Finite Group Databasebeta

← #501634

Subgroups — #501634

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2215C212yesmaximal, complemented
2C2×#2206C211yesmaximal, characteristic, complemented
3C22×C4⋊C16C211yesmaximal, characteristic, complemented
4#33171C211yesmaximal, characteristic, complemented
5C4×#466C211yesmaximal, characteristic, complemented
6#13474C212yesmaximal, complemented
7#13448C212yesmaximal, complemented
8#13252C211yesmaximal, characteristic, complemented
9#13176C211yesmaximal, characteristic, complemented
10#13059C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C23×C16C2212yes—
14C2×#493C2211yescharacteristic
15C2×#466—21no—
16C2×#466C2212yes—
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2856—21no—
21C4×#121—21no—
22#2215—28no—
23#2206—24no—
24C2×#366C2212yes—
25C2×C4⋊C16—24no—
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2212yes—
28C2×C4⋊C16C2211yescharacteristic
29C2×#357C2212yes—
30C2×#357C2212yes—
31C2×#357C2212yes—
32C2×#357C2212yes—
33#1970—21no—
34#1970C2211yescharacteristic
35#1970C2212yes—
36#1957—21no—
37#1942C2211yescharacteristic
38#1934—21no—
39#1893—21no—
40#1845C2211yescharacteristic
41#1808C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#102—21no—
44#1380C2212yes—
45#1380C2212yes—
46C22×D8—21no—
47C22×D8C2312yes—
48C22×D8C2311yescharacteristic
49C23×C8D411yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#127—21no—
52C2×#121—21no—
53C2×#121—22no—
54C2×#121—41no—
55C2×#121—41no—
56C2×C4×D4—21no—
57C2×C4×D4—41no—
58C22×C42D411yescharacteristic
59C2×D16—24no—
60C2×D16—24no—
61C2×D16—42no—
62C22×C16—24no—
63C22×C16—24no—
64C22×C16—42no—
65C22×C16C2312yes—
66C22×C16C2312yes—
67#493—22no—
68#493—22no—
69#466—22no—
70#466—24no—
71#466—24no—
72#466—44no—
73#466D414yescomplemented
74C4×D8—22no—
75C4×D8—22no—
76C2×#107—21no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2312yes—
80C2×#107C2311yescharacteristic
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#390—22no—
93#390—21no—
94#390—41no—
95#390—41no—
96#389—21no—
97#388—41no—
98#386—41no—
99#384—21no—
100#380—21no—
101C4×C4⋊C4—21no—
102C4×C22⋊C4—41no—
103#366—24no—
104C4⋊C16—24no—
105C4⋊C16—22no—
106C4⋊C16D414yescomplemented
107#357—24no—
108#357—24no—
109#357—24no—
110#357—42no—
111#340—21no—
112#340—21no—
113#340—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—41no—
117C22×D4—42no—
118C22×D4—41no—
119C22×D4—41no—
120C23×C4—41no—
121C23×C4D811yescharacteristic
122C23×C4D811yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×D8—21no—
125C2×D8—22no—
126C2×D8—22no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—21no—
132C2×D8—22no—
133C2×D8—21no—
134C2×D8—22no—
135C2×D8—42no—
136C2×D8—42no—
137C2×D8—42no—
138C2×D8C2×D412yes—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—41no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D412yes—
150C22×C8#4112yes—
151C22×C8C2411yescharacteristic
152#127—42no—
153#127—42no—
154#121—24no—
155#121—42no—
156#121—44no—
157#121—44no—
158#121—84no—
159#121—84no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—84no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—41no—
181C2×C42D811yescharacteristic
182C2×C42D811yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185D16—48no—
186C2×C16—22no—
187C2×C16—22no—
188C2×C16—22no—
189C2×C16—22no—
190C2×C16—22no—
191C2×C16—48no—
192C2×C16C2×D414yes—
193#107—21no—
194#107—21no—
195#107—22no—
196#107—21no—
197#107—22no—
198#107—21no—
199#107—21no—
200#107—42no—
201#107C2×D412yes—
202#102—24no—
203#102—42no—
204#102—44no—
205#102—42no—
206#102—44no—
207C4×C8—22no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—22no—
211C4×C8C2×D412yes—
212C4×C8#4112yes—
213#95—41no—
214#95—41no—
215#95—41no—
216C24—41no—
217C24—42no—
218C24C2×D811yescharacteristic
219C2×D4—22no—
220C2×D4—44no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—82no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—84no—
237C2×D4—84no—
238C2×D4—84no—
239C2×D4—82no—
240C2×D4—82no—
241C2×D4—82no—
242C2×D4—82no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4D1614yescomplemented
260C22×C4#12112yes—
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4AGL(1,ℤ/8)12yes—
267C22×C4C22×D411yescharacteristic
268D8—22no—
269D8—22no—
270D8—42no—
271D8—42no—
272D8—41no—
273D8—42no—
274D8—41no—
275D8—42no—
276D8—84no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—44no—
289C2×C8#12112yes—
290C2×C8#12112yes—
291C2×C8C22×D411yescharacteristic
292C2×C8C22×D411yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C2×C82−1+411yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C22⋊C4—44no—
309C22⋊C4—82no—
310C22⋊C4—84no—
311C22⋊C4—82no—
312C22⋊C4—84no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—84no—
316C42—21no—
317C42—21no—
318C42—22no—
319C42—22no—
320C42—82no—
321C42#12112yes—
322C42#13512yes—
323C16—24no—
324C16—42no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—42no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—82no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23#45912yes—
344C23C2×D1612yes—
345C23#50912yes—
346C23C22×D811yescharacteristic
347D4—42no—
348D4—42no—
349D4—88no—
350D4—82no—
351D4—81no—
352D4—82no—
353D4—82no—
354D4—81no—
355D4—82no—
356D4—164no—
357D4—168no—
358D4—164no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—164no—
387C2×C4#45912yes—
388C2×C4#46612yes—
389C2×C4#46612yes—
390C2×C4C2×D1612yes—
391C2×C4C2×D1612yes—
392C2×C4#50812yes—
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×#12111yescharacteristic
395C2×C4#53711yescharacteristic
396C2×C4C2×#13511yescharacteristic
397C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
398C2×C4#57511yescharacteristic
399C8—22no—
400C8—22no—
401C8—41no—
402C8D4211yescharacteristic
403C8#54711yescharacteristic
404C8#54811yescharacteristic
405C8#55311yescharacteristic
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—162no—
420C22—162no—
421C22—162no—
422C22—162no—
423C22—162no—
424C22—162no—
425C22—162no—
426C22#220614yescomplemented
427C22C2×#45911yescharacteristic
428C22C2×#46611yescharacteristic
429C22#324511yescharacteristic
430C22C22×D1611yescharacteristic
431C22C2×#50811yescharacteristic
432C22C2×#50911yescharacteristic
433C22#341411yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—41no—
438C4—82no—
439C4—161no—
440C4—162no—
441C4—161no—
442C4#221514yescomplemented
443C4#241611yescharacteristic
444C4#303511yescharacteristic
445C4#305511yescharacteristic
446C4#324711yescharacteristic
447C2—24no—
448C2—162no—
449C2—322no—
450C2#2475011yescharacteristic
451C2C2×#220611yescharacteristic
452C2#3357911yescharacteristic
453C2C2×#221511yescharacteristic
454C2#3360511yescharacteristic
455C2#3381111yescharacteristic
456C2#3381311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2215C2semidirect
2. C2×#2206C2semidirect
3. C22×C4⋊C16C2semidirect
4. #33171C2semidirect
5. C4×#466C2semidirect
6. #13474C2semidirect
7. #13448C2semidirect
8. #13252C2semidirect
9. #13176C2semidirect
10. #13059C2semidirect
11. #12811C2semidirect
73. #466D4semidirect
106. C4⋊C16D4semidirect
259. C22×C4D16semidirect
426. C22#2206semidirect
442. C4#2215semidirect

Maximal subgroups

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

subgroupindex
C2×#22152
C2×#22062
C22×C4⋊C162
#331712
C4×#4662
#134742
#134482
#132522
#131762
#130592
#128112