Finite Group Databasebeta

← #302369

Subgroups — #302369

Order 512.

1158 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#33312C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic
4#28689C211yesmaximal, characteristic, complemented
5#10573C214yesmaximal, complemented
6#9898C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8882C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×D4×Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#500—21no—
13C2×#500—22no—
14C2×#474—21no—
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#2083—22no—
29#1990—21no—
30#1988—21no—
31#1947—21no—
32#1947C2214yes—
33#1947C2212yes—
34#1845—22no—
35#1845C2214yes—
36#1842C2211yescharacteristic
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#500—48no—
72#500—44no—
73#474—44no—
74C2×#107—24no—
75C2×#107—28no—
76C2×#107—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#102—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#384D411yescharacteristic
105#384D412yes—
106#384C2312yes—
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—22no—
120#340—22no—
121#340—24no—
122#336—21no—
123D4⋊C8Q818yescomplemented
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—48no—
128C22×Q8—48no—
129C22×Q8—41no—
130C22×Q8—84no—
131C22×D4C2×D411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—44no—
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—42no—
155C4⋊Q8—42no—
156C4⋊Q8D814yes—
157C22⋊Q8—416no—
158C22⋊Q8—48no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4D814yes—
167C4×D4Q1614yes—
168C4×D4C2×Q814yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D414yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—24no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213#107—416no—
214#107—44no—
215#107—48no—
216C4⋊C8—22no—
217C4⋊C8—24no—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C4⋊C8C2×Q814yescomplemented
221#102—44no—
222C4×C8—24no—
223C4×C8—24no—
224C4×C8—22no—
225C4×C8—42no—
226C4×C8C2×Q814yescomplemented
227#95—24no—
228#95—24no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—22no—
234C24—21no—
235C2×Q8—24no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—84no—
244C2×Q8—832no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C4⋊Q812yes—
252C2×D4C2×D812yes—
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—24no—
283C2×C8—24no—
284C2×C8—24no—
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—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4C4⋊Q812yes—
319C4⋊C4C2×D814yes—
320C4⋊C4C2×D812yes—
321C4⋊C4C2×Q1612yes—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—48no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—41no—
330C42—42no—
331C42—41no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42C2×D812yes—
336C42C2×D812yes—
337C42C2×Q1612yes—
338C42AGL(1,ℤ/8)12yes—
339C42AGL(1,ℤ/8)12yes—
340C42#13712yes—
341C42C22×Q812yes—
342C422−1+412yes—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23C2×#12011yescharacteristic
351Q8—48no—
352Q8—816no—
353Q8—88no—
354Q8—164no—
355D4—22no—
356D4—22no—
357D4—22no—
358D4#50014yescomplemented
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
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—42no—
382C2×C4—44no—
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#44712yes—
394C2×C4#44712yes—
395C2×C4#45112yes—
396C2×C4#45112yes—
397C2×C4#45312yes—
398C2×C4#45612yes—
399C2×C4C2×C4⋊Q811yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4D4×Q812yes—
402C2×C4D4×Q812yes—
403C2×C4C22×D811yescharacteristic
404C2×C4C22×D811yescharacteristic
405C2×C4C22×Q1611yescharacteristic
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
408C2×C4C2×#13711yescharacteristic
409C2×C4#57812yes—
410C8—28no—
411C8—44no—
412C8—82no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22#243411yescharacteristic
421C22C2×#44711yescharacteristic
422C22C2×#44711yescharacteristic
423C22C2×#45111yescharacteristic
424C22#301111yescharacteristic
425C22C2×#45311yescharacteristic
426C22C2×#45611yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—42no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—84no—
435C4—82no—
436C4—162no—
437C4#161914yes—
438C4#164914yes—
439C4C2×#50011yescharacteristic
440C4#316511yescharacteristic
441C4#322611yescharacteristic
442C4#323011yescharacteristic
443C4Q8×D812yes—
444C4#338612yes—
445C2—44no—
446C2C2×#161911yescharacteristic
447C2C2×#164911yescharacteristic
448C2#1862311yescharacteristic
449C2#1865011yescharacteristic
450C2#2581311yescharacteristic
451C2#2583011yescharacteristic
452C2#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. #33312C2semidirect
4. #28689C2semidirect
5. #10573C2semidirect
6. #9898C2semidirect
7. #9870C2semidirect
123. D4⋊C8Q8semidirect
220. C4⋊C8C2×Q8semidirect
226. C4×C8C2×Q8semidirect
358. D4#500semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#333122
#331952
#286892
#105732
#98982
#98702
#88822
C4×D4⋊C82