Finite Group Databasebeta

← #318176

Subgroups — #318176

Order 512.

646 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28206C211yesmaximal, characteristic
2#28161C211yesmaximal, characteristic
3C4×#498C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5C2×#1672C212yesmaximal
6#10483C212yesmaximal
7#10478C212yesmaximal
8#8807C211yesmaximal, characteristic
9#8643C212yesmaximal
10#8591C211yesmaximal, characteristic
11#8590C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#477C2212yes—
14C2×#462C2212yes—
15C2×C8×Q8C2212yes—
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2083—21no—
20#2072—21no—
21#2043—21no—
22#2016—21no—
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1984C2212yes—
26#1941—21no—
27#1941—21no—
28#1939—21no—
29#1939—21no—
30#1928—22no—
31#1878—22no—
32#1844C2211yescharacteristic
33#1842C2211yescharacteristic
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1803C2212yes—
37#1803C2212yes—
38C4×#106—21no—
39C4×#106—21no—
40C8×C4⋊C4—21no—
41C8×C4⋊C4—21no—
42C42×C8C2211yescharacteristic
43#1672—28no—
44C2×#335C2212yes—
45C2×#334C2212yes—
46C2×#333C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50C22×Q16—22no—
51C2×C4⋊Q8D411yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2312yes—
57#498—22no—
58#498D414yes—
59#477—24no—
60#477—42no—
61#462—24no—
62#462—42no—
63C8×Q8—24no—
64C8×Q8—24no—
65C2×#107—22no—
66C2×#106—21no—
67C2×#106—21no—
68C2×#106—21no—
69C2×#106—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×C4×C8—22no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#391—21no—
88#391—21no—
89#391D412yes—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#384D411yescharacteristic
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C43C2311yescharacteristic
106#340—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110#340—22no—
111#340—22no—
112#336—22no—
113#336—22no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#335—42no—
119#334—24no—
120#333—24no—
121#332—24no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8C2×D412yes—
125C2×Q16—44no—
126C2×Q16—44no—
127C2×Q16—82no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C4⋊Q8—24no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C2×D412yes—
143C4⋊Q8C2×D412yes—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—82no—
149C4×Q8—82no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4C2×D412yes—
168C2×C42—22no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178#107—44no—
179#106—24no—
180#106—24no—
181#106—42no—
182#106—42no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—42no—
190C4⋊C8—44no—
191C4⋊C8—42no—
192#103—44no—
193#103—44no—
194#103—44no—
195#103—82no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—41no—
201C4×C8—44no—
202C4×C8Q1614yescomplemented
203C4×C8C2×D412yes—
204C4×C8#4112yes—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—81no—
223C2×Q8—81no—
224C2×Q8#13514yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C2×#4111yescharacteristic
238Q16—88no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8#12112yes—
256C2×C8#12112yes—
257C2×C8C2×Q1612yes—
258C2×C8C2×Q1612yes—
259C2×C8#13712yes—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C42—22no—
287C42—21no—
288C42—21no—
289C42—22no—
290C42—22no—
291C42—41no—
292C42—42no—
293C42—42no—
294C42—82no—
295C42—82no—
296C42C2×Q1612yes—
297C42#13712yes—
298C42C22×D411yescharacteristic
299C42C2×#4111yescharacteristic
300C42H(5,𝔽2)11yescharacteristic
301C422−1+411yescharacteristic
302C23C2×#12111yescharacteristic
303Q8—24no—
304Q8—24no—
305Q8—42no—
306Q8—42no—
307Q8—44no—
308Q8—82no—
309Q8—82no—
310Q8—82no—
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—21no—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4#46212yes—
341C2×C4#46212yes—
342C2×C4#47112yes—
343C2×C4C2×#12111yescharacteristic
344C2×C4#53711yescharacteristic
345C2×C4D4211yescharacteristic
346C2×C4#54711yescharacteristic
347C2×C4#54711yescharacteristic
348C2×C4#55511yescharacteristic
349C2×C4C22×Q1611yescharacteristic
350C2×C4C2×#13512yes—
351C2×C4C2×#13711yescharacteristic
352C2×C4C2×#13711yescharacteristic
353C2×C4#57511yescharacteristic
354C2×C4#57712yes—
355C8—22no—
356C8—22no—
357C8—41no—
358C8—82no—
359C8—82no—
360C8—82no—
361C8#46214yes—
362C22#253711yescharacteristic
363C22C2×#46211yescharacteristic
364C22#303711yescharacteristic
365C22#305011yescharacteristic
366C22#305711yescharacteristic
367C22C2×#47111yescharacteristic
368C22#306711yescharacteristic
369C4—22no—
370C4—22no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—82no—
377C4—82no—
378C4—82no—
379C4—161no—
380C4—161no—
381C4C2×#46211yescharacteristic
382C4#303711yescharacteristic
383C4#324911yescharacteristic
384C4#325511yescharacteristic
385C4#328212yes—
386C4#333712yes—
387C2#1911811yescharacteristic
388C2#1920711yescharacteristic
389C2#1926311yescharacteristic
390C2#1934011yescharacteristic
391C2#2822411yescharacteristic
392C2#2823911yescharacteristic
393C2#2825511yescharacteristic
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
#282062
#281612
C4×#4982
C8×C4⋊Q82
C2×#16722
#104832
#104782
#88072
#86432
#85912
#85902