Finite Group Databasebeta

← #330448

Subgroups — #330448

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29984C211yesmaximal, characteristic
3#29869C211yesmaximal, characteristic
4#21341C211yesmaximal, characteristic
5#10593C212yesmaximal
6#10589C212yesmaximal
7#10483C212yesmaximal
8#10482C212yesmaximal
9#9903C211yesmaximal, characteristic
10#9702C211yesmaximal, characteristic
11#8807C211yesmaximal, characteristic
12C2×#477C2212yes—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2295C2211yescharacteristic
16#2100—21no—
17#2095—21no—
18#2083—21no—
19#2083—21no—
20#2072—21no—
21#2072—21no—
22#2057—21no—
23#2024—21no—
24#1990—21no—
25#1990—21no—
26#1990C2212yes—
27#1990C2212yes—
28#1987C2212yes—
29#1984C2211yescharacteristic
30#1984C2211yescharacteristic
31#1947C2212yes—
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1916—21no—
35#1916—21no—
36#1834—21no—
37#1834—21no—
38#1831C2211yescharacteristic
39#1831C2211yescharacteristic
40#1803C2212yes—
41#1768—21no—
42#1767C2211yescharacteristic
43#1746—21no—
44C42×C8C2211yescharacteristic
45C2×#335C2212yes—
46C2×#334C2212yes—
47C2×#333C2212yes—
48C2×#333C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2312yes—
57C2×C4×Q8C2311yescharacteristic
58#477—42no—
59#477—42no—
60C2×#107—22no—
61C2×#107—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×#103—22no—
69C2×#103—22no—
70C2×#97—21no—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391D411yescharacteristic
78#391D411yescharacteristic
79#389D411yescharacteristic
80#389D411yescharacteristic
81#387—21no—
82#387—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2311yescharacteristic
93#376—21no—
94C43C2311yescharacteristic
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—21no—
104#336—21no—
105#336—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#335—42no—
115#334—42no—
116#333—42no—
117#333—42no—
118C22×Q8—22no—
119C22×Q8—41no—
120C22×Q8C2×D411yescharacteristic
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—82no—
136C4⋊Q8—82no—
137C4⋊Q8—82no—
138C4⋊Q8—82no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C4×Q8—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42C2411yescharacteristic
170#107—44no—
171#107—82no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182#103—44no—
183#103—82no—
184#97—42no—
185C4×C8—22no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—41no—
191C4×C8—41no—
192C4×C8Q1614yes—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—82no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C2×#4111yescharacteristic
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8C2×Q1612yes—
247C2×C8C2×Q1612yes—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42C2×Q1612yes—
283C42#13712yes—
284C42H(5,𝔽2)11yescharacteristic
285C42H(5,𝔽2)11yescharacteristic
286C422−1+411yescharacteristic
287C422−1+411yescharacteristic
288C23C2×#12111yescharacteristic
289Q8—42no—
290Q8—42no—
291Q8—162no—
292Q8—162no—
293Q8—164no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#46212yes—
323C2×C4#46212yes—
324C2×C4#47112yes—
325C2×C4#47212yes—
326C2×C4#53511yescharacteristic
327C2×C4#53611yescharacteristic
328C2×C4#53711yescharacteristic
329C2×C4#53711yescharacteristic
330C2×C4#54211yescharacteristic
331C2×C4#54211yescharacteristic
332C2×C4C22×Q1611yescharacteristic
333C2×C4C2×#13711yescharacteristic
334C2×C4#57511yescharacteristic
335C2×C4#57511yescharacteristic
336C2×C4#57712yes—
337C2×C4#57812yes—
338C8—22no—
339C8—22no—
340C8—41no—
341C8—41no—
342C8—82no—
343C8—82no—
344C8—81no—
345C8—81no—
346C22#262011yescharacteristic
347C22C2×#46211yescharacteristic
348C22#304011yescharacteristic
349C22#305711yescharacteristic
350C22#305911yescharacteristic
351C22C2×#47111yescharacteristic
352C22C2×#47211yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—21no—
357C4—21no—
358C4—22no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—162no—
364C4—162no—
365C4#322512yes—
366C4#324212yes—
367C4#324911yescharacteristic
368C4#324911yescharacteristic
369C4#325211yescharacteristic
370C4#325211yescharacteristic
371C2#1920711yescharacteristic
372C2#1920911yescharacteristic
373C2#1936211yescharacteristic
374C2#1936511yescharacteristic
375C2#2983011yescharacteristic
376C2#2984811yescharacteristic
377C2#2985811yescharacteristic
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
#332892
#299842
#298692
#213412
#105932
#105892
#104832
#104822
#99032
#97022
#88072