Finite Group Databasebeta

← #277392

Subgroups — #277392

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28682C211yesmaximal, characteristic
2#24817C212yesmaximal
3Q8×#106C211yesmaximal, characteristic
4C2×#1676C211yesmaximal, characteristic
5C2×#1584C211yesmaximal, characteristic
6#10601C212yesmaximal
7#10415C212yesmaximal
8#9836C211yesmaximal, characteristic
9#9596C211yesmaximal, characteristic
10#8975C212yesmaximal
11#8651C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×#446C2211yescharacteristic
15C2×C8×Q8C2211yescharacteristic
16C2×C4×Q16C2211yescharacteristic
17#2567C2212yes—
18Q8×C4⋊C4C2212yes—
19#2086—22no—
20#2083—22no—
21#2068—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1956C2212yes—
25#1948—22no—
26#1948C2212yes—
27#1944—21no—
28#1944—21no—
29#1939C2212yes—
30#1878—22no—
31#1844—21no—
32#1844C2211yescharacteristic
33#1803—22no—
34#1803C2212yes—
35C4×#106—21no—
36C4×#106C2211yescharacteristic
37#1676—24no—
38#1584—24no—
39C2×#335C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#333C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×#322C2211yescharacteristic
45#1296C2212yes—
46#1279C2212yes—
47C22×Q16D411yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×C4×Q8—22no—
50C2×C4×Q8C2311yescharacteristic
51C2×C4×Q8C2311yescharacteristic
52#500—22no—
53#500—22no—
54#477—24no—
55#477—24no—
56#446—22no—
57#446—22no—
58C8×Q8—22no—
59C8×Q8D414yes—
60C4×Q16—22no—
61C4×Q16—22no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#107D411yescharacteristic
66C2×#106—21no—
67C2×#106—21no—
68C2×#106—22no—
69C2×#106D411yescharacteristic
70C2×#106D411yescharacteristic
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—42no—
80C2×#103D411yescharacteristic
81C2×#103D411yescharacteristic
82C2×#97—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#395—42no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—42no—
93#384—22no—
94#384—22no—
95#384C2312yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2312yes—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340D412yes—
105#336—22no—
106#335—22no—
107#335—22no—
108#333—24no—
109Q8⋊C8—22no—
110Q8⋊C8—22no—
111#322—22no—
112C22×Q8—22no—
113C22×Q8C2×D411yescharacteristic
114C22×Q8C2×D411yescharacteristic
115C2×Q16—22no—
116C2×Q16—24no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4×Q8—21no—
130C4×Q8—21no—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—44no—
136C4×Q8—82no—
137C4×Q8C2×D412yes—
138C4×Q8#4112yes—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D412yes—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—42no—
158C2×C42C2411yescharacteristic
159#107—22no—
160#107—42no—
161#107—42no—
162#107—44no—
163#106—22no—
164#106—42no—
165#106—82no—
166#106—81no—
167#106Q1614yescomplemented
168C4⋊C8—24no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—42no—
175C4⋊C8—82no—
176C4⋊C8C2×D412yes—
177#103—22no—
178#103—22no—
179#103—44no—
180#103—82no—
181#103—84no—
182#97—42no—
183C4×C8—24no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8C2×D412yes—
189C4×C8#4112yes—
190#95—22no—
191#95—42no—
192#95—42no—
193#95—42no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×Q8#12112yes—
205C2×Q8#13714yes—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4H(5,𝔽2)11yescharacteristic
217Q16—44no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—22no—
222C2×C8—21no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8#12112yes—
236C2×C8#12112yes—
237C2×C8C2×Q1612yes—
238C2×C8#13712yes—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—84no—
258C4⋊C4—82no—
259C4⋊C4—162no—
260C4⋊C4—162no—
261C4⋊C4—162no—
262C4⋊C4#12112yes—
263C4⋊C4C2×Q1614yes—
264C42—22no—
265C42—22no—
266C42—41no—
267C42—41no—
268C42—42no—
269C42—42no—
270C42—84no—
271C42C22×D411yescharacteristic
272C42C2×#4111yescharacteristic
273C42C2×#4111yescharacteristic
274C422−1+411yescharacteristic
275C23D4211yescharacteristic
276Q8—24no—
277Q8—22no—
278Q8—42no—
279Q8—41no—
280Q8—41no—
281Q8—84no—
282Q8—162no—
283Q8#47014yes—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4C2×#12111yescharacteristic
306C2×C4C2×#12111yescharacteristic
307C2×C4#53711yescharacteristic
308C2×C4#54711yescharacteristic
309C2×C4#54811yescharacteristic
310C2×C4#55311yescharacteristic
311C2×C4C22×Q1611yescharacteristic
312C2×C4C2×#13712yes—
313C2×C4C2×#13711yescharacteristic
314C2×C4#57712yes—
315C2×C4#57711yescharacteristic
316C2×C4#57811yescharacteristic
317C8—22no—
318C8—42no—
319C8—41no—
320C8—41no—
321C8—81no—
322C8—81no—
323C8—82no—
324C8#46214yes—
325C22#241611yescharacteristic
326C22#326612yes—
327C22#327811yescharacteristic
328C22D4×Q1611yescharacteristic
329C22#328812yes—
330C4—24no—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—42no—
335C4—81no—
336C4—82no—
337C4—81no—
338C4—162no—
339C4—162no—
340C4C2×#46211yescharacteristic
341C4#304611yescharacteristic
342C4C2×#47011yescharacteristic
343C4#306011yescharacteristic
344C4#334412yes—
345C4#336311yescharacteristic
346C4#337411yescharacteristic
347C2#1981312yes—
348C2#2018812yes—
349C2#2473211yescharacteristic
350C2#2478311yescharacteristic
351C2#2479911yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
167. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#286822
#248172
Q8×#1062
C2×#16762
C2×#15842
#106012
#104152
#98362
#95962
#89752
#86512