Finite Group Databasebeta

← #302389

Subgroups — #302389

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 449 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#33197C211yesmaximal, characteristic
4#29208C211yesmaximal, characteristic, complemented
5#10575C214yesmaximal, complemented
6#9898C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8890C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×D4×Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#476—21no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2857—21no—
16#2551—21no—
17#2551C2212yes—
18#2546—21no—
19#2440—22no—
20#2440—22no—
21#2434—22no—
22D4×C4⋊C4—21no—
23D4×C4⋊C4C2212yes—
24C4×C4⋊Q8C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#2057—22no—
27#1990—22no—
28#1990—22no—
29#1990—21no—
30#1987—21no—
31#1948—21no—
32#1948C2212yes—
33#1947C2214yes—
34#1845—21no—
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#476—44no—
72C2×#107—24no—
73C2×#106—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#102—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8C2312yes—
83#398—24no—
84#395—24no—
85#395—24no—
86#391—22no—
87#391—21no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—21no—
93#389D412yes—
94#385—24no—
95#385—24no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384D411yescharacteristic
102#384D412yes—
103#384C2312yes—
104#384C2312yes—
105#380—22no—
106#380—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D411yescharacteristic
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4C2312yes—
111C4×C4⋊C4C2311yescharacteristic
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C43C2311yescharacteristic
116#340—24no—
117#340—22no—
118#340—22no—
119#340—28no—
120#336—21no—
121D4⋊C8Q818yescomplemented
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—48no—
126C22×Q8—48no—
127C22×Q8—41no—
128C22×Q8—84no—
129C22×D4C2×D411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—24no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—44no—
139C4⋊Q8—44no—
140C4⋊Q8—44no—
141C4⋊Q8—48no—
142C4⋊Q8—44no—
143C4⋊Q8—48no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—82no—
153C4⋊Q8SD1614yes—
154C22⋊Q8—416no—
155C22⋊Q8—48no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4D814yes—
164C4×D4Q1614yes—
165C4×D4C2×Q814yes—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D414yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—24no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210#107—84no—
211#106—44no—
212C4⋊C8—22no—
213C4⋊C8—24no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8C2×Q814yescomplemented
217#102—44no—
218C4×C8—22no—
219C4×C8—42no—
220C4×C8—42no—
221C4×C8—42no—
222C4×C8C2×Q814yescomplemented
223#95—24no—
224#95—24no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—21no—
229#95—22no—
230C24—21no—
231C2×Q8—24no—
232C2×Q8—48no—
233C2×Q8—48no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8—832no—
243C2×Q8—82no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4C4⋊Q812yes—
248C2×D4C2×D812yes—
249C2×D4C2×Q1612yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—48no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—48no—
268C22×C4—42no—
269C22×C4—84no—
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4C4⋊Q812yes—
314C4⋊C4C2×D812yes—
315C4⋊C4C2×SD1614yes—
316C4⋊C4C2×Q1612yes—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—48no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—42no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42C2×D812yes—
331C42C2×SD1612yes—
332C42C2×Q1612yes—
333C42AGL(1,ℤ/8)12yes—
334C42AGL(1,ℤ/8)12yes—
335C42#13712yes—
336C42C22×Q812yes—
337C422−1+412yes—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23C2×#12011yescharacteristic
346Q8—48no—
347Q8—816no—
348Q8—88no—
349Q8—164no—
350D4—22no—
351D4—22no—
352D4—22no—
353D4#50014yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—816no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#44712yes—
387C2×C4#44712yes—
388C2×C4#45012yes—
389C2×C4#45012yes—
390C2×C4#45112yes—
391C2×C4#45112yes—
392C2×C4#45412yes—
393C2×C4#45512yes—
394C2×C4C2×C4⋊Q811yescharacteristic
395C2×C4#53611yescharacteristic
396C2×C4D4×Q812yes—
397C2×C4D4×Q812yes—
398C2×C4C22×D811yescharacteristic
399C2×C4C22×SD1611yescharacteristic
400C2×C4C22×Q1611yescharacteristic
401C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
402C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4#57712yes—
405C8—44no—
406C8—44no—
407C8—82no—
408C22—24no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22#243411yescharacteristic
416C22C2×#44711yescharacteristic
417C22C2×#45011yescharacteristic
418C22C2×#45111yescharacteristic
419C22#301011yescharacteristic
420C22C2×#45511yescharacteristic
421C22C2×#45411yescharacteristic
422C4—22no—
423C4—22no—
424C4—22no—
425C4—42no—
426C4—44no—
427C4—42no—
428C4—42no—
429C4—84no—
430C4—82no—
431C4—162no—
432C4#163714yes—
433C4#163914yes—
434C4C2×#50011yescharacteristic
435C4#316511yescharacteristic
436C4#322611yescharacteristic
437C4#323011yescharacteristic
438C4Q8×SD1612yes—
439C4#338412yes—
440C2—44no—
441C2C2×#163911yescharacteristic
442C2C2×#163711yescharacteristic
443C2#1863311yescharacteristic
444C2#1864011yescharacteristic
445C2#2581311yescharacteristic
446C2#2582911yescharacteristic
447C2#2583411yescharacteristic
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. #29208C2semidirect
5. #10575C2semidirect
6. #9898C2semidirect
7. #9870C2semidirect
121. D4⋊C8Q8semidirect
216. C4⋊C8C2×Q8semidirect
222. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#332892
#331972
#292082
#105752
#98982
#98702
#88902
C4×D4⋊C82