Finite Group Databasebeta

← #352856

Subgroups — #352856

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33290C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic
3#30305C211yesmaximal, characteristic
4#30262C211yesmaximal, characteristic
5#30230C211yesmaximal, characteristic
6#30205C211yesmaximal, characteristic
7#9840C212yesmaximal
8#9839C212yesmaximal
9#9835C212yesmaximal
10#9826C212yesmaximal
11#7843C211yesmaximal, characteristic
12C2×#494C2212yes—
13C2×#487C2212yes—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2862C2211yescharacteristic
17#2859C2211yescharacteristic
18#2100—22no—
19#2100—22no—
20#2084—22no—
21#2072—22no—
22#2058—22no—
23#2024—22no—
24#2016—22no—
25#2016—22no—
26#1991—22no—
27#1990—22no—
28#1990C2212yes—
29#1990C2212yes—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2212yes—
35#1990C2212yes—
36#1990C2212yes—
37#1956C2212yes—
38#1844C2212yes—
39#1835—22no—
40#1834—22no—
41#1834C2212yes—
42#1744C2211yescharacteristic
43#1744C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46C2×#322C2212yes—
47C22×Q16—24no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2312yes—
56C2×#125—21no—
57#494—24no—
58#494—24no—
59#487—24no—
60#487—24no—
61C2×#106—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×C4×C8D412yes—
65C2×C4×C8D412yes—
66C2×C4×C8C2312yes—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2312yes—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—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#384—21no—
85#384C2311yescharacteristic
86#384C2312yes—
87#384C2311yescharacteristic
88#382—21no—
89#382—21no—
90#382—21no—
91C43C2311yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#336D412yes—
103#336D412yes—
104#336D412yes—
105#336D412yes—
106#322—24no—
107#322—24no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—41no—
112C2×Q16—48no—
113C2×Q16—84no—
114C2×Q16—84no—
115C22×C8C2×D414yes—
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C22×C8C2×D414yes—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—44no—
128C4⋊Q8—82no—
129C4⋊Q8—82no—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132#125—44no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
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×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2411yescharacteristic
164#106—84no—
165#103—48no—
166#103—84no—
167C4×C8—24no—
168C4×C8—24no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8#4114yes—
174C4×C8#4114yes—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—44no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—82no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4#12711yescharacteristic
201C22×C4#12711yescharacteristic
202C22×C4#12711yescharacteristic
203C22×C4#12711yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207Q16—816no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8#12418yes—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42#13714yes—
253C42#13714yes—
254C42C2×#4112yes—
255C42C2×#4112yes—
256C23C2×#12711yescharacteristic
257Q8—84no—
258Q8—84no—
259Q8—84no—
260Q8—84no—
261Q8—164no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4#49712yes—
289C2×C4#49712yes—
290C2×C4#49712yes—
291C2×C4#49712yes—
292C2×C4C2×#12412yes—
293C2×C4#53212yes—
294C2×C4#53211yescharacteristic
295C2×C4#53211yescharacteristic
296C2×C4C2×#13712yes—
297C2×C4C2×#13712yes—
298C2×C4#57612yes—
299C2×C4#57712yes—
300C2×C4#57712yes—
301C2×C4#57812yes—
302C8—28no—
303C8—44no—
304C8—42no—
305C8—42no—
306C22#285811yescharacteristic
307C22C2×#49711yescharacteristic
308C22C2×#49711yescharacteristic
309C22#315411yescharacteristic
310C22#315411yescharacteristic
311C22#315511yescharacteristic
312C22#315611yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—162no—
318C4—161no—
319C4—161no—
320C4—161no—
321C4—161no—
322C4—162no—
323C4#313614yes—
324C4#317814yes—
325C4#318512yes—
326C4#318512yes—
327C2#1970311yescharacteristic
328C2#1970411yescharacteristic
329C2#1970511yescharacteristic
330C2#1970611yescharacteristic
331C2#3322811yescharacteristic
332C2#3324411yescharacteristic
333C2#3325511yescharacteristic
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

Maximal subgroups

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

subgroupindex
#332902
#332892
#303052
#302622
#302302
#302052
#98402
#98392
#98352
#98262
#78432