Finite Group Databasebeta

← #276529

Subgroups — #276529

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33196C211yesmaximal, characteristic
2#29208C211yesmaximal, characteristic
3#22828C211yesmaximal, characteristic
4#21616C211yesmaximal, characteristic
5#10618C212yesmaximal
6#10483C212yesmaximal
7#9903C211yesmaximal, characteristic
8#9615C212yesmaximal
9#9604C211yesmaximal, characteristic
10#9567C212yesmaximal
11#8590C211yesmaximal, characteristic
12C2×#477C2211yescharacteristic
13C2×#475—21no—
14#2868C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#2057—21no—
18#2057—21no—
19#1990C2212yes—
20#1988—21no—
21#1987C2211yescharacteristic
22#1984C2212yes—
23#1948—21no—
24#1948C2212yes—
25#1948C2211yescharacteristic
26#1947—22no—
27#1918—21no—
28#1916—22no—
29#1865—21no—
30#1845C2212yes—
31#1842C2211yescharacteristic
32#1839—21no—
33#1831C2211yescharacteristic
34#1803—21no—
35#1803C2211yescharacteristic
36#1803C2212yes—
37#1776—21no—
38C8×C4⋊C4C2212yes—
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1763—21no—
42#1744—21no—
43C2×#335C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57#477—22no—
58#477—22no—
59#475—44no—
60C2×#107—21no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#106—22no—
64C2×#106—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×#103—21no—
73C2×#103—22no—
74C2×#103—21no—
75C2×#97—21no—
76C2×C4×C8—22no—
77C2×C4×C8C2312yes—
78#391—21no—
79#389—21no—
80#389—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384C2311yescharacteristic
85#384C2312yes—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4C2311yescharacteristic
94C43—21no—
95C43C2311yescharacteristic
96#340—22no—
97#340—22no—
98#340—22no—
99#340—21no—
100#340—22no—
101#340—21no—
102#340—21no—
103#340—21no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336D411yescharacteristic
111#336D411yescharacteristic
112#335—24no—
113#333—24no—
114C22×Q8—21no—
115C22×Q8—41no—
116C22×Q8C2×D411yescharacteristic
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8—82no—
129C4⋊Q8—82no—
130C4⋊Q8#4112yes—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—41no—
138C4×Q8—82no—
139C4×Q8Q1614yes—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—41no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#107—24no—
173#107—82no—
174#107—82no—
175#106—42no—
176#106—44no—
177C4⋊C8—24no—
178C4⋊C8—21no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186C4⋊C8—42no—
187C4⋊C8#4112yes—
188#103—24no—
189#103—42no—
190#103—44no—
191#97—42no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—44no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—41no—
199#95—41no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—81no—
210C2×Q8—81no—
211C2×Q8—82no—
212C2×Q8C2×Q1612yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
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—42no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8#13714yes—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—81no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4C2×Q1612yes—
271C4⋊C4AGL(1,ℤ/8)12yes—
272C42—21no—
273C42—22no—
274C42—21no—
275C42—22no—
276C42—24no—
277C42—22no—
278C42—41no—
279C42—42no—
280C42—41no—
281C42—41no—
282C42—42no—
283C42—42no—
284C42—81no—
285C42—82no—
286C42C2×Q1612yes—
287C42#13512yes—
288C42C2×#4111yescharacteristic
289C42C2×#4111yescharacteristic
290C422−1+411yescharacteristic
291C422−1+411yescharacteristic
292C23C2×#12111yescharacteristic
293Q8—22no—
294Q8—22no—
295Q8—41no—
296Q8—41no—
297Q8—82no—
298Q8—82no—
299Q8—164no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4#46712yes—
333C2×C4#46712yes—
334C2×C4#47212yes—
335C2×C4#52511yescharacteristic
336C2×C4#53611yescharacteristic
337C2×C4#54711yescharacteristic
338C2×C4#54811yescharacteristic
339C2×C4#55211yescharacteristic
340C2×C4#55611yescharacteristic
341C2×C4C22×Q1611yescharacteristic
342C2×C4C2×#13511yescharacteristic
343C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
344C2×C4C2×#13712yes—
345C2×C4#57511yescharacteristic
346C2×C4#57712yes—
347C8—24no—
348C8—42no—
349C8—42no—
350C8—82no—
351C8—82no—
352C8—81no—
353C22#241511yescharacteristic
354C22#303911yescharacteristic
355C22#304311yescharacteristic
356C22C2×#46711yescharacteristic
357C22#305511yescharacteristic
358C22C2×#47211yescharacteristic
359C22#306911yescharacteristic
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—22no—
365C4—42no—
366C4—41no—
367C4—42no—
368C4—42no—
369C4—81no—
370C4—82no—
371C4—161no—
372C4—162no—
373C4#310411yescharacteristic
374C4#310611yescharacteristic
375C4#323011yescharacteristic
376C4#323311yescharacteristic
377C4#334412yes—
378C4#336912yes—
379C2#1927211yescharacteristic
380C2#1931411yescharacteristic
381C2#1939911yescharacteristic
382C2#1947011yescharacteristic
383C2#2469811yescharacteristic
384C2#2470211yescharacteristic
385C2#2472011yescharacteristic
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
#331962
#292082
#228282
#216162
#106182
#104832
#99032
#96152
#96042
#95672
#85902