Finite Group Databasebeta

← #315028

Subgroups — #315028

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28814C211yesmaximal, characteristic
2#28675C211yesmaximal, characteristic
3#27749C212yesmaximal
4#10418C212yesmaximal
5#10357C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9562C212yesmaximal
9#8645C211yesmaximal, characteristic
10C8×#107C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#499—21no—
14C2×C8×Q8C2211yescharacteristic
15#2552C2212yes—
16#2546C2212yes—
17#2086—22no—
18#2072—22no—
19#2059—22no—
20#2043—22no—
21#1990C2212yes—
22#1988C2212yes—
23#1981C2212yes—
24#1956—22no—
25#1956C2212yes—
26#1948C2212yes—
27#1942—21no—
28#1942—21no—
29#1942—21no—
30#1933—22no—
31#1845C2211yescharacteristic
32#1803C2212yes—
33C4×#107C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#332C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2312yes—
47C2×#125—22no—
48C2×C4×Q8C2311yescharacteristic
49#500—22no—
50#500—22no—
51#499—44no—
52C8×Q8—22no—
53C8×Q8—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107D411yescharacteristic
59C2×#107D411yescharacteristic
60C2×#106—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103D411yescharacteristic
71C2×#103D411yescharacteristic
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#398—42no—
79#398—42no—
80#391—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#384—22no—
86#384C2312yes—
87C4×C4⋊C4C2312yes—
88C4×C4⋊C4C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—42no—
94#340D412yes—
95#340D411yescharacteristic
96#340D411yescharacteristic
97#336—21no—
98#336—22no—
99#335—22no—
100#332—24no—
101Q8⋊C8—22no—
102C82—22no—
103C22×Q8—22no—
104C22×Q8C2×D411yescharacteristic
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—44no—
118C4⋊Q8—82no—
119#125—44no—
120#125—82no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D412yes—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42C2411yescharacteristic
145#107—42no—
146#107—42no—
147#107—42no—
148#107—81no—
149#107SD1614yes—
150#107Q1614yes—
151#106—44no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—42no—
158C4⋊C8#4112yes—
159#103—22no—
160#103—22no—
161#103—44no—
162#103—82no—
163#103—82no—
164#103—82no—
165#103—82no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—21no—
169C4×C8—21no—
170C4×C8—42no—
171C4×C8—42no—
172C4×C8—82no—
173C4×C8#4112yes—
174C4×C8#4112yes—
175#95—22no—
176#95—21no—
177#95—42no—
178#95—42no—
179#95—42no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8#13512yes—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C22×D411yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4H(5,𝔽2)11yescharacteristic
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—21no—
209C2×C8—22no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8#12412yes—
223C2×C8#12412yes—
224C2×C8C2×SD1612yes—
225C2×C8C2×Q1612yes—
226C2×C8#13512yes—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C4⋊C4—82no—
250C4⋊C4—162no—
251C4⋊C4C2×SD1614yes—
252C4⋊C4C2×Q1614yes—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—42no—
257C42—42no—
258C42—41no—
259C42C2×#4111yescharacteristic
260C42C2×#4111yescharacteristic
261C42C2×#4111yescharacteristic
262C422−1+411yescharacteristic
263C23D4211yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—41no—
267Q8—41no—
268Q8—82no—
269Q8—82no—
270Q8—82no—
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4C2×#12411yescharacteristic
299C2×C4#53211yescharacteristic
300C2×C4#54211yescharacteristic
301C2×C4#54711yescharacteristic
302C2×C4#54811yescharacteristic
303C2×C4#55511yescharacteristic
304C2×C4C22×SD1611yescharacteristic
305C2×C4C22×Q1611yescharacteristic
306C2×C4C2×#13511yescharacteristic
307C2×C4C2×#13511yescharacteristic
308C2×C4#57611yescharacteristic
309C2×C4#57611yescharacteristic
310C2×C4#57711yescharacteristic
311C2×C4#57811yescharacteristic
312C8—22no—
313C8—22no—
314C8—42no—
315C8—41no—
316C8—42no—
317C8—81no—
318C8—82no—
319C8#48714yes—
320C22#249111yescharacteristic
321C22#328011yescharacteristic
322C22D4×SD1611yescharacteristic
323C22D4×Q1611yescharacteristic
324C22#328711yescharacteristic
325C22#329011yescharacteristic
326C22#329311yescharacteristic
327C4—21no—
328C4—21no—
329C4—44no—
330C4—44no—
331C4—42no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—162no—
336C4—162no—
337C4C2×#48711yescharacteristic
338C4#313711yescharacteristic
339C4#316811yescharacteristic
340C4#317611yescharacteristic
341C4#333911yescharacteristic
342C4#333911yescharacteristic
343C4#335911yescharacteristic
344C4#336311yescharacteristic
345C2#1997911yescharacteristic
346C2#1998511yescharacteristic
347C2#1999311yescharacteristic
348C2#2001011yescharacteristic
349C2#2776911yescharacteristic
350C2#2779011yescharacteristic
351C2#2780811yescharacteristic
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
#288142
#286752
#277492
#104182
#103572
#98232
#97312
#95622
#86452
C8×#1072
C8×#1032