Finite Group Databasebeta

← #277315

Subgroups — #277315

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28675C211yesmaximal, characteristic
2#24732C212yesmaximal
3Q8×#107C211yesmaximal, characteristic
4C2×#1674C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic
6#10592C212yesmaximal
7#10419C212yesmaximal
8#9823C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8987C212yesmaximal
11#8645C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#462—22no—
14C2×#462C2212yes—
15C2×C8×Q8C2211yescharacteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18#2546C2212yes—
19Q8×C4⋊C4C2212yes—
20#2086—22no—
21#2068—22no—
22#2043—22no—
23#1990C2212yes—
24#1988C2212yes—
25#1956C2212yes—
26#1947—22no—
27#1947C2212yes—
28#1942—21no—
29#1942—21no—
30#1939C2212yes—
31#1878—22no—
32#1845—21no—
33#1845C2211yescharacteristic
34#1803C2212yes—
35C4×#107—21no—
36C4×#107C2211yescharacteristic
37#1674—24no—
38C8×Q16—24no—
39C2×#335C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×C82C2211yescharacteristic
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—21no—
68C2×#107—21no—
69C2×#107—22no—
70C2×#107D411yescharacteristic
71C2×#107D411yescharacteristic
72C2×#107D411yescharacteristic
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103D411yescharacteristic
84C2×#103D411yescharacteristic
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#398—42no—
91#391—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—42no—
96#384—22no—
97#384C2312yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2312yes—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—42no—
105#340D412yes—
106#336—22no—
107#335—22no—
108#335—22no—
109#332—24no—
110Q8⋊C8—22no—
111Q8⋊C8—22no—
112C82—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—21no—
124C22×C8—22no—
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—22no—
168#107—42no—
169#107—42no—
170#107—42no—
171#107—82no—
172#107—81no—
173#107Q1614yescomplemented
174C4⋊C8—24no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—22no—
180C4⋊C8—42no—
181C4⋊C8—44no—
182C4⋊C8C2×D412yes—
183#103—22no—
184#103—22no—
185#103—44no—
186#103—44no—
187#103—44no—
188#103—82no—
189C4×C8—24no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—42no—
195C4×C8C2×D412yes—
196C4×C8#4112yes—
197#95—22no—
198#95—42no—
199#95—42no—
200#95—42no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8#12112yes—
216C2×Q8C2×D812yes—
217C2×Q8C2×D812yes—
218C2×Q8#13512yes—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4H(5,𝔽2)11yescharacteristic
230Q16—28no—
231Q16—88no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—22no—
237C2×C8—21no—
238C2×C8—22no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8#12112yes—
252C2×C8#12112yes—
253C2×C8C2×D812yes—
254C2×C8C2×Q1612yes—
255C2×C8#13712yes—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—82no—
278C4⋊C4—162no—
279C4⋊C4#12112yes—
280C4⋊C4C2×Q1614yes—
281C42—22no—
282C42—22no—
283C42—41no—
284C42—41no—
285C42—42no—
286C42—42no—
287C42—84no—
288C42C22×D411yescharacteristic
289C42C2×#4111yescharacteristic
290C42C2×#4111yescharacteristic
291C422−1+411yescharacteristic
292C23D4211yescharacteristic
293Q8—22no—
294Q8—22no—
295Q8—22no—
296Q8—22no—
297Q8—22no—
298Q8—41no—
299Q8—41no—
300Q8—82no—
301Q8—82no—
302Q8—82no—
303Q8—82no—
304Q8#46614yes—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4C2×#12111yescharacteristic
332C2×C4C2×#12111yescharacteristic
333C2×C4#53711yescharacteristic
334C2×C4#54711yescharacteristic
335C2×C4#54811yescharacteristic
336C2×C4#55311yescharacteristic
337C2×C4C22×D811yescharacteristic
338C2×C4C22×Q1611yescharacteristic
339C2×C4C2×#13511yescharacteristic
340C2×C4C2×#13711yescharacteristic
341C2×C4#57611yescharacteristic
342C2×C4#57711yescharacteristic
343C2×C4#57811yescharacteristic
344C2×C4#57811yescharacteristic
345C8—22no—
346C8—22no—
347C8—42no—
348C8—42no—
349C8—41no—
350C8—81no—
351C8—82no—
352C8#46214yescomplemented
353C22#241611yescharacteristic
354C22#327811yescharacteristic
355C22D4×D811yescharacteristic
356C22#328511yescharacteristic
357C22D4×Q1611yescharacteristic
358C22#329011yescharacteristic
359C22#329311yescharacteristic
360C4—24no—
361C4—21no—
362C4—21no—
363C4—44no—
364C4—42no—
365C4—42no—
366C4—82no—
367C4—81no—
368C4—162no—
369C4—162no—
370C4C2×#46211yescharacteristic
371C4#304611yescharacteristic
372C4C2×#46611yescharacteristic
373C4#306111yescharacteristic
374C4#333411yescharacteristic
375C4#333911yescharacteristic
376C4#336311yescharacteristic
377C4#337411yescharacteristic
378C2#1988911yescharacteristic
379C2#1994411yescharacteristic
380C2D8×Q1611yescharacteristic
381C2#2000811yescharacteristic
382C2#2473211yescharacteristic
383C2#2478011yescharacteristic
384C2#2479811yescharacteristic
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

Complements

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

NH ≅ G/Ntype
173. #107Q16semidirect
352. C8#462semidirect

Maximal subgroups

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

subgroupindex
#286752
#247322
Q8×#1072
C2×#16742
C2×C8×Q162
#105922
#104192
#98232
#95942
#89872
#86452