Finite Group Databasebeta

← #288053

Subgroups — #288053

Order 512.

684 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#28675C211yesmaximal, characteristic
2#25276C212yesmaximal
3#23184C211yesmaximal, characteristic
4C2×#1707C211yesmaximal, characteristic
5C2×#1670C211yesmaximal, characteristic
6#10542C212yesmaximal
7#10075C212yesmaximal
8#9731C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#9059C212yesmaximal
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#498C2212yes—
14C2×#494C2211yescharacteristic
15C2×#487C2212yes—
16C2×#475C2212yes—
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16—21no—
19#2546C2212yes—
20#2365C2212yes—
21#2086—22no—
22#2066—22no—
23#2043—22no—
24#1988C2212yes—
25#1981—22no—
26#1981C2212yes—
27#1981C2212yes—
28#1969—21no—
29#1942—21no—
30#1942—21no—
31#1939C2212yes—
32#1927—22no—
33#1878—22no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1707—24no—
38#1670—24no—
39C2×#335C2211yescharacteristic
40C2×#332C2212yes—
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×C82C2211yescharacteristic
44#1279C2211yescharacteristic
45#1279C2212yes—
46#1277C2211yescharacteristic
47C22×Q16—21no—
48C22×Q16—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2312yes—
51C2×C4⋊Q8C2312yes—
52C2×C4×Q8—22no—
53C2×C4×Q8C2311yescharacteristic
54#500—22no—
55#500—22no—
56#498—24no—
57#498—24no—
58#494—22no—
59#494D414yes—
60#487—24no—
61#487—42no—
62#475—24no—
63#475—42no—
64C8×Q8—22no—
65C8×Q8—22no—
66C4×Q16—44no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107D411yescharacteristic
70C2×#107D411yescharacteristic
71C2×#106—22no—
72C2×#106—42no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103D411yescharacteristic
86C2×#103D411yescharacteristic
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#398—42no—
93#391—22no—
94#391—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#387—42no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2311yescharacteristic
101C4×C4⋊C4C2312yes—
102#340—22no—
103#340—22no—
104#340D411yescharacteristic
105#340D412yes—
106#340D411yescharacteristic
107#336—21no—
108#336—22no—
109#335—22no—
110#332—24no—
111#332—42no—
112Q8⋊C8—22no—
113C82—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8C2×D411yescharacteristic
117C2×Q16—24no—
118C2×Q16—24no—
119C2×Q16—42no—
120C2×Q16—42no—
121C2×Q16—81no—
122C2×Q16—81no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—44no—
139C4⋊Q8—82no—
140C4⋊Q8C2×D414yes—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—44no—
146C4×Q8—82no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—42no—
164C2×C42C2411yescharacteristic
165#107—22no—
166#107—42no—
167#107—42no—
168#107SD1614yes—
169#106—44no—
170#106—48no—
171C4⋊C8—21no—
172C4⋊C8—22no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178C4⋊C8—82no—
179#103—22no—
180#103—22no—
181#103—44no—
182#103—44no—
183#103—44no—
184#103—44no—
185#103—44no—
186#103—82no—
187#103—82no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—21no—
191C4×C8—42no—
192C4×C8—42no—
193C4×C8C2×D412yes—
194C4×C8#4112yes—
195C4×C8#4112yes—
196#95—21no—
197#95—22no—
198#95—42no—
199#95—42no—
200#95—42no—
201#95—42no—
202C2×Q8—21no—
203C2×Q8—24no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8#13512yes—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4H(5,𝔽2)11yescharacteristic
231Q16—48no—
232Q16—84no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8#12112yes—
254C2×C8#12112yes—
255C2×C8C2×SD1612yes—
256C2×C8#13712yes—
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—21no—
262C4⋊C4—24no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—84no—
278C4⋊C4C2×SD1614yes—
279C4⋊C4#13512yes—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—42no—
284C42—42no—
285C42—41no—
286C42—84no—
287C42C22×D411yescharacteristic
288C42C2×#4111yescharacteristic
289C42C2×#4111yescharacteristic
290C42H(5,𝔽2)11yescharacteristic
291C23D4211yescharacteristic
292Q8—22no—
293Q8—22no—
294Q8—41no—
295Q8—41no—
296Q8—44no—
297Q8—44no—
298Q8—82no—
299Q8—82no—
300Q8—82no—
301Q8—82no—
302Q8—82no—
303Q8—82no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4C2×#12111yescharacteristic
331C2×C4#53211yescharacteristic
332C2×C4#54011yescharacteristic
333C2×C4D4211yescharacteristic
334C2×C4#54611yescharacteristic
335C2×C4#55011yescharacteristic
336C2×C4C22×SD1611yescharacteristic
337C2×C4C2×#13511yescharacteristic
338C2×C4C2×#13511yescharacteristic
339C2×C4C2×#13711yescharacteristic
340C2×C4#57611yescharacteristic
341C2×C4#57711yescharacteristic
342C2×C4#57711yescharacteristic
343C2×C4#57811yescharacteristic
344C8—24no—
345C8—22no—
346C8—41no—
347C8—41no—
348C8—81no—
349C8—82no—
350C8—81no—
351C8#46114yes—
352C22#242811yescharacteristic
353C22#327811yescharacteristic
354C22#328011yescharacteristic
355C22D4×SD1611yescharacteristic
356C22#328511yescharacteristic
357C22#329311yescharacteristic
358C22#329311yescharacteristic
359C4—21no—
360C4—21no—
361C4—44no—
362C4—42no—
363C4—42no—
364C4—81no—
365C4—84no—
366C4—81no—
367C4—82no—
368C4—162no—
369C4C2×#46111yescharacteristic
370C4#304211yescharacteristic
371C4#316811yescharacteristic
372C4#317511yescharacteristic
373C4#328511yescharacteristic
374C4#328711yescharacteristic
375C4#339211yescharacteristic
376C4#340011yescharacteristic
377C2#1990711yescharacteristic
378C2#1994411yescharacteristic
379C2#1997911yescharacteristic
380C2#1998311yescharacteristic
381C2#2536811yescharacteristic
382C2#2541211yescharacteristic
383C2#2543511yescharacteristic
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
#286752
#252762
#231842
C2×#17072
C2×#16702
#105422
#100752
#97312
#95942
#90592
C8×#1032