Finite Group Databasebeta

← #317605

Subgroups — #317605

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29286C211yesmaximal, characteristic
2#28284C211yesmaximal, characteristic
3#22307C211yesmaximal, characteristic
4#22218C211yesmaximal, characteristic
5#10653C212yesmaximal
6#10542C212yesmaximal
7#9566C212yesmaximal
8#9561C212yesmaximal
9#8787C211yesmaximal, characteristic
10#8591C211yesmaximal, characteristic
11#8589C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13C2×#477—21no—
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2086—22no—
18#2068—22no—
19#1990C2211yescharacteristic
20#1988—21no—
21#1988C2212yes—
22#1984—22no—
23#1981C2212yes—
24#1956C2211yescharacteristic
25#1947—21no—
26#1947C2212yes—
27#1947C2212yes—
28#1939C2212yes—
29#1927—22no—
30#1916—22no—
31#1865—22no—
32#1844C2211yescharacteristic
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
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#487—22no—
52#487—22no—
53#477—44no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—21no—
73#391—21no—
74#391—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389D412yes—
81#384D411yescharacteristic
82#384C2311yescharacteristic
83#384C2311yescharacteristic
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2312yes—
90C43C2311yescharacteristic
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#336—22no—
97#336—22no—
98#336—21no—
99#336—21no—
100#336D412yes—
101#336D412yes—
102#332—24no—
103#332—24no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8C2×D412yes—
111C22×C8C2×D412yes—
112C4⋊Q8—22no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8SD1614yes—
120C4⋊Q8#4112yes—
121C4⋊Q8#4112yes—
122C4×Q8—42no—
123C4×Q8—42no—
124C4×Q8—82no—
125C4×Q8—82no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
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—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C42—22no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42C2411yescharacteristic
154#107—44no—
155#107—82no—
156#106—82no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—81no—
166C4⋊C8—81no—
167C4⋊C8—82no—
168#103—28no—
169#103—44no—
170#103—82no—
171#103—82no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8—21no—
175C4×C8—21no—
176C4×C8—82no—
177#95—22no—
178#95—22no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—41no—
194C2×Q8—81no—
195C2×Q8—81no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C22×D411yescharacteristic
208C22×C4C2×#4111yescharacteristic
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—22no—
227C4⋊C4—22no—
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—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
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—
249C4⋊C4C2×SD1614yes—
250C42—21no—
251C42—22no—
252C42—22no—
253C42—21no—
254C42—41no—
255C42—41no—
256C42—42no—
257C42—42no—
258C42—82no—
259C42—82no—
260C42C2×SD1612yes—
261C42#13512yes—
262C42C2×#4111yescharacteristic
263C42C2×#4111yescharacteristic
264C422−1+411yescharacteristic
265C422−1+411yescharacteristic
266C23C2×#12111yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—44no—
270Q8—84no—
271Q8—81no—
272Q8—81no—
273Q8—82no—
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—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4#46512yes—
305C2×C4#46512yes—
306C2×C4#52511yescharacteristic
307C2×C4#53611yescharacteristic
308C2×C4#54811yescharacteristic
309C2×C4#54811yescharacteristic
310C2×C4#55211yescharacteristic
311C2×C4#55211yescharacteristic
312C2×C4C22×SD1611yescharacteristic
313C2×C4C2×#13511yescharacteristic
314C2×C4#57611yescharacteristic
315C2×C4#57612yes—
316C2×C4#57811yescharacteristic
317C2×C4#57812yes—
318C8—42no—
319C8—42no—
320C8—82no—
321C8—82no—
322C8—82no—
323C22#253611yescharacteristic
324C22#304111yescharacteristic
325C22#304411yescharacteristic
326C22C2×#46511yescharacteristic
327C22#306111yescharacteristic
328C22#306911yescharacteristic
329C22#306911yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—44no—
337C4—82no—
338C4—82no—
339C4—82no—
340C4—161no—
341C4—161no—
342C4#312412yes—
343C4#324312yes—
344C4#335911yescharacteristic
345C4#336111yescharacteristic
346C4#336211yescharacteristic
347C4#336411yescharacteristic
348C2#1938511yescharacteristic
349C2#1940211yescharacteristic
350C2#1951111yescharacteristic
351C2#1953411yescharacteristic
352C2#2818611yescharacteristic
353C2#2821111yescharacteristic
354C2#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
#292862
#282842
#223072
#222182
#106532
#105422
#95662
#95612
#87872
#85912
#85892