Finite Group Databasebeta

← #287231

Subgroups — #287231

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29208C211yesmaximal, characteristic
2#28717C211yesmaximal, characteristic
3#28255C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5#10617C212yesmaximal
6#10478C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9601C211yesmaximal, characteristic
9#9218C212yesmaximal
10#8645C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#487—21no—
13C2×#462—21no—
14C2×#462C2212yes—
15C2×C8×Q8C2212yes—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2072—22no—
20#2057—22no—
21#2056—22no—
22#1990—21no—
23#1990C2212yes—
24#1984C2212yes—
25#1947—21no—
26#1947—21no—
27#1939—21no—
28#1916—22no—
29#1845C2212yes—
30#1842C2211yescharacteristic
31#1803—21no—
32#1803C2212yes—
33#1803C2212yes—
34#1803C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42#1296C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1277C2212yes—
47C22×Q16—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8D411yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8C2312yes—
55#487—44no—
56#462—24no—
57#462—42no—
58#462—44no—
59C8×Q8—24no—
60C8×Q8—42no—
61C2×#107—22no—
62C2×#107—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×C4×C8—22no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#391—21no—
82#391D412yes—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384D411yescharacteristic
91#384C2311yescharacteristic
92#384C2312yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C43C2311yescharacteristic
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—22no—
104#336—22no—
105#336—21no—
106#336—21no—
107#336D412yes—
108#335—42no—
109#332—22no—
110#332—22no—
111C22×Q8—21no—
112C22×Q8—41no—
113C22×Q8C2×D412yes—
114C2×Q16—44no—
115C2×Q16—44no—
116C2×Q16—82no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8C2×D412yes—
123C22×C8C2×D412yes—
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—42no—
129C4⋊Q8—42no—
130C4⋊Q8—81no—
131C4⋊Q8—81no—
132C4⋊Q8—82no—
133C4⋊Q8D814yes—
134C4⋊Q8#4112yes—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—42no—
140C4×Q8—42no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C2×D412yes—
159C2×C42—22no—
160C2×C42—41no—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42C2411yescharacteristic
168#107—82no—
169#107—82no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—22no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—44no—
181C4⋊C8—42no—
182C4⋊C8—81no—
183C4⋊C8Q1614yescomplemented
184C4⋊C8#4112yes—
185#103—44no—
186#103—44no—
187#103—44no—
188#103—44no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—44no—
192C4×C8—42no—
193C4×C8—42no—
194#95—22no—
195#95—22no—
196#95—41no—
197#95—41no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8C2×D814yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225Q16—88no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8C2×Q1614yes—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—22no—
278C42—41no—
279C42—42no—
280C42—42no—
281C42—82no—
282C42C2×D812yes—
283C42C2×Q1612yes—
284C42#13512yes—
285C42#13712yes—
286C42C2×#4111yescharacteristic
287C42C2×#4111yescharacteristic
288C42H(5,𝔽2)11yescharacteristic
289C422−1+411yescharacteristic
290C23C2×#12111yescharacteristic
291Q8—24no—
292Q8—24no—
293Q8—42no—
294Q8—42no—
295Q8—42no—
296Q8—42no—
297Q8—81no—
298Q8—81no—
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—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4#46212yes—
325C2×C4#46212yes—
326C2×C4#46612yes—
327C2×C4#46612yes—
328C2×C4#47312yes—
329C2×C4#52911yescharacteristic
330C2×C4#54211yescharacteristic
331C2×C4#54611yescharacteristic
332C2×C4#54611yescharacteristic
333C2×C4#54711yescharacteristic
334C2×C4#54811yescharacteristic
335C2×C4C22×D811yescharacteristic
336C2×C4C22×Q1611yescharacteristic
337C2×C4C2×#13511yescharacteristic
338C2×C4C2×#13711yescharacteristic
339C2×C4#57711yescharacteristic
340C2×C4#57711yescharacteristic
341C2×C4#57711yescharacteristic
342C2×C4#57811yescharacteristic
343C8—24no—
344C8—42no—
345C8—42no—
346C8—81no—
347C8—81no—
348C8—81no—
349C8—82no—
350C22#242711yescharacteristic
351C22C2×#46211yescharacteristic
352C22#304311yescharacteristic
353C22C2×#46611yescharacteristic
354C22#306311yescharacteristic
355C22C2×#47311yescharacteristic
356C22#306811yescharacteristic
357C4—21no—
358C4—21no—
359C4—22no—
360C4—22no—
361C4—41no—
362C4—44no—
363C4—82no—
364C4—82no—
365C4—161no—
366C4—162no—
367C4#167414yes—
368C4#329411yescharacteristic
369C4#329911yescharacteristic
370C4#330211yescharacteristic
371C4#333111yescharacteristic
372C4#333411yescharacteristic
373C4#334011yescharacteristic
374C4#336311yescharacteristic
375C4#336911yescharacteristic
376C2C2×#167411yescharacteristic
377C2#1924111yescharacteristic
378C2#1939811yescharacteristic
379C2#1956711yescharacteristic
380C2#2532411yescharacteristic
381C2#2533811yescharacteristic
382C2#2536011yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
183. C4⋊C8Q16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#292082
#287172
#282552
#231492
#106172
#104782
#98962
#96012
#92182
#86452
C4×#3322