Finite Group Databasebeta

← #302386

Subgroups — #302386

Order 512.

954 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic
4#29206C211yesmaximal, characteristic, complemented
5#10582C214yesmaximal, complemented
6#9896C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8894C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×#554—21no—
11C2×#476—21no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14#2857—21no—
15#2567—21no—
16#2552—21no—
17#2552C2212yes—
18#2502—22no—
19#2440—22no—
20#2440—22no—
21D4×C4⋊C4—21no—
22D4×C4⋊C4C2212yes—
23C4×C4⋊Q8C2211yescharacteristic
24C2×#384—21no—
25C42×D4C2211yescharacteristic
26#2058—22no—
27#1990—22no—
28#1990—22no—
29#1990—21no—
30#1987—21no—
31#1948—21no—
32#1948C2214yes—
33#1948C2212yes—
34#1844—21no—
35#1842C2211yescharacteristic
36#1834C2214yes—
37#1802—21no—
38#1802C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42#1296C2214yes—
43#1279C2212yes—
44#1279C2214yes—
45#554—44no—
46#554—44no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D412yes—
54C2×#125—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—24no—
57C2×C4×Q8—21no—
58C2×C4×D4—22no—
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C42—21no—
66#476—44no—
67C2×#106—22no—
68C2×#106—24no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#102—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8C2312yes—
78#398—24no—
79#398—24no—
80#395—24no—
81#391—22no—
82#391—21no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—21no—
88#389D412yes—
89#385—24no—
90#385—24no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384D411yescharacteristic
100#384D412yes—
101#384C2312yes—
102#384C2312yes—
103#380—22no—
104#380—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4C2312yes—
109C4×C4⋊C4C2311yescharacteristic
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C43C2311yescharacteristic
114#340—22no—
115#340—22no—
116#340—24no—
117#340—28no—
118#336—21no—
119D4⋊C8Q818yescomplemented
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×D4C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—24no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—82no—
140C4⋊Q8—82no—
141#125—44no—
142#125—44no—
143C22⋊Q8—416no—
144C22⋊Q8—48no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—44no—
151C4×D4—44no—
152C4×D4SD1614yes—
153C4×D4SD1614yes—
154C4×D4C2×Q814yes—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4—84no—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D414yes—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—24no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2411yescharacteristic
200#106—44no—
201#106—84no—
202C4⋊C8—22no—
203C4⋊C8—24no—
204C4⋊C8—42no—
205C4⋊C8—42no—
206C4⋊C8C2×Q814yes—
207#102—44no—
208C4×C8—22no—
209C4×C8—42no—
210C4×C8—42no—
211C4×C8—42no—
212C4×C8C2×Q814yes—
213#95—24no—
214#95—24no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95—21no—
219#95—22no—
220C24—21no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4C4⋊Q812yes—
234C2×D4C2×SD1612yes—
235C2×D4C2×SD1612yes—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—48no—
253C22×C4—42no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—44no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4C4⋊Q812yes—
301C4⋊C4C2×SD1612yes—
302C4⋊C4C2×SD1612yes—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—48no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—42no—
311C42—41no—
312C42—42no—
313C42—41no—
314C42—42no—
315C42—42no—
316C42C2×SD1612yes—
317C42C2×SD1612yes—
318C42#13512yes—
319C42AGL(1,ℤ/8)12yes—
320C42#13712yes—
321C42#13712yes—
322C42C22×Q812yes—
323C422−1+412yes—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23C2×#12011yescharacteristic
332Q8—84no—
333Q8—164no—
334Q8—164no—
335D4—22no—
336D4—22no—
337D4—22no—
338D4#49814yes—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—168no—
371C2×C4#44812yes—
372C2×C4#44812yes—
373C2×C4#45012yes—
374C2×C4#45012yes—
375C2×C4#45212yes—
376C2×C4#45212yes—
377C2×C4#45412yes—
378C2×C4#45512yes—
379C2×C4C2×C4⋊Q811yescharacteristic
380C2×C4#53611yescharacteristic
381C2×C4D4×Q812yes—
382C2×C4D4×Q812yes—
383C2×C4C22×SD1611yescharacteristic
384C2×C4C22×SD1611yescharacteristic
385C2×C4C2×#13511yescharacteristic
386C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
387C2×C4C2×#13711yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4#57612yes—
390C8—44no—
391C8—44no—
392C8—82no—
393C22—24no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22#243411yescharacteristic
401C22C2×#45011yescharacteristic
402C22C2×#44811yescharacteristic
403C22C2×#45211yescharacteristic
404C22#300811yescharacteristic
405C22C2×#45511yescharacteristic
406C22C2×#45411yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—82no—
414C4—82no—
415C4—84no—
416C4—162no—
417C4#163514yes—
418C4#164314yes—
419C4C2×#49811yescharacteristic
420C4#316511yescharacteristic
421C4#322711yescharacteristic
422C4#322911yescharacteristic
423C4#338012yes—
424C4#338512yes—
425C2—44no—
426C2C2×#164311yescharacteristic
427C2C2×#163511yescharacteristic
428C2#1863111yescharacteristic
429C2#1864211yescharacteristic
430C2#2581111yescharacteristic
431C2#2582911yescharacteristic
432C2#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. #29206C2semidirect
5. #10582C2semidirect
6. #9896C2semidirect
7. #9870C2semidirect
119. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#361802
#332892
#331972
#292062
#105822
#98962
#98702
#88942
C4×D4⋊C82