Finite Group Databasebeta

← #317594

Subgroups — #317594

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28284C211yesmaximal, characteristic
2#28283C211yesmaximal, characteristic
3#22307C211yesmaximal, characteristic
4#22247C211yesmaximal, characteristic
5#10655C212yesmaximal
6#10545C212yesmaximal
7#9566C212yesmaximal
8#9565C212yesmaximal
9#8787C211yesmaximal, characteristic
10#8589C211yesmaximal, characteristic
11#8589C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477—21no—
14C2×#475—21no—
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2068—22no—
19#2066—22no—
20#1990C2211yescharacteristic
21#1990C2211yescharacteristic
22#1984—22no—
23#1981C2212yes—
24#1981C2212yes—
25#1948C2212yes—
26#1947C2212yes—
27#1939C2212yes—
28#1918—22no—
29#1918—22no—
30#1916—22no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1834C2211yescharacteristic
34#1831C2211yescharacteristic
35#1803—21no—
36#1803—21no—
37C8×C4⋊C4C2212yes—
38#1744C2211yescharacteristic
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51#500—22no—
52#500—22no—
53#477—44no—
54#475—44no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#106—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×C4×C8—22no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—21no—
74#391—21no—
75#391—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389D412yes—
82#384D411yescharacteristic
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2312yes—
91C43C2311yescharacteristic
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#336—22no—
98#336—22no—
99#336—21no—
100#336—21no—
101#336D412yes—
102#336D412yes—
103#332—24no—
104#332—24no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8D814yes—
121C4⋊Q8#4112yes—
122C4⋊Q8#4112yes—
123C4×Q8—42no—
124C4×Q8—42no—
125C4×Q8—42no—
126C4×Q8—42no—
127C4×Q8—82no—
128C2×C4⋊C4—21no—
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—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C42—22no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156#107—28no—
157#107—44no—
158#106—44no—
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—44no—
171#103—44no—
172#103—82no—
173#103—82no—
174C4×C8—21no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—82no—
179#95—22no—
180#95—22no—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185C2×Q8—24no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—81no—
199C2×Q8—81no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C2×#4111yescharacteristic
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—82no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4C2×D814yes—
254C42—21no—
255C42—21no—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C42—42no—
261C42—42no—
262C42—82no—
263C42—82no—
264C42C2×D812yes—
265C42#13512yes—
266C42C2×#4111yescharacteristic
267C42C2×#4111yescharacteristic
268C422−1+411yescharacteristic
269C422−1+411yescharacteristic
270C23C2×#12111yescharacteristic
271Q8—42no—
272Q8—42no—
273Q8—42no—
274Q8—42no—
275Q8—84no—
276Q8—81no—
277Q8—81no—
278Q8—81no—
279Q8—81no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#46612yes—
311C2×C4#46612yes—
312C2×C4#52511yescharacteristic
313C2×C4#53611yescharacteristic
314C2×C4#54811yescharacteristic
315C2×C4#54811yescharacteristic
316C2×C4#55211yescharacteristic
317C2×C4#55211yescharacteristic
318C2×C4C22×D811yescharacteristic
319C2×C4C2×#13511yescharacteristic
320C2×C4#57711yescharacteristic
321C2×C4#57711yescharacteristic
322C2×C4#57712yes—
323C2×C4#57812yes—
324C8—42no—
325C8—42no—
326C8—82no—
327C8—82no—
328C8—82no—
329C22#253611yescharacteristic
330C22#304211yescharacteristic
331C22#304411yescharacteristic
332C22C2×#46611yescharacteristic
333C22#306011yescharacteristic
334C22#306911yescharacteristic
335C22#306911yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—44no—
343C4—82no—
344C4—82no—
345C4—82no—
346C4—161no—
347C4—161no—
348C4#312112yes—
349C4#324212yes—
350C4#335811yescharacteristic
351C4#335811yescharacteristic
352C4#336011yescharacteristic
353C4#336011yescharacteristic
354C2#1939211yescharacteristic
355C2#1940411yescharacteristic
356C2#1951711yescharacteristic
357C2#1953311yescharacteristic
358C2#2818511yescharacteristic
359C2#2821311yescharacteristic
360C2#2822211yescharacteristic
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

Maximal subgroups

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

subgroupindex
#282842
#282832
#223072
#222472
#106552
#105452
#95662
#95652
#87872
#85892