Finite Group Databasebeta

← #307346

Subgroups — #307346

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28532C211yesmaximal, characteristic
2#26317C212yesmaximal
3#22808C211yesmaximal, characteristic
4#10487C212yesmaximal
5#10423C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9656C211yesmaximal, characteristic
9#9615C211yesmaximal, characteristic
10#8982C212yesmaximal
11C8×#107C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#475C2212yes—
14C2×C4×Q16C2211yescharacteristic
15#2568C2212yes—
16#2355C2212yes—
17#2097—22no—
18#2097—22no—
19#2074—22no—
20#1990C2212yes—
21#1984C2212yes—
22#1981C2212yes—
23#1969—21no—
24#1969—21no—
25#1956C2212yes—
26#1948C2212yes—
27#1928—22no—
28#1928—22no—
29#1927—22no—
30#1846—21no—
31#1845C2211yescharacteristic
32#1803C2212yes—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×Q8⋊C8C2212yes—
41C2×C82C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2211yescharacteristic
45C22×Q16D411yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8C2312yes—
48#500—22no—
49#500—22no—
50#475—24no—
51#475—42no—
52C4×Q16—22no—
53C4×Q16—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107D411yescharacteristic
59C2×#107D411yescharacteristic
60C2×#107D411yescharacteristic
61C2×#106—22no—
62C2×#106—22no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103D412yes—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#400—42no—
79#391—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#387—42no—
84#384C2312yes—
85#382—22no—
86#382—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4C2311yescharacteristic
90C4×C4⋊C4C2311yescharacteristic
91#340—22no—
92#340—22no—
93#340—42no—
94#340D411yescharacteristic
95#340D411yescharacteristic
96#336—22no—
97#336—21no—
98#336—22no—
99#336—21no—
100#335—22no—
101#335—22no—
102#332—42no—
103Q8⋊C8—42no—
104C82—22no—
105C22×Q8—22no—
106C22×Q8C2×D412yes—
107C2×Q16—22no—
108C2×Q16—22no—
109C2×Q16—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—42no—
141C2×C42C2411yescharacteristic
142#107—22no—
143#107—22no—
144#107—22no—
145#107—42no—
146#107—42no—
147#107—42no—
148#107—81no—
149#106—44no—
150#106—82no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—22no—
155C4⋊C8—21no—
156C4⋊C8#4112yes—
157#103—24no—
158#103—44no—
159#103—44no—
160#103—44no—
161#103—82no—
162#103—82no—
163#103—82no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—42no—
169C4×C8—42no—
170C4×C8#4112yes—
171C4×C8#4112yes—
172#95—21no—
173#95—21no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179#95—42no—
180#95—42no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4H(5,𝔽2)11yescharacteristic
204Q16—44no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—22no—
210C2×C8—21no—
211C2×C8—22no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—41no—
224C2×C8#12312yes—
225C2×C8#12312yes—
226C2×C8#13512yes—
227C2×C8#13512yes—
228C2×C8#13712yes—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—82no—
249C4⋊C4#13512yes—
250C42—42no—
251C42—42no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—162no—
257C42C2×#4111yescharacteristic
258C42C2×#4111yescharacteristic
259C42C2×#4111yescharacteristic
260C42H(5,𝔽2)11yescharacteristic
261C23D4211yescharacteristic
262Q8—42no—
263Q8—42no—
264Q8—42no—
265Q8—42no—
266Q8—82no—
267Q8—82no—
268Q8—82no—
269Q8—82no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4C2×#12311yescharacteristic
297C2×C4#53211yescharacteristic
298C2×C4#54011yescharacteristic
299C2×C4#54611yescharacteristic
300C2×C4#55011yescharacteristic
301C2×C4#55911yescharacteristic
302C2×C4C2×#13511yescharacteristic
303C2×C4C2×#13511yescharacteristic
304C2×C4C2×#13511yescharacteristic
305C2×C4C2×#13711yescharacteristic
306C2×C4#57611yescharacteristic
307C2×C4#57711yescharacteristic
308C2×C4#57811yescharacteristic
309C2×C4#57811yescharacteristic
310C8—22no—
311C8—22no—
312C8—42no—
313C8—41no—
314C8—42no—
315C8—82no—
316C8—81no—
317C8#48514yes—
318C22#244511yescharacteristic
319C22#327811yescharacteristic
320C22#328311yescharacteristic
321C22#328511yescharacteristic
322C22#328711yescharacteristic
323C22#329011yescharacteristic
324C22#329311yescharacteristic
325C4—21no—
326C4—21no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—82no—
332C4—82no—
333C4—162no—
334C4—162no—
335C4C2×#48511yescharacteristic
336C4#309711yescharacteristic
337C4#316811yescharacteristic
338C4#317511yescharacteristic
339C4#330311yescharacteristic
340C4#332811yescharacteristic
341C4#339311yescharacteristic
342C4#339611yescharacteristic
343C2#1994411yescharacteristic
344C2#1996611yescharacteristic
345C2#2001111yescharacteristic
346C2#2002011yescharacteristic
347C2#2623111yescharacteristic
348C2#2625311yescharacteristic
349C2#2627111yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ 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
#285322
#263172
#228082
#104872
#104232
#98232
#97312
#96562
#96152
#89822
C8×#1072