Finite Group Databasebeta

← #323362

Subgroups — #323362

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29208C211yesmaximal, characteristic
4#29208C211yesmaximal, characteristic
5#29206C211yesmaximal, characteristic
6C4×#494C211yesmaximal, characteristic
7#10680C212yesmaximal
8#10679C212yesmaximal
9#9835C212yesmaximal
10#8820C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×C4×Q16—22no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2100—22no—
19#2058—22no—
20#2057—22no—
21#2057—22no—
22#1990—22no—
23#1990—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1987—22no—
29#1969—22no—
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948—22no—
33#1947—22no—
34#1947—22no—
35#1845C2212yes—
36#1845C2212yes—
37#1845C2211yescharacteristic
38#1844C2212yes—
39#1834C2212yes—
40#1834C2212yes—
41#1834C2212yes—
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C22×Q16—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58#494—22no—
59#494Q814yescomplemented
60C4×Q16—48no—
61C2×#107—22no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#106—24no—
65C2×C4⋊C8D412yes—
66C2×C4⋊C8D412yes—
67C2×#103—22no—
68C2×#103—22no—
69C2×C4×C8D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2312yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75#391—21no—
76#391—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89#384C2312yes—
90#384C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C43C2311yescharacteristic
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336D412yes—
104#336D412yes—
105#322—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—41no—
109C22×Q8—41no—
110C2×Q16—28no—
111C2×Q16—44no—
112C2×Q16—44no—
113C22×C8C2×D414yes—
114C22×C8C2×D412yes—
115C22×C8C2×D412yes—
116C22×C8C2×D414yes—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—82no—
127C4⋊Q8—82no—
128C4⋊Q8—82no—
129C4⋊Q8C2×Q812yes—
130C4⋊Q8C2×Q812yes—
131C4×Q8—44no—
132C4×Q8—44no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2411yescharacteristic
166#107—44no—
167#107—84no—
168#107—84no—
169#106—84no—
170C4⋊C8—24no—
171C4⋊C8Q1618yescomplemented
172#103—44no—
173#103—44no—
174C4×C8—24no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—22no—
181C4×C8—42no—
182C4×C8C2×Q812yes—
183C4×C8#4112yes—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4#12711yescharacteristic
209C22×C4#12711yescharacteristic
210C22×C4#12711yescharacteristic
211C22×C4#12711yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215Q16—416no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—24no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—24no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8C22⋊Q814yes—
227C2×C8C2×Q1618yes—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—82no—
260C42—82no—
261C42C2×Q1614yes—
262C42AGL(1,ℤ/8)14yes—
263C42#13714yes—
264C42C22×Q811yescharacteristic
265C42C2×#4111yescharacteristic
266C422−1+411yescharacteristic
267C422−1+411yescharacteristic
268C23C2×#12711yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—82no—
272Q8—82no—
273Q8—164no—
274Q8—164no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4#49412yes—
300C2×C4#49412yes—
301C2×C4#49612yes—
302C2×C4#49612yes—
303C2×C4#49712yes—
304C2×C4#49712yes—
305C2×C4C2×C22⋊Q811yescharacteristic
306C2×C4#53611yescharacteristic
307C2×C4#54711yescharacteristic
308C2×C4#54711yescharacteristic
309C2×C4D4×Q811yescharacteristic
310C2×C4D4×Q811yescharacteristic
311C2×C4C22×Q1612yes—
312C2×C4C2×AGL(1,ℤ/8)12yes—
313C2×C4C2×#13712yes—
314C2×C4#57512yes—
315C2×C4#57612yes—
316C2×C4#57712yes—
317C8—24no—
318C8—28no—
319C8—44no—
320C8—42no—
321C22#254211yescharacteristic
322C22C2×#49411yescharacteristic
323C22C2×#49611yescharacteristic
324C22C2×#49711yescharacteristic
325C22#315311yescharacteristic
326C22#315411yescharacteristic
327C22#315511yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—82no—
332C4—82no—
333C4—162no—
334C4—161no—
335C4—161no—
336C4—162no—
337C4#172314yes—
338C4#307712yes—
339C4#323512yes—
340C4#333812yes—
341C4#335612yes—
342C4Q8×Q1612yes—
343C4#338712yes—
344C2C2×#172311yescharacteristic
345C2#1963411yescharacteristic
346C2#1968211yescharacteristic
347C2#1970111yescharacteristic
348C2#2852111yescharacteristic
349C2#2853111yescharacteristic
350C2#2854211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
59. #494Q8semidirect
171. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332892
#292182
#292082
#292062
C4×#4942
#106802
#106792
#98352
#88202
#82612