Finite Group Databasebeta

← #302368

Subgroups — #302368

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#33310C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#28689C211yesmaximal, characteristic
5#10580C214yesmaximal
6#9896C211yesmaximal, characteristic
7#9870C211yesmaximal, characteristic
8#8887C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×#500—21no—
11C2×#498—22no—
12C2×#475—21no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2859—21no—
16#2582—21no—
17#2567—21no—
18#2552C2212yes—
19#2550—22no—
20#2507—22no—
21#2451—22no—
22#2369—21no—
23#2355C2212yes—
24Q8×C4⋊C4—21no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C42×Q8C2211yescharacteristic
28#2083—22no—
29#1990—21no—
30#1988—21no—
31#1948—21no—
32#1948C2212yes—
33#1947C2214yes—
34#1844—22no—
35#1844C2214yes—
36#1842C2211yescharacteristic
37#1803—21no—
38#1803C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊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×#125—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—21no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58#500—44no—
59#498—48no—
60#475—44no—
61C2×#107—24no—
62C2×#106—22no—
63C2×#106—28no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×#103—22no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8C2312yes—
73#398—24no—
74#398—24no—
75#395—24no—
76#391—22no—
77#391—24no—
78#391—22no—
79#391—21no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—24no—
86#389—22no—
87#389—22no—
88#389—21no—
89#389D412yes—
90#387—24no—
91#384—22no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384D411yescharacteristic
98#384D412yes—
99#384C2312yes—
100#384C2312yes—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2311yescharacteristic
113C43—21no—
114C43C2311yescharacteristic
115#340—22no—
116#340—22no—
117#340—24no—
118#336—21no—
119Q8⋊C8Q818yescomplemented
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×Q8C2×D411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—82no—
137C4⋊Q8SD1614yes—
138#125—84no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—84no—
147C4×Q8C2×Q814yes—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D414yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—24no—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2411yescharacteristic
190#107—48no—
191#106—44no—
192#106—416no—
193C4⋊C8—22no—
194C4⋊C8—24no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197C4⋊C8C2×Q814yes—
198#103—44no—
199C4×C8—24no—
200C4×C8—24no—
201C4×C8—22no—
202C4×C8—42no—
203C4×C8C2×Q814yes—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—21no—
209#95—21no—
210#95—22no—
211#95—21no—
212#95—22no—
213C2×Q8—21no—
214C2×Q8—24no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8C4⋊Q812yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—21no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4C4⋊Q812yes—
285C4⋊C4C2×SD1614yes—
286C42—24no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—84no—
298C42C2×SD1612yes—
299C42#13512yes—
300C42#13512yes—
301C42AGL(1,ℤ/8)12yes—
302C42#13712yes—
303C42#13712yes—
304C42C22×Q812yes—
305C422−1+412yes—
306C23C2×#12011yescharacteristic
307Q8—22no—
308Q8—22no—
309Q8—48no—
310Q8—41no—
311Q8—41no—
312Q8—164no—
313Q8—164no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4#44812yes—
345C2×C4#44812yes—
346C2×C4#44912yes—
347C2×C4#44912yes—
348C2×C4#45212yes—
349C2×C4#45212yes—
350C2×C4#45312yes—
351C2×C4#45612yes—
352C2×C4C2×C4⋊Q811yescharacteristic
353C2×C4#53611yescharacteristic
354C2×C4D4×Q812yes—
355C2×C4D4×Q812yes—
356C2×C4C22×SD1611yescharacteristic
357C2×C4C2×#13511yescharacteristic
358C2×C4C2×#13511yescharacteristic
359C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
360C2×C4C2×#13711yescharacteristic
361C2×C4C2×#13711yescharacteristic
362C2×C4#57712yes—
363C8—28no—
364C8—44no—
365C8—82no—
366C22#243411yescharacteristic
367C22C2×#44811yescharacteristic
368C22C2×#44911yescharacteristic
369C22C2×#45211yescharacteristic
370C22#300911yescharacteristic
371C22C2×#45311yescharacteristic
372C22C2×#45611yescharacteristic
373C4—22no—
374C4—22no—
375C4—22no—
376C4—44no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—84no—
381C4—81no—
382C4—82no—
383C4—162no—
384C4#162214yes—
385C4#165314yes—
386C4#316011yescharacteristic
387C4#316511yescharacteristic
388C4#322811yescharacteristic
389C4#323111yescharacteristic
390C4Q8×SD1612yes—
391C4#338712yes—
392C2C2×#162211yescharacteristic
393C2C2×#165311yescharacteristic
394C2#1862511yescharacteristic
395C2#1865611yescharacteristic
396C2#2581211yescharacteristic
397C2#2583011yescharacteristic
398C2#2583711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
119. Q8⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#333102
#331962
#286892
#105802
#98962
#98702
#88872
C4×Q8⋊C82