Finite Group Databasebeta

← #318824

Subgroups — #318824

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29289C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3#21673C211yesmaximal, characteristic
4#10597C212yesmaximal
5#10589C212yesmaximal
6#9739C211yesmaximal, characteristic
7#9739C211yesmaximal, characteristic
8#9621C211yesmaximal, characteristic
9#9620C211yesmaximal, characteristic
10#8805C211yesmaximal, 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—
35#1771—21no—
36#1771—21no—
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39C42×C8C2211yescharacteristic
40C2×#334C2211yescharacteristic
41C2×#334C2211yescharacteristic
42C2×#334C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C2×C4⋊Q8C2312yes—
48C2×#125D411yescharacteristic
49C2×C4×Q8—22no—
50#500—22no—
51#500Q814yes—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#106—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#97—21no—
67C2×#97—21no—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#389—22no—
76#389—22no—
77#389D411yescharacteristic
78#389D411yescharacteristic
79#387—21no—
80#387—21no—
81#384C2312yes—
82#382D411yescharacteristic
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2311yescharacteristic
88C4×C4⋊C4C2311yescharacteristic
89C43C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—21no—
104#334—22no—
105#334—22no—
106#334—42no—
107#333—42no—
108#333—42no—
109C22×Q8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8C2×Q814yes—
122#125—24no—
123C4×Q8—84no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—42no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42C2411yescharacteristic
148#107—24no—
149#107—24no—
150#107—42no—
151#107—42no—
152#107—82no—
153#106—82no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—44no—
165C4⋊C8—82no—
166#97—42no—
167#97—42no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—41no—
174C4×C8C2×Q812yes—
175C4×C8#4112yes—
176#95—21no—
177#95—21no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—42no—
183#95—42no—
184C2×Q8—24no—
185C2×Q8—42no—
186C2×Q8—82no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C2×#4111yescharacteristic
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8C22⋊Q812yes—
221C2×C8C22⋊Q812yes—
222C2×C8#13512yes—
223C2×C8#13712yes—
224C2×C8#13712yes—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C4⋊C4—84no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—22no—
248C42—22no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42—84no—
255C42#13714yes—
256C42C22×Q811yescharacteristic
257C42C2×#4111yescharacteristic
258C42H(5,𝔽2)11yescharacteristic
259C422−1+411yescharacteristic
260C23C2×#12111yescharacteristic
261Q8—84no—
262Q8—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#47012yes—
293C2×C4#47012yes—
294C2×C4#47212yes—
295C2×C4C2×C22⋊Q811yescharacteristic
296C2×C4#53711yescharacteristic
297C2×C4D4×Q811yescharacteristic
298C2×C4#55011yescharacteristic
299C2×C4#55411yescharacteristic
300C2×C4#55611yescharacteristic
301C2×C4C2×#13511yescharacteristic
302C2×C4C2×#13711yescharacteristic
303C2×C4C2×#13711yescharacteristic
304C2×C4C2×#13712yes—
305C2×C4#57511yescharacteristic
306C2×C4#57712yes—
307C8—22no—
308C8—22no—
309C8—41no—
310C8—82no—
311C8—82no—
312C8—81no—
313C8—81no—
314C8#47914yes—
315C22#253811yescharacteristic
316C22#303511yescharacteristic
317C22#303911yescharacteristic
318C22C2×#47011yescharacteristic
319C22#306011yescharacteristic
320C22C2×#47211yescharacteristic
321C22#306711yescharacteristic
322C4—22no—
323C4—22no—
324C4—21no—
325C4—21no—
326C4—22no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—84no—
331C4—81no—
332C4—81no—
333C4—162no—
334C4C2×#47911yescharacteristic
335C4#307811yescharacteristic
336C4#324811yescharacteristic
337C4#325211yescharacteristic
338C4#338712yes—
339C4#340012yes—
340C2#1915811yescharacteristic
341C2#1922711yescharacteristic
342C2#1929711yescharacteristic
343C2#1936011yescharacteristic
344C2#2827411yescharacteristic
345C2#2828711yescharacteristic
346C2#2829511yescharacteristic
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
#216732
#105972
#105892
#97392
#96212
#96202
#88052
#85902