Finite Group Databasebeta

← #286675

Subgroups — #286675

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28748C211yesmaximal, characteristic
2#28689C211yesmaximal, characteristic
3#28283C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5#10602C212yesmaximal
6#10545C212yesmaximal
7#9731C212yesmaximal
8#9692C211yesmaximal, characteristic
9#9601C211yesmaximal, characteristic
10#9565C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#487—21no—
14C2×#475—21no—
15C2×C8×Q8—21no—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2083—22no—
20#2066—22no—
21#2043—22no—
22#1990C2212yes—
23#1984—21no—
24#1981C2212yes—
25#1981C2212yes—
26#1981C2212yes—
27#1956—21no—
28#1948C2212yes—
29#1947—21no—
30#1947—21no—
31#1939C2212yes—
32#1916—22no—
33#1842C2211yescharacteristic
34#1803—21no—
35C8×C4⋊C4C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#332C2211yescharacteristic
41C2×#332C2212yes—
42C2×#332C2211yescharacteristic
43C2×#332C2212yes—
44#1296C2211yescharacteristic
45#1296C2211yescharacteristic
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54#500—24no—
55#500—42no—
56#487—44no—
57#475—44no—
58C8×Q8—44no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#106—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8C2312yes—
82#391—21no—
83#391—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389D412yes—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384D411yescharacteristic
93#384C2311yescharacteristic
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4C2312yes—
98C43C2311yescharacteristic
99#340—22no—
100#340—22no—
101#340—22no—
102#336—22no—
103#336—22no—
104#336—21no—
105#336—21no—
106#336D412yes—
107#335—42no—
108#332—22no—
109#332—22no—
110#332—24no—
111#332—42no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8C2×D412yes—
121C22×C8C2×D412yes—
122C4⋊Q8—22no—
123C4⋊Q8—21no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—21no—
127C4⋊Q8—22no—
128C4⋊Q8—21no—
129C4⋊Q8—22no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—81no—
133C4⋊Q8—81no—
134C4⋊Q8D814yes—
135C4⋊Q8#4112yes—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×Q8—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42C2411yescharacteristic
167#107—44no—
168#107—44no—
169#106—44no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—44no—
181C4⋊C8—42no—
182C4⋊C8—81no—
183C4⋊C8SD1614yes—
184C4⋊C8#4112yes—
185#103—44no—
186#103—44no—
187#103—44no—
188#103—82no—
189#103—82no—
190#103—82no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—44no—
194C4×C8—42no—
195C4×C8—42no—
196#95—22no—
197#95—22no—
198#95—41no—
199#95—41no—
200C2×Q8—24no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—41no—
208C2×Q8—42no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—41no—
214C2×Q8—42no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C2×#4111yescharacteristic
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—42no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8C2×SD1614yes—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4C2×D814yes—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—22no—
279C42—41no—
280C42—42no—
281C42—42no—
282C42—82no—
283C42C2×D812yes—
284C42C2×SD1612yes—
285C42#13512yes—
286C42#13712yes—
287C42C2×#4111yescharacteristic
288C42C2×#4111yescharacteristic
289C42H(5,𝔽2)11yescharacteristic
290C422−1+411yescharacteristic
291C23C2×#12111yescharacteristic
292Q8—48no—
293Q8—42no—
294Q8—42no—
295Q8—81no—
296Q8—81no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300Q8—82no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4#46112yes—
326C2×C4#46112yes—
327C2×C4#46612yes—
328C2×C4#46612yes—
329C2×C4#47312yes—
330C2×C4#52911yescharacteristic
331C2×C4#54211yescharacteristic
332C2×C4#54611yescharacteristic
333C2×C4#54811yescharacteristic
334C2×C4#54811yescharacteristic
335C2×C4#55011yescharacteristic
336C2×C4C22×D811yescharacteristic
337C2×C4C22×SD1611yescharacteristic
338C2×C4C2×#13511yescharacteristic
339C2×C4C2×#13711yescharacteristic
340C2×C4#57611yescharacteristic
341C2×C4#57711yescharacteristic
342C2×C4#57811yescharacteristic
343C2×C4#57811yescharacteristic
344C8—24no—
345C8—42no—
346C8—42no—
347C8—81no—
348C8—81no—
349C8—82no—
350C8—81no—
351C22#242611yescharacteristic
352C22C2×#46111yescharacteristic
353C22#304411yescharacteristic
354C22C2×#46611yescharacteristic
355C22#306211yescharacteristic
356C22C2×#47311yescharacteristic
357C22#306911yescharacteristic
358C4—22no—
359C4—22no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—44no—
364C4—82no—
365C4—82no—
366C4—161no—
367C4—162no—
368C4#167314yes—
369C4#329811yescharacteristic
370C4#332811yescharacteristic
371C4#335811yescharacteristic
372C4#335911yescharacteristic
373C4#336111yescharacteristic
374C4#337411yescharacteristic
375C4#339111yescharacteristic
376C4#339411yescharacteristic
377C2C2×#167311yescharacteristic
378C2#1923211yescharacteristic
379C2#1940411yescharacteristic
380C2#1957211yescharacteristic
381C2#2527611yescharacteristic
382C2#2529211yescharacteristic
383C2#2531411yescharacteristic
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
#287482
#286892
#282832
#231492
#106022
#105452
#97312
#96922
#96012
#95652
C4×#3322