Finite Group Databasebeta

← #302391

Subgroups — #302391

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#29206C211yesmaximal, characteristic
5#10578C214yesmaximal
6#9898C211yesmaximal, characteristic
7#9870C211yesmaximal, characteristic
8#8891C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×#477—21no—
11#2868C2211yescharacteristic
12#2868C2211yescharacteristic
13#2862—21no—
14#2582—21no—
15#2571C2212yes—
16#2570—22no—
17#2568—21no—
18#2511—22no—
19#2446—22no—
20#2365—21no—
21#2355—21no—
22#2355C2212yes—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C42×Q8C2211yescharacteristic
26#2058—22no—
27#1990—22no—
28#1990—22no—
29#1990—21no—
30#1987—21no—
31#1948C2214yes—
32#1947—21no—
33#1947C2212yes—
34#1844—21no—
35#1842C2211yescharacteristic
36#1834C2214yes—
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×C4×Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57#477—44no—
58C2×#107—22no—
59C2×#106—24no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8C2312yes—
69#400—24no—
70#400—24no—
71#398—24no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391—22no—
76#391—22no—
77#391—22no—
78#391—21no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—21no—
83#389D412yes—
84#387—24no—
85#387—24no—
86#387—24no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384D411yescharacteristic
93#384D412yes—
94#384C2312yes—
95#384C2312yes—
96#382—22no—
97#382—22no—
98#382—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4C2312yes—
109C4×C4⋊C4C2311yescharacteristic
110C43—21no—
111C43C2311yescharacteristic
112#340—24no—
113#340—22no—
114#340—22no—
115#340—28no—
116#336—21no—
117Q8⋊C8Q818yescomplemented
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—41no—
121C22×Q8C2×D411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—42no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—82no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—84no—
144C4×Q8C2×Q814yes—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
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×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D414yes—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—24no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2411yescharacteristic
185#107—44no—
186#106—84no—
187C4⋊C8—22no—
188C4⋊C8—24no—
189C4⋊C8—42no—
190C4⋊C8—42no—
191C4⋊C8C2×Q814yes—
192#103—44no—
193C4×C8—22no—
194C4×C8—42no—
195C4×C8—42no—
196C4×C8—42no—
197C4×C8C2×Q814yes—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—21no—
205#95—21no—
206#95—22no—
207#95—21no—
208#95—22no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8C4⋊Q812yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—21no—
249C4⋊C4—24no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—84no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4C4⋊Q812yes—
280C42—24no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—84no—
292C42#13512yes—
293C42#13512yes—
294C42#13512yes—
295C42AGL(1,ℤ/8)12yes—
296C42#13712yes—
297C42#13712yes—
298C42C22×Q812yes—
299C422−1+412yes—
300C23C2×#12011yescharacteristic
301Q8—22no—
302Q8—22no—
303Q8—41no—
304Q8—41no—
305Q8—84no—
306Q8—88no—
307Q8—164no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4#44912yes—
339C2×C4#44912yes—
340C2×C4#45212yes—
341C2×C4#45212yes—
342C2×C4#45212yes—
343C2×C4#45212yes—
344C2×C4#45412yes—
345C2×C4#45512yes—
346C2×C4C2×C4⋊Q811yescharacteristic
347C2×C4#53611yescharacteristic
348C2×C4D4×Q812yes—
349C2×C4D4×Q812yes—
350C2×C4C2×#13511yescharacteristic
351C2×C4C2×#13511yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57712yes—
357C8—44no—
358C8—44no—
359C8—82no—
360C22#243411yescharacteristic
361C22C2×#44911yescharacteristic
362C22C2×#45211yescharacteristic
363C22C2×#45211yescharacteristic
364C22#301011yescharacteristic
365C22C2×#45511yescharacteristic
366C22C2×#45411yescharacteristic
367C4—22no—
368C4—22no—
369C4—22no—
370C4—42no—
371C4—42no—
372C4—84no—
373C4—82no—
374C4—81no—
375C4—82no—
376C4—81no—
377C4—162no—
378C4#163814yes—
379C4#164014yes—
380C4#316011yescharacteristic
381C4#316511yescharacteristic
382C4#322811yescharacteristic
383C4#323111yescharacteristic
384C4#338112yes—
385C4#338712yes—
386C2C2×#164011yescharacteristic
387C2C2×#163811yescharacteristic
388C2#1863711yescharacteristic
389C2#1864411yescharacteristic
390C2#2581211yescharacteristic
391C2#2582911yescharacteristic
392C2#2583611yescharacteristic
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
117. Q8⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#332892
#331982
#292062
#105782
#98982
#98702
#88912
C4×Q8⋊C82