Finite Group Databasebeta

← #317560

Subgroups — #317560

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29289C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3C8×#125C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic
5#10650C211yesmaximal, characteristic
6#10543C211yesmaximal, characteristic
7#10543C211yesmaximal, characteristic
8#9568C212yesmaximal
9#9567C212yesmaximal
10#8784C211yesmaximal, characteristic
11#8590C212yesmaximal
12C2×#500C2211yescharacteristic
13C4×C4⋊Q8C2212yes—
14C4×#125C2211yescharacteristic
15#2088—22no—
16#2086—22no—
17#1990C2212yes—
18#1989—22no—
19#1987C2212yes—
20#1987C2212yes—
21#1949—22no—
22#1948C2212yes—
23#1947C2212yes—
24#1942—21no—
25#1942—21no—
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1916—21no—
29#1916—21no—
30#1845C2211yescharacteristic
31#1842C2212yes—
32#1831C2211yescharacteristic
33C4×#107—21no—
34C4×#107—21no—
35C8×C4⋊C4—21no—
36C8×C4⋊C4—21no—
37C8×C4⋊C4C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
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⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—22no—
74#389—22no—
75#389—22no—
76#389D411yescharacteristic
77#389D411yescharacteristic
78#387—21no—
79#387—21no—
80#384C2312yes—
81#382D411yescharacteristic
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2311yescharacteristic
87C4×C4⋊C4C2311yescharacteristic
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#336—21no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
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—22no—
119#125Q1614yes—
120C4×Q8—84no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—42no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42C2411yescharacteristic
145#107—24no—
146#107—24no—
147#107—42no—
148#107—42no—
149#107—82no—
150#106—82no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—42no—
158C4⋊C8—42no—
159C4⋊C8—44no—
160C4⋊C8—82no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—41no—
167C4×C8—42no—
168C4×C8—42no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—42no—
176#95—42no—
177C2×Q8—24no—
178C2×Q8—42no—
179C2×Q8—82no—
180C2×Q8—82no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C2×#4111yescharacteristic
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8#13512yes—
216C2×C8#13712yes—
217C2×C8#13712yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C4⋊C4C2×Q1612yes—
237C4⋊C4C2×Q1612yes—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—22no—
243C42—22no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—84no—
250C42C2×Q1612yes—
251C42#13512yes—
252C42C2×#4112yes—
253C422−1+412yes—
254C23C2×#12111yescharacteristic
255Q8—84no—
256Q8—84no—
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—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#46712yes—
287C2×C4#46712yes—
288C2×C4#47212yes—
289C2×C4#52511yescharacteristic
290C2×C4#53611yescharacteristic
291C2×C4#54812yes—
292C2×C4#55212yes—
293C2×C4C22×Q1611yescharacteristic
294C2×C4C2×#13511yescharacteristic
295C2×C4C2×#13511yescharacteristic
296C2×C4C2×#13711yescharacteristic
297C2×C4C2×#13711yescharacteristic
298C2×C4#57511yescharacteristic
299C2×C4#57712yes—
300C8—22no—
301C8—22no—
302C8—22no—
303C8—41no—
304C8—81no—
305C8—82no—
306C8—82no—
307C8—81no—
308C22#253611yescharacteristic
309C22#303511yescharacteristic
310C22#303911yescharacteristic
311C22C2×#46711yescharacteristic
312C22#306011yescharacteristic
313C22C2×#47211yescharacteristic
314C22#306711yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—41no—
321C4—42no—
322C4—81no—
323C4—84no—
324C4—81no—
325C4—162no—
326C4#310811yescharacteristic
327C4#311711yescharacteristic
328C4#323111yescharacteristic
329C4#323611yescharacteristic
330C4#336312yes—
331C4#336512yes—
332C2#1912611yescharacteristic
333C2#1922711yescharacteristic
334C2#1927211yescharacteristic
335C2#1936011yescharacteristic
336C2#2818711yescharacteristic
337C2#2820911yescharacteristic
338C2#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
#292892
C4×#5002
C8×#1252
#106522
#106502
#105432
#95682
#95672
#87842
#85902