Finite Group Databasebeta

← #353070

Subgroups — #353070

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic
2#33297C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#33256C211yesmaximal, characteristic
5#30305C211yesmaximal, characteristic
6#30304C211yesmaximal, characteristic
7#8821C212yesmaximal
8#8821C212yesmaximal
9#8821C212yesmaximal
10#8819C212yesmaximal
11#7842C211yesmaximal, characteristic
12C2×#501—22no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16#2859C2211yescharacteristic
17#2084—22no—
18#2083—22no—
19#2058—22no—
20#2057—22no—
21#1993—22no—
22#1992—22no—
23#1990—22no—
24#1990—22no—
25#1941—22no—
26#1941—22no—
27#1847—22no—
28#1845C2212yes—
29#1845C2212yes—
30#1844C2212yes—
31#1844C2212yes—
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38#1834C2212yes—
39#1834C2212yes—
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46C2×#322C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×#125—21no—
54C2×#125—21no—
55#501—48no—
56C2×#107—24no—
57C2×#107—24no—
58C2×#106—24no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—24no—
62C2×#97D412yes—
63C2×#97D412yes—
64C2×C4×C8D412yes—
65C2×C4×C8D412yes—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384C2311yescharacteristic
85#384C2312yes—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#384C2312yes—
89#384C2311yescharacteristic
90#382—21no—
91#382—21no—
92C43C2311yescharacteristic
93#340—24no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#336D412yes—
102#336D412yes—
103#322Q818yescomplemented
104#322Q818yescomplemented
105C22×Q8—41no—
106C22×Q8—41no—
107C22×C8C2×D414yes—
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C22×C8C2×D414yes—
111C4⋊Q8—44no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C4⋊Q8—82no—
116C4⋊Q8—82no—
117#125—44no—
118#125—44no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
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—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2411yescharacteristic
154#107—48no—
155#107—84no—
156#106—44no—
157#106—44no—
158#106—48no—
159#106—84no—
160#97—24no—
161#97—24no—
162C4×C8—42no—
163C4×C8—42no—
164C4×C8C2×Q814yes—
165C4×C8C2×Q814yes—
166C4×C8C2×Q814yes—
167C4×C8C2×Q814yes—
168C4×C8C2×Q814yes—
169C4×C8C2×Q814yes—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174C2×Q8—82no—
175C2×Q8—82no—
176C2×Q8—82no—
177C2×Q8—82no—
178C2×Q8—82no—
179C2×Q8—82no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12711yescharacteristic
187C22×C4#12711yescharacteristic
188C22×C4#12711yescharacteristic
189C22×C4#12711yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8C4⋊Q814yes—
202C2×C8C4⋊Q818yes—
203C2×C8C4⋊Q818yes—
204C2×C8C4⋊Q814yes—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42AGL(1,ℤ/8)14yes—
238C42#13714yes—
239C42C22×Q812yes—
240C42C22×Q812yes—
241C23C2×#12711yescharacteristic
242Q8—164no—
243Q8—164no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4#49612yes—
271C2×C4#49612yes—
272C2×C4#49712yes—
273C2×C4#49712yes—
274C2×C4C2×C4⋊Q811yescharacteristic
275C2×C4C2×C4⋊Q812yes—
276C2×C4C2×C4⋊Q812yes—
277C2×C4C2×C4⋊Q811yescharacteristic
278C2×C4C2×AGL(1,ℤ/8)12yes—
279C2×C4C2×#13712yes—
280C2×C4#57512yes—
281C2×C4#57512yes—
282C2×C4#57712yes—
283C2×C4#57712yes—
284C8—24no—
285C8—28no—
286C8—28no—
287C8—24no—
288C22#286811yescharacteristic
289C22C2×#49611yescharacteristic
290C22C2×#49711yescharacteristic
291C22#315311yescharacteristic
292C22#315311yescharacteristic
293C22#315411yescharacteristic
294C22#315411yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—22no—
299C4—161no—
300C4—162no—
301C4—161no—
302C4—162no—
303C4—161no—
304C4—161no—
305C4#316212yes—
306C4#316314yes—
307C4#316514yes—
308C4#316612yes—
309C2#1967711yescharacteristic
310C2#1967811yescharacteristic
311C2#1968511yescharacteristic
312C2#1968611yescharacteristic
313C2#3331411yescharacteristic
314C2#3331711yescharacteristic
315C2#3331811yescharacteristic
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
103. #322Q8semidirect
104. #322Q8semidirect

Maximal subgroups

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

subgroupindex
#333162
#332972
#332892
#332562
#303052
#303042
#88212
#88192
#78422