Finite Group Databasebeta

← #317575

Subgroups — #317575

Order 512.

572 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29277C211yesmaximal, characteristic
2#29259C211yesmaximal, characteristic
3C4×#498C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10653C212yesmaximal
6#10544C212yesmaximal
7#9568C212yesmaximal
8#9566C212yesmaximal
9#8785C211yesmaximal, characteristic
10#8591C211yesmaximal, characteristic
11#8590C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×C8×Q8—21no—
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2100—21no—
18#2095—21no—
19#2057—21no—
20#2024—21no—
21#1990C2211yescharacteristic
22#1990C2211yescharacteristic
23#1987C2212yes—
24#1984—21no—
25#1981C2212yes—
26#1947C2212yes—
27#1947C2212yes—
28#1941—21no—
29#1941—21no—
30#1939—21no—
31#1939C2212yes—
32#1921—22no—
33#1863—22no—
34#1844C2211yescharacteristic
35#1842C2211yescharacteristic
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38C4×#106—21no—
39C4×#106—21no—
40C8×C4⋊C4—21no—
41C8×C4⋊C4C2212yes—
42C42×C8C2211yescharacteristic
43C2×#333C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1277C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55#498—22no—
56#498—22no—
57C8×Q8—44no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#106—21no—
61C2×#106—21no—
62C2×#106—21no—
63C2×#106—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×#103—22no—
71C2×#103—22no—
72C2×C4×C8—22no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#391—21no—
80#391—21no—
81#391—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389D412yes—
88#384D411yescharacteristic
89#384C2311yescharacteristic
90#384C2311yescharacteristic
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4C2312yes—
97C43C2311yescharacteristic
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—22no—
107#336—22no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#333—24no—
113#332—24no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C4⋊Q8—24no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8#4112yes—
132C4⋊Q8#4112yes—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—82no—
136C4×Q8—82no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42C2411yescharacteristic
165#107—44no—
166#107—82no—
167#106—24no—
168#106—24no—
169#106—42no—
170#106—42no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178C4⋊C8—44no—
179C4⋊C8—42no—
180#103—44no—
181#103—82no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—41no—
189C4×C8—44no—
190C4×C8Q1614yes—
191#95—22no—
192#95—22no—
193#95—41no—
194#95—41no—
195#95—41no—
196#95—41no—
197C2×Q8—24no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—81no—
207C2×Q8—81no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C2×#4111yescharacteristic
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8C2×Q1612yes—
240C2×C8C2×Q1612yes—
241C2×C8AGL(1,ℤ/8)12yes—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4#13514yes—
268C42—22no—
269C42—21no—
270C42—21no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—42no—
275C42—42no—
276C42—82no—
277C42—82no—
278C42C2×Q1612yes—
279C42#13712yes—
280C42C2×#4111yescharacteristic
281C42C2×#4111yescharacteristic
282C422−1+411yescharacteristic
283C422−1+411yescharacteristic
284C23C2×#12111yescharacteristic
285Q8—48no—
286Q8—44no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4#46212yes—
320C2×C4#46212yes—
321C2×C4#47112yes—
322C2×C4#52511yescharacteristic
323C2×C4#53611yescharacteristic
324C2×C4#54811yescharacteristic
325C2×C4#54811yescharacteristic
326C2×C4#55211yescharacteristic
327C2×C4#55211yescharacteristic
328C2×C4C22×Q1611yescharacteristic
329C2×C4C2×#13512yes—
330C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
331C2×C4C2×#13711yescharacteristic
332C2×C4#57511yescharacteristic
333C2×C4#57712yes—
334C8—22no—
335C8—22no—
336C8—22no—
337C8—41no—
338C8—82no—
339C8—82no—
340C8—82no—
341C22#253611yescharacteristic
342C22C2×#46211yescharacteristic
343C22#303611yescharacteristic
344C22#305011yescharacteristic
345C22#305711yescharacteristic
346C22C2×#47111yescharacteristic
347C22#306711yescharacteristic
348C4—22no—
349C4—22no—
350C4—21no—
351C4—21no—
352C4—22no—
353C4—44no—
354C4—41no—
355C4—82no—
356C4—82no—
357C4—82no—
358C4—161no—
359C4—161no—
360C4#310411yescharacteristic
361C4#311211yescharacteristic
362C4#323011yescharacteristic
363C4#323711yescharacteristic
364C4#336012yes—
365C4#336512yes—
366C2#1911811yescharacteristic
367C2#1920711yescharacteristic
368C2#1925711yescharacteristic
369C2#1933411yescharacteristic
370C2#2818911yescharacteristic
371C2#2820611yescharacteristic
372C2#2821811yescharacteristic
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
#292772
#292592
C4×#4982
C8×C4⋊Q82
#106532
#105442
#95682
#95662
#87852
#85912
#85902