Finite Group Databasebeta

← #307364

Subgroups — #307364

Order 512.

592 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#28532C211yesmaximal, characteristic
2#26320C212yesmaximal
3C4⋊C4×Q16C211yesmaximal, characteristic
4#10486C212yesmaximal
5#10423C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9656C211yesmaximal, characteristic
9#9615C211yesmaximal, characteristic
10#8983C212yesmaximal
11C8×#106C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#475C2212yes—
14C2×C4×Q16—21no—
15C2×C4×Q16C2211yescharacteristic
16#2568C2212yes—
17Q8×C4⋊C4C2212yes—
18#2097—22no—
19#2096—22no—
20#2072—22no—
21#1990C2212yes—
22#1984C2212yes—
23#1981C2212yes—
24#1969—21no—
25#1956C2212yes—
26#1948C2212yes—
27#1942—21no—
28#1927—22no—
29#1927—22no—
30#1863—22no—
31#1846—21no—
32#1845C2211yescharacteristic
33#1803C2212yes—
34C4×#106C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#334C2211yescharacteristic
39C2×#332C2212yes—
40C2×Q8⋊C8C2212yes—
41C2×C82C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2211yescharacteristic
45C22×Q16D411yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8C2312yes—
49#500—22no—
50#500—22no—
51#475—24no—
52#475—42no—
53C4×Q16—22no—
54C4×Q16—22no—
55C4×Q16—44no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107D411yescharacteristic
61C2×#106—22no—
62C2×#106—22no—
63C2×#106D411yescharacteristic
64C2×#106D411yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103D412yes—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#400—42no—
81#391—22no—
82#391—22no—
83#391—22no—
84#389—22no—
85#389—42no—
86#384—22no—
87#384C2312yes—
88#382—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4C2311yescharacteristic
91C4×C4⋊C4C2311yescharacteristic
92#340—22no—
93#340—22no—
94#340—42no—
95#340D411yescharacteristic
96#340D411yescharacteristic
97#336—22no—
98#336—22no—
99#336—21no—
100#336—21no—
101#335—22no—
102#334—22no—
103#332—42no—
104Q8⋊C8—42no—
105C82—22no—
106C22×Q8—22no—
107C22×Q8C2×D412yes—
108C2×Q16—22no—
109C2×Q16—22no—
110C2×Q16SD1614yes—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—44no—
128C4×Q8—82no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—21no—
145C2×C42—42no—
146C2×C42C2411yescharacteristic
147#107—22no—
148#107—42no—
149#107—42no—
150#107—42no—
151#107—81no—
152#106—22no—
153#106—22no—
154#106—44no—
155#106—82no—
156C4⋊C8—21no—
157C4⋊C8—22no—
158C4⋊C8—21no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8#4112yes—
162#103—24no—
163#103—44no—
164#103—44no—
165#103—44no—
166#103—44no—
167#103—82no—
168#103—82no—
169C4×C8—21no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—42no—
174C4×C8—42no—
175C4×C8#4112yes—
176C4×C8#4112yes—
177#95—21no—
178#95—21no—
179#95—42no—
180#95—42no—
181#95—42no—
182#95—42no—
183#95—42no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8C2×SD1614yes—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4H(5,𝔽2)11yescharacteristic
207Q16—28no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—22no—
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—42no—
223C2×C8—41no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—41no—
227C2×C8#12312yes—
228C2×C8#12312yes—
229C2×C8C2×SD1612yes—
230C2×C8#13512yes—
231C2×C8#13712yes—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—84no—
252C4⋊C4—82no—
253C4⋊C4—162no—
254C4⋊C4#13512yes—
255C42—42no—
256C42—42no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—84no—
262C42C2×#4111yescharacteristic
263C42C2×#4111yescharacteristic
264C42C2×#4111yescharacteristic
265C42H(5,𝔽2)11yescharacteristic
266C23D4211yescharacteristic
267Q8—24no—
268Q8—24no—
269Q8—42no—
270Q8—42no—
271Q8—82no—
272Q8—82no—
273Q8—82no—
274Q8—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4C2×#12311yescharacteristic
302C2×C4#53211yescharacteristic
303C2×C4#54011yescharacteristic
304C2×C4#54611yescharacteristic
305C2×C4#55011yescharacteristic
306C2×C4#55911yescharacteristic
307C2×C4C22×SD1611yescharacteristic
308C2×C4C2×#13511yescharacteristic
309C2×C4C2×#13511yescharacteristic
310C2×C4C2×#13711yescharacteristic
311C2×C4#57611yescharacteristic
312C2×C4#57711yescharacteristic
313C2×C4#57811yescharacteristic
314C2×C4#57811yescharacteristic
315C8—22no—
316C8—22no—
317C8—42no—
318C8—42no—
319C8—41no—
320C8—82no—
321C8—81no—
322C8#48314yes—
323C22#244511yescharacteristic
324C22#327611yescharacteristic
325C22D4×SD1611yescharacteristic
326C22#328511yescharacteristic
327C22#328711yescharacteristic
328C22#329011yescharacteristic
329C22#329311yescharacteristic
330C4—21no—
331C4—21no—
332C4—42no—
333C4—44no—
334C4—81no—
335C4—82no—
336C4—81no—
337C4—81no—
338C4—162no—
339C4—162no—
340C4C2×#48311yescharacteristic
341C4#309711yescharacteristic
342C4#316811yescharacteristic
343C4#317511yescharacteristic
344C4#329811yescharacteristic
345C4#333111yescharacteristic
346C4#339311yescharacteristic
347C4#339611yescharacteristic
348C2#1994011yescharacteristic
349C2#1996411yescharacteristic
350C2#1998311yescharacteristic
351C2#1998511yescharacteristic
352C2#2623411yescharacteristic
353C2#2625311yescharacteristic
354C2#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
#263202
C4⋊C4×Q162
#104862
#104232
#98232
#97312
#96562
#96152
#89832
C8×#1062