Finite Group Databasebeta

← #318872

Subgroups — #318872

Order 512.

544 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#29287C211yesmaximal, characteristic
2#29286C211yesmaximal, characteristic
3#22311C211yesmaximal, characteristic
4#22270C211yesmaximal, characteristic
5#10596C212yesmaximal
6#10588C212yesmaximal
7#9742C212yesmaximal
8#9622C212yesmaximal
9#8809C211yesmaximal, characteristic
10#8591C211yesmaximal, characteristic
11#8591C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2088—22no—
17#2086—22no—
18#1988—21no—
19#1988C2212yes—
20#1988C2212yes—
21#1987—21no—
22#1985—22no—
23#1956C2211yescharacteristic
24#1956C2211yescharacteristic
25#1948—21no—
26#1948C2212yes—
27#1947—21no—
28#1947C2212yes—
29#1939C2212yes—
30#1917—22no—
31#1878—22no—
32#1878—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1834C2211yescharacteristic
36#1831C2211yescharacteristic
37#1770C2212yes—
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×Q16—22no—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52#494—22no—
53#494Q814yescomplemented
54C2×#107—22no—
55C2×#106—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#97—22no—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#391—21no—
70#391—21no—
71#391—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389D412yes—
78#384D411yescharacteristic
79#384C2311yescharacteristic
80#384C2311yescharacteristic
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2312yes—
87C43C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—21no—
98#336—21no—
99#336D412yes—
100#336D412yes—
101#334—24no—
102#334—42no—
103#332—42no—
104#332—42no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C2×Q16—28no—
109C2×Q16—82no—
110C2×Q16—82no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D412yes—
115C22×C8C2×D412yes—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C2×Q812yes—
125C4⋊Q8C2×Q812yes—
126C4×Q8—82no—
127C4×Q8—82no—
128C4×Q8—82no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
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—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C42—22no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—82no—
158#106—82no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—81no—
168C4⋊C8—81no—
169C4⋊C8—82no—
170#103—82no—
171#103—82no—
172#97—82no—
173C4×C8—21no—
174C4×C8—21no—
175C4×C8C2×Q812yes—
176C4×C8#4112yes—
177#95—22no—
178#95—22no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—81no—
193C2×Q8—81no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C2×#4111yescharacteristic
207Q16—88no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8C22⋊Q814yes—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—82no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C42—22no—
250C42—21no—
251C42—21no—
252C42—22no—
253C42—41no—
254C42—41no—
255C42—42no—
256C42—42no—
257C42—82no—
258C42—82no—
259C42#13712yes—
260C42#13712yes—
261C42C22×Q811yescharacteristic
262C42C2×#4111yescharacteristic
263C42H(5,𝔽2)11yescharacteristic
264C422−1+411yescharacteristic
265C23C2×#12111yescharacteristic
266Q8—44no—
267Q8—44no—
268Q8—84no—
269Q8—82no—
270Q8—82no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4#47012yes—
302C2×C4#47012yes—
303C2×C4C2×C22⋊Q811yescharacteristic
304C2×C4#53711yescharacteristic
305C2×C4D4×Q811yescharacteristic
306C2×C4#55011yescharacteristic
307C2×C4#55411yescharacteristic
308C2×C4#55611yescharacteristic
309C2×C4C2×#13711yescharacteristic
310C2×C4C2×#13711yescharacteristic
311C2×C4#57612yes—
312C2×C4#57712yes—
313C2×C4#57711yescharacteristic
314C2×C4#57711yescharacteristic
315C8—24no—
316C8—42no—
317C8—82no—
318C8—82no—
319C8—82no—
320C22#253811yescharacteristic
321C22#304111yescharacteristic
322C22#304311yescharacteristic
323C22C2×#47011yescharacteristic
324C22#306011yescharacteristic
325C22#306911yescharacteristic
326C22#306911yescharacteristic
327C4—21no—
328C4—21no—
329C4—22no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4—82no—
335C4—82no—
336C4—82no—
337C4—161no—
338C4—161no—
339C4#308112yes—
340C4#325712yes—
341C4#338711yescharacteristic
342C4#338711yescharacteristic
343C4#340011yescharacteristic
344C4#340011yescharacteristic
345C2#1942311yescharacteristic
346C2#1943511yescharacteristic
347C2#1950811yescharacteristic
348C2#1952411yescharacteristic
349C2#2827411yescharacteristic
350C2#2829011yescharacteristic
351C2#2830011yescharacteristic
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
53. #494Q8semidirect

Maximal subgroups

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

subgroupindex
#292872
#292862
#223112
#222702
#105962
#105882
#97422
#96222
#88092
#85912