Finite Group Databasebeta

← #277322

Subgroups — #277322

Order 512.

642 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#28737C211yesmaximal, characteristic
2#24818C212yesmaximal
3Q8×#107C211yesmaximal, characteristic
4C2×#1670C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic
6#10594C212yesmaximal
7#10422C212yesmaximal
8#9825C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8983C212yesmaximal
11#8645C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#475C2212yes—
14C2×C8×Q8C2211yescharacteristic
15C2×C8×Q8C2211yescharacteristic
16C2×C4×Q16C2211yescharacteristic
17#2568C2212yes—
18Q8×C4⋊C4C2212yes—
19#2085—22no—
20#2066—22no—
21#2056—22no—
22#1990C2212yes—
23#1990C2212yes—
24#1981C2212yes—
25#1969—21no—
26#1969—21no—
27#1947—22no—
28#1947C2212yes—
29#1939C2212yes—
30#1878—22no—
31#1845—21no—
32#1845C2211yescharacteristic
33#1805—22no—
34#1803C2212yes—
35C4×#107—21no—
36C4×#107C2211yescharacteristic
37#1670—24no—
38C8×Q16—24no—
39C2×#335C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×C82C2211yescharacteristic
45#1296C2212yes—
46#1279C2212yes—
47C22×Q16—21no—
48C22×Q16—21no—
49C22×Q16D411yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8C2311yescharacteristic
53C2×C4×Q8C2311yescharacteristic
54#494—22no—
55#494—22no—
56#475—24no—
57#475—24no—
58C8×Q8—22no—
59C8×Q8—22no—
60C8×Q8—22no—
61C8×Q8D414yes—
62C4×Q16—22no—
63C4×Q16—22no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107D411yescharacteristic
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×#106—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103D411yescharacteristic
82C2×#103D411yescharacteristic
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#400—42no—
89#391—22no—
90#391—22no—
91#391—22no—
92#389—22no—
93#389—42no—
94#384—22no—
95#384C2312yes—
96#382—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2312yes—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—42no—
104#340D412yes—
105#336—22no—
106#335—22no—
107#335—22no—
108#332—24no—
109Q8⋊C8—22no—
110Q8⋊C8—22no—
111C82—22no—
112C22×Q8—22no—
113C22×Q8C2×D411yescharacteristic
114C22×Q8C2×D411yescharacteristic
115C2×Q16—22no—
116C2×Q16—22no—
117C2×Q16—42no—
118C2×Q16—42no—
119C2×Q16—42no—
120C2×Q16—42no—
121C2×Q16—81no—
122C2×Q16—81no—
123C2×Q16Q1614yescomplemented
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
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⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—21no—
162C2×C42—42no—
163C2×C42C2411yescharacteristic
164#107—22no—
165#107—22no—
166#107—22no—
167#107—42no—
168#107—82no—
169#107—81no—
170#106—44no—
171C4⋊C8—24no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178C4⋊C8—82no—
179C4⋊C8C2×D412yes—
180#103—22no—
181#103—22no—
182#103—44no—
183#103—82no—
184#103—82no—
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—
198#95—42no—
199#95—42no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8#12112yes—
215C2×Q8C2×Q1612yes—
216C2×Q8C2×Q1612yes—
217C2×Q8#13512yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D411yescharacteristic
228C22×C4H(5,𝔽2)11yescharacteristic
229Q16—28no—
230Q16—84no—
231Q16—84no—
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—41no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8#12112yes—
252C2×C8#12112yes—
253C2×C8C2×Q1612yes—
254C2×C8#13512yes—
255C2×C8#13712yes—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—84no—
276C4⋊C4—82no—
277C4⋊C4—162no—
278C4⋊C4#12112yes—
279C42—22no—
280C42—22no—
281C42—41no—
282C42—41no—
283C42—42no—
284C42—42no—
285C42—84no—
286C42C22×D411yescharacteristic
287C42C2×#4111yescharacteristic
288C42C2×#4111yescharacteristic
289C422−1+411yescharacteristic
290C23D4211yescharacteristic
291Q8—22no—
292Q8—22no—
293Q8—22no—
294Q8—22no—
295Q8—22no—
296Q8—41no—
297Q8—41no—
298Q8—82no—
299Q8—82no—
300Q8—82no—
301Q8—82no—
302Q8#46714yes—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C2×#12111yescharacteristic
330C2×C4C2×#12111yescharacteristic
331C2×C4#53711yescharacteristic
332C2×C4#54711yescharacteristic
333C2×C4#54811yescharacteristic
334C2×C4#55311yescharacteristic
335C2×C4C22×Q1611yescharacteristic
336C2×C4C2×#13511yescharacteristic
337C2×C4C2×#13511yescharacteristic
338C2×C4C2×#13711yescharacteristic
339C2×C4#57611yescharacteristic
340C2×C4#57611yescharacteristic
341C2×C4#57711yescharacteristic
342C2×C4#57811yescharacteristic
343C8—22no—
344C8—22no—
345C8—42no—
346C8—42no—
347C8—41no—
348C8—81no—
349C8—82no—
350C8#46414yes—
351C22#241611yescharacteristic
352C22#327811yescharacteristic
353C22#328011yescharacteristic
354C22D4×Q1611yescharacteristic
355C22#328711yescharacteristic
356C22#329011yescharacteristic
357C22#329311yescharacteristic
358C4—24no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—42no—
363C4—82no—
364C4—82no—
365C4—81no—
366C4—162no—
367C4—162no—
368C4C2×#46411yescharacteristic
369C4#304511yescharacteristic
370C4C2×#46711yescharacteristic
371C4#306111yescharacteristic
372C4#333611yescharacteristic
373C4#333811yescharacteristic
374C4#336511yescharacteristic
375C4#337511yescharacteristic
376C2#1990711yescharacteristic
377C2#1993211yescharacteristic
378C2#1999511yescharacteristic
379C2#2001011yescharacteristic
380C2#2473311yescharacteristic
381C2#2478111yescharacteristic
382C2#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
123. C2×Q16Q16semidirect

Maximal subgroups

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

subgroupindex
#287372
#248182
Q8×#1072
C2×#16702
C2×C8×Q162
#105942
#104222
#98252
#95942
#89832
#86452