yes—
106#384C2312yes—
107#380—22no—
108#380—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4C2312yes—
113C4×C4⋊C4C2311yescharacteristic
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C43C2311yescharacteristic
118#340—22no—
119#340—22no—
120#340—28no—
121#336—21no—
122D4⋊C8—24no—
123C22×Q8—24no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—48no—
128C22×Q8—48no—
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—24no—
140C4⋊Q8—24no—
141C4⋊Q8—44no—
142C4⋊Q8—44no—
143C4⋊Q8—48no—
144C4⋊Q8—44no—
145C4⋊Q8—48no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—82no—
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×D4Q1614yes—
167C4×D4#4114yes—
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—44no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—41no—
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—
213C4⋊C8—22no—
214C4⋊C8—24no—
215C4⋊C8—22no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218#103—48no—
219#103—84no—
220#102—44no—
221C4×C8—24no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—42no—
225C4×C8—42no—
226#95—24no—
227#95—24no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—21no—
232#95—22no—
233C24—21no—
234C2×Q8—24no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—44no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—84no—
247C2×Q8—832no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4#12412yes—
252C2×D4C2×SD1612yes—
253C2×D4C2×Q1612yes—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—48no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—84no—
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—44no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4#12412yes—
318C4⋊C4C2×D814yes—
319C4⋊C4C2×SD1612yes—
320C4⋊C4C2×Q1612yes—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—48no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—41no—
329C42—42no—
330C42—41no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42C2×D812yes—
335C42C2×SD1612yes—
336C42C2×Q1612yes—
337C42AGL(1,ℤ/8)12yes—
338C42AGL(1,ℤ/8)12yes—
339C42#13712yes—
340C42C2×#4112yes—
341C422−1+412yes—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C2×#12011yescharacteristic
350Q8—48no—
351Q8—816no—
352Q8—88no—
353Q8—84no—
354Q8—84no—
355D4—22no—
356D4—22no—
357D4—22no—
358D4#48714yescomplemented
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—816no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#44712yes—
392C2×C4#44712yes—
393C2×C4#45012yes—
394C2×C4#45012yes—
395C2×C4#45112yes—
396C2×C4#45112yes—
397C2×C4#45512yes—
398C2×C4#45612yes—
399C2×C4C2×#12411yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4#54812yes—
402C2×C4#54812yes—
403C2×C4C22×D811yescharacteristic
404C2×C4C22×SD1611yescharacteristic
405C2×C4C22×Q1611yescharacteristic
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
408C2×C4C2×#13711yescharacteristic
409C2×C4#57712yes—
410C8—44no—
411C8—82no—
412C8—82no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22#243611yescharacteristic
421C22C2×#44711yescharacteristic
422C22C2×#45011yescharacteristic
423C22C2×#45111yescharacteristic
424C22#301011yescharacteristic
425C22C2×#45511yescharacteristic
426C22C2×#45611yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—42no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—84no—
435C4—82no—
436C4—162no—
437C4#163914yes—
438C4#164914yes—
439C4C2×#48711yescharacteristic
440C4#314111yescharacteristic
441C4#322911yescharacteristic
442C4#323011yescharacteristic
443C4#335812yes—
444C4#337312yes—
445C2—44no—
446C2C2×#163911yescharacteristic
447C2C2×#164911yescharacteristic
448C2#1863311yescharacteristic
449C2#1865111yescharacteristic
450C2#2590211yescharacteristic
451C2#2592311yescharacteristic
452C2#2593311yescharacteristic
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. #28748C2semidirect
5. #10350C2semidirect
6. #9898C2semidirect
7. #9870C2semidirect
358. D4#487semidirect

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
#287482
#103502
#98982
#98702
#89812
C4×D4⋊C82