Finite Group Databasebeta

← #317672

Subgroups — #317672

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29275C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3#21673C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic
5#10650C211yesmaximal, characteristic
6#10543C211yesmaximal, characteristic
7#10543C211yesmaximal, characteristic
8#9576C212yesmaximal
9#9575C212yesmaximal
10#8785C211yesmaximal, characteristic
11#8594C212yesmaximal
12C2×#500C2211yescharacteristic
13C4×C4⋊Q8C2212yes—
14C4×#125C2211yescharacteristic
15#2101—22no—
16#2059—22no—
17#1990C2212yes—
18#1987C2212yes—
19#1987C2212yes—
20#1948C2212yes—
21#1947C2212yes—
22#1942—21no—
23#1942—21no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1920—22no—
27#1918—22no—
28#1916—21no—
29#1916—21no—
30#1845C2211yescharacteristic
31#1831C2212yes—
32#1831C2211yescharacteristic
33C4×#107—21no—
34C4×#107—21no—
35#1771—21no—
36#1771—21no—
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39C42×C8C2211yescharacteristic
40C2×#333C2212yes—
41C2×#333C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C2×C4⋊Q8C2312yes—
48C2×#125D411yescharacteristic
49C2×C4×Q8—22no—
50#500—24no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#97—21no—
64C2×#97—21no—
65C2×C4×C8D411yescharacteristic
66C2×C4×C8D411yescharacteristic
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D411yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#391—22no—
72#389—22no—
73#389—22no—
74#389D411yescharacteristic
75#389D411yescharacteristic
76#387—21no—
77#387—21no—
78#384C2312yes—
79#382D411yescharacteristic
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2311yescharacteristic
85C4×C4⋊C4C2311yescharacteristic
86C43C2311yescharacteristic
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336—22no—
100#336—21no—
101#336—21no—
102#336—22no—
103#333—24no—
104#333—24no—
105C22×Q8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8#4114yes—
118#125—42no—
119C4×Q8—84no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—42no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42C2411yescharacteristic
144#107—24no—
145#107—24no—
146#107—44no—
147#107—42no—
148#107—42no—
149#106—44no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—42no—
157C4⋊C8—42no—
158#97—42no—
159#97—42no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—41no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—42no—
173#95—42no—
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—82no—
177C2×Q8—82no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8#13512yes—
213C2×C8#13712yes—
214C2×C8#13712yes—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—22no—
240C42—21no—
241C42—21no—
242C42—22no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—84no—
249C42C2×#4112yes—
250C422−1+412yes—
251C23C2×#12111yescharacteristic
252Q8—84no—
253Q8—84no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#47212yes—
286C2×C4#52511yescharacteristic
287C2×C4#53611yescharacteristic
288C2×C4#54812yes—
289C2×C4#55212yes—
290C2×C4C2×#13511yescharacteristic
291C2×C4C2×#13711yescharacteristic
292C2×C4C2×#13711yescharacteristic
293C2×C4#57511yescharacteristic
294C2×C4#57512yes—
295C2×C4#57811yescharacteristic
296C2×C4#57811yescharacteristic
297C8—22no—
298C8—22no—
299C8—22no—
300C8—41no—
301C8—81no—
302C8—81no—
303C8—82no—
304C8—82no—
305C22#253611yescharacteristic
306C22#303511yescharacteristic
307C22#303911yescharacteristic
308C22#305411yescharacteristic
309C22#305911yescharacteristic
310C22C2×#47211yescharacteristic
311C22#306711yescharacteristic
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—84no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—162no—
323C4#310811yescharacteristic
324C4#311711yescharacteristic
325C4#323111yescharacteristic
326C4#323611yescharacteristic
327C4#337712yes—
328C4#337712yes—
329C2#1916011yescharacteristic
330C2#1922011yescharacteristic
331C2#1929811yescharacteristic
332C2#1935911yescharacteristic
333C2#2819311yescharacteristic
334C2#2820911yescharacteristic
335C2#2821711yescharacteristic
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
#292752
C4×#5002
#216732
#106522
#106502
#105432
#95762
#95752
#87852
#85942