Finite Group Databasebeta

← #277495

Subgroups — #277495

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28675C211yesmaximal, characteristic
2#24732C212yesmaximal
3#22740C211yesmaximal, characteristic
4C2×#1674C211yesmaximal, characteristic
5C2×#1579C211yesmaximal, characteristic
6#10588C212yesmaximal
7#10419C212yesmaximal
8#9831C211yesmaximal, characteristic
9#9598C211yesmaximal, characteristic
10#8987C212yesmaximal
11#8649C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#462—22no—
14C2×#462C2212yes—
15C2×C8×Q8C2211yescharacteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18#2546C2212yes—
19Q8×C4⋊C4C2212yes—
20#2086—21no—
21#2086—21no—
22#2068—22no—
23#2043—22no—
24#1990C2212yes—
25#1988C2212yes—
26#1956C2212yes—
27#1947—22no—
28#1947C2212yes—
29#1942—22no—
30#1939C2212yes—
31#1878—22no—
32#1834—21no—
33#1834C2211yescharacteristic
34#1803C2212yes—
35C4×#107—21no—
36C4×#107C2211yescharacteristic
37#1674—24no—
38#1579—24no—
39C2×#335C2211yescharacteristic
40C2×#334C2211yescharacteristic
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×#322C2211yescharacteristic
45#1296C2212yes—
46#1279C2212yes—
47C22×Q16—22no—
48C22×Q16D411yescharacteristic
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8C2311yescharacteristic
53C2×C4×Q8C2311yescharacteristic
54#500—22no—
55#500—22no—
56#462—24no—
57#462—24no—
58#462—48no—
59C8×Q8—22no—
60C8×Q8—22no—
61C8×Q8—22no—
62C8×Q8D414yes—
63C4×Q16—22no—
64C4×Q16—22no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—22no—
68C2×#107D411yescharacteristic
69C2×#107D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103D411yescharacteristic
81C2×#103D411yescharacteristic
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#398—42no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—42no—
93#384—22no—
94#384C2312yes—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2312yes—
97#340—21no—
98#340—21no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—42no—
104#340D411yescharacteristic
105#340D411yescharacteristic
106#336—22no—
107#335—22no—
108#334—22no—
109#332—24no—
110Q8⋊C8—22no—
111Q8⋊C8—22no—
112#322—22no—
113C22×Q8—22no—
114C22×Q8C2×D411yescharacteristic
115C22×Q8C2×D411yescharacteristic
116C2×Q16—22no—
117C2×Q16—22no—
118C2×Q16—44no—
119C2×Q16—44no—
120C2×Q16—44no—
121C2×Q16D814yes—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—44no—
135C4⋊Q8—82no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C4×Q8—44no—
143C4×Q8—82no—
144C4×Q8C2×D412yes—
145C4×Q8#4112yes—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D412yes—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42—42no—
165C2×C42C2411yescharacteristic
166#107—22no—
167#107—24no—
168#107—42no—
169#107—42no—
170#107—44no—
171C4⋊C8—24no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178C4⋊C8—44no—
179C4⋊C8C2×D412yes—
180#103—22no—
181#103—22no—
182#103—44no—
183#103—44no—
184#103—44no—
185#103—82no—
186C4×C8—24no—
187C4×C8—21no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—21no—
191C4×C8—42no—
192C4×C8C2×D412yes—
193C4×C8#4112yes—
194#95—22no—
195#95—42no—
196#95—42no—
197#95—42no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8#12112yes—
213C2×Q8C2×D812yes—
214C2×Q8C2×D812yes—
215C2×Q8#13512yes—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4H(5,𝔽2)11yescharacteristic
227Q16—28no—
228Q16—88no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—21no—
234C2×C8—22no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8#12112yes—
246C2×C8#12112yes—
247C2×C8C2×D812yes—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4—82no—
270C4⋊C4—162no—
271C4⋊C4#12112yes—
272C4⋊C4#13714yes—
273C42—22no—
274C42—22no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42—42no—
279C42—84no—
280C42C22×D411yescharacteristic
281C42C2×#4111yescharacteristic
282C42C2×#4111yescharacteristic
283C422−1+411yescharacteristic
284C23D4211yescharacteristic
285Q8—22no—
286Q8—22no—
287Q8—22no—
288Q8—22no—
289Q8—22no—
290Q8—41no—
291Q8—41no—
292Q8—82no—
293Q8—82no—
294Q8—82no—
295Q8—82no—
296Q8#46614yes—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4C2×#12111yescharacteristic
321C2×C4C2×#12111yescharacteristic
322C2×C4#53711yescharacteristic
323C2×C4#54711yescharacteristic
324C2×C4#54811yescharacteristic
325C2×C4#55311yescharacteristic
326C2×C4C22×D811yescharacteristic
327C2×C4C2×#13511yescharacteristic
328C2×C4C2×#13712yes—
329C2×C4#57611yescharacteristic
330C2×C4#57811yescharacteristic
331C2×C4#57812yes—
332C8—22no—
333C8—22no—
334C8—22no—
335C8—42no—
336C8—41no—
337C8—81no—
338C8—81no—
339C8—82no—
340C22#241611yescharacteristic
341C22#327012yes—
342C22D4×D811yescharacteristic
343C22#328511yescharacteristic
344C22#328912yes—
345C4—24no—
346C4—21no—
347C4—21no—
348C4—44no—
349C4—42no—
350C4—42no—
351C4—82no—
352C4—81no—
353C4—162no—
354C4—162no—
355C4#303911yescharacteristic
356C4#304411yescharacteristic
357C4C2×#46611yescharacteristic
358C4#306111yescharacteristic
359C4#333411yescharacteristic
360C4#333911yescharacteristic
361C4#337412yes—
362C2#1976112yes—
363C2#1984512yes—
364C2#2474611yescharacteristic
365C2#2478011yescharacteristic
366C2#2480011yescharacteristic
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
#247322
#227402
C2×#16742
C2×#15792
#105882
#104192
#98312
#95982
#89872
#86492