Finite Group Databasebeta

← #350925

Subgroups — #350925

Order 512.

602 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#33289C211yesmaximal, characteristic
2#29984C211yesmaximal, characteristic
3#29799C211yesmaximal, characteristic
4#20516C211yesmaximal, characteristic
5#9903C211yesmaximal, characteristic
6#9703C211yesmaximal, characteristic
7#9574C212yesmaximal
8#9566C212yesmaximal
9#9226C212yesmaximal
10#9218C212yesmaximal
11#8785C211yesmaximal, characteristic
12C2×#462C2212yes—
13C2×#446C2211yescharacteristic
14C2×C8×Q8C2211yescharacteristic
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2295C2211yescharacteristic
18#2100—21no—
19#2095—21no—
20#2057—21no—
21#2043—21no—
22#2043—21no—
23#2024—21no—
24#2016—21no—
25#2016—21no—
26#1990—21no—
27#1990—21no—
28#1990C2212yes—
29#1990C2212yes—
30#1981C2212yes—
31#1947C2212yes—
32#1834—21no—
33#1834—21no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1803C2212yes—
37#1771—21no—
38#1770C2211yescharacteristic
39C8×C4⋊C4—21no—
40C8×C4⋊C4C2211yescharacteristic
41#1768—21no—
42#1767C2211yescharacteristic
43#1726—21no—
44C42×C8C2211yescharacteristic
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1277C2212yes—
50#1277C2212yes—
51C22×Q16—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2312yes—
58C2×C4×Q8C2311yescharacteristic
59#462—24no—
60#462—24no—
61#446—22no—
62#446—22no—
63C8×Q8—22no—
64C8×Q8—22no—
65C2×#107—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#391D411yescharacteristic
88#391D411yescharacteristic
89#389D411yescharacteristic
90#389D411yescharacteristic
91#387—21no—
92#387—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384C2312yes—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2311yescharacteristic
103#376—21no—
104C43C2311yescharacteristic
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#332—24no—
119#332—24no—
120C22×Q8—22no—
121C22×Q8—41no—
122C22×Q8C2×D411yescharacteristic
123C2×Q16—44no—
124C2×Q16—82no—
125C2×Q16—82no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—82no—
141C4⋊Q8—82no—
142C4⋊Q8—82no—
143C4⋊Q8—82no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8#4112yes—
147C4×Q8#4112yes—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42C2411yescharacteristic
175#107—82no—
176C4⋊C8—24no—
177C4⋊C8—24no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8#4112yes—
185C4⋊C8#4112yes—
186#103—44no—
187#103—82no—
188#103—82no—
189#103—82no—
190#97—24no—
191#97—42no—
192#97—42no—
193C4×C8—24no—
194C4×C8—22no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—41no—
200C4×C8—41no—
201C4×C8—42no—
202C4×C8Q1614yescomplemented
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8#12412yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4111yescharacteristic
236Q16—88no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8C2×Q1612yes—
260C2×C8C2×Q1612yes—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4#12412yes—
285C42—22no—
286C42—22no—
287C42—21no—
288C42—21no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—41no—
294C42—41no—
295C42C2×Q1612yes—
296C42#13712yes—
297C42C2×#4111yescharacteristic
298C42C2×#4111yescharacteristic
299C42C2×#4111yescharacteristic
300C42C2×#4111yescharacteristic
301C23C2×#12111yescharacteristic
302Q8—24no—
303Q8—42no—
304Q8—82no—
305Q8—82no—
306Q8—82no—
307Q8—82no—
308Q8—164no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#46212yes—
338C2×C4#46212yes—
339C2×C4#47112yes—
340C2×C4#47312yes—
341C2×C4#52511yescharacteristic
342C2×C4#52511yescharacteristic
343C2×C4C2×#12411yescharacteristic
344C2×C4#52911yescharacteristic
345C2×C4#52911yescharacteristic
346C2×C4#53211yescharacteristic
347C2×C4C22×Q1611yescharacteristic
348C2×C4C2×#13711yescharacteristic
349C2×C4#57511yescharacteristic
350C2×C4#57511yescharacteristic
351C2×C4#57612yes—
352C2×C4#57712yes—
353C8—22no—
354C8—22no—
355C8—41no—
356C8—41no—
357C8—42no—
358C8—42no—
359C8—82no—
360C8—82no—
361C22#285011yescharacteristic
362C22C2×#46211yescharacteristic
363C22#304011yescharacteristic
364C22#305711yescharacteristic
365C22#305811yescharacteristic
366C22C2×#47111yescharacteristic
367C22C2×#47311yescharacteristic
368C4—22no—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—162no—
379C4—162no—
380C4#310411yescharacteristic
381C4#310411yescharacteristic
382C4#311611yescharacteristic
383C4#311611yescharacteristic
384C4#313612yes—
385C4#317912yes—
386C2#1920711yescharacteristic
387C2#1920811yescharacteristic
388C2#1936311yescharacteristic
389C2#1936411yescharacteristic
390C2#3302711yescharacteristic
391C2#3304511yescharacteristic
392C2#3306111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
202. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332892
#299842
#297992
#205162
#99032
#97032
#95742
#95662
#92262
#92182
#87852