Finite Group Databasebeta

← #302398

Subgroups — #302398

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic
4#29208C211yesmaximal, characteristic, complemented
5#10578C214yesmaximal, complemented
6#9896C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8897C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×#476—21no—
11#2868C2211yescharacteristic
12#2868C2211yescharacteristic
13#2857—21no—
14#2571C2212yes—
15#2563—21no—
16#2538—21no—
17#2497—22no—
18#2492—22no—
19#2439—22no—
20#2369—21no—
21#2355—21no—
22#2354C2212yes—
23C4×C4⋊Q8C2211yescharacteristic
24#2285—21no—
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—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D412yes—
52C2×C22⋊Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C2×#117—22no—
58C22×C42—21no—
59#476—44no—
60C2×#107—24no—
61C2×#106—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#102—22no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2312yes—
71#400—24no—
72#396—24no—
73#393—24no—
74#391—22no—
75#391—24no—
76#391—21no—
77#390—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—21no—
82#389D412yes—
83#388—24no—
84#387—24no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384D411yescharacteristic
91#384D412yes—
92#384C2312yes—
93#384C2312yes—
94#382—22no—
95#382—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D411yescharacteristic
105C4×C4⋊C4D411yescharacteristic
106C4×C4⋊C4C2312yes—
107C4×C4⋊C4C2311yescharacteristic
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C43C2311yescharacteristic
112#340—24no—
113#340—22no—
114#340—22no—
115#340—28no—
116#336—21no—
117D4⋊C8Q818yescomplemented
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—41no—
121C22×D4C2×D411yescharacteristic
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—82no—
137C4⋊Q8—82no—
138C22⋊Q8—84no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4C2×Q814yes—
147#117—84no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D414yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2411yescharacteristic
189#107—84no—
190#106—44no—
191C4⋊C8—22no—
192C4⋊C8—24no—
193C4⋊C8—42no—
194C4⋊C8—42no—
195C4⋊C8C2×Q814yes—
196#102—44no—
197C4×C8—22no—
198C4×C8—42no—
199C4×C8—42no—
200C4×C8—42no—
201C4×C8C2×Q814yes—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—21no—
210#95—22no—
211C24—21no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4C4⋊Q812yes—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4C4⋊Q812yes—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—84no—
292C22⋊C4—84no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—42no—
298C42—42no—
299C42—41no—
300C42—42no—
301C42—42no—
302C42—84no—
303C42#13512yes—
304C42#13512yes—
305C42#13512yes—
306C42AGL(1,ℤ/8)12yes—
307C42#13712yes—
308C42#13712yes—
309C42C22×Q812yes—
310C422−1+412yes—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23C2×#12011yescharacteristic
319Q8—84no—
320Q8—164no—
321Q8—164no—
322D4—22no—
323D4—22no—
324D4—41no—
325D4—41no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#44912yes—
358C2×C4#44912yes—
359C2×C4#45212yes—
360C2×C4#45212yes—
361C2×C4#45212yes—
362C2×C4#45212yes—
363C2×C4#45412yes—
364C2×C4#45512yes—
365C2×C4C2×C4⋊Q811yescharacteristic
366C2×C4#53611yescharacteristic
367C2×C4D4×Q812yes—
368C2×C4D4×Q812yes—
369C2×C4C2×#13511yescharacteristic
370C2×C4C2×#13511yescharacteristic
371C2×C4C2×#13511yescharacteristic
372C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
373C2×C4C2×#13711yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C2×C4#57612yes—
376C8—44no—
377C8—44no—
378C8—82no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#243411yescharacteristic
387C22C2×#44911yescharacteristic
388C22C2×#45211yescharacteristic
389C22C2×#45211yescharacteristic
390C22#300811yescharacteristic
391C22C2×#45511yescharacteristic
392C22C2×#45411yescharacteristic
393C4—22no—
394C4—22no—
395C4—22no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—82no—
400C4—84no—
401C4—82no—
402C4—162no—
403C4#163814yes—
404C4#164614yes—
405C4#316011yescharacteristic
406C4#316511yescharacteristic
407C4#322811yescharacteristic
408C4#323111yescharacteristic
409C4#338012yes—
410C4#338512yes—
411C2—82no—
412C2C2×#163811yescharacteristic
413C2C2×#164611yescharacteristic
414C2#1863511yescharacteristic
415C2#1864611yescharacteristic
416C2#2581211yescharacteristic
417C2#2582911yescharacteristic
418C2#2583911yescharacteristic
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. #10578C2semidirect
6. #9896C2semidirect
7. #9870C2semidirect
117. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
#332892
#331972
#292082
#105782
#98962
#98702
#88972
C4×D4⋊C82