Finite Group Databasebeta

← #501637

Subgroups — #501637

Order 512.

780 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C212yesmaximal, complemented
2C2×#2207C211yesmaximal, characteristic, complemented
3C22×C4⋊C16C211yesmaximal, characteristic, complemented
4#33171C211yesmaximal, characteristic
5C4×#467C211yesmaximal, characteristic, complemented
6#13476C212yesmaximal, complemented
7#13449C212yesmaximal, complemented
8#13252C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13059C212yesmaximal, complemented
11#12812C211yesmaximal, characteristic, complemented
12C22×SD32C2212yes—
13C23×C16C2212yes—
14C2×#493C2211yescharacteristic
15C2×#467C2211yescharacteristic
16C2×#466—21no—
17C2×#466C2212yes—
18C2×C4×Q16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2856—21no—
21C4×C22⋊Q8—21no—
22#2217—28no—
23#2207—24no—
24C2×#367C2212yes—
25C2×C4⋊C16—24no—
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2212yes—
28C2×C4⋊C16C2211yescharacteristic
29C2×#358C2212yes—
30C2×#357C2212yes—
31C2×#357C2212yes—
32C2×#357C2212yes—
33#1970—21no—
34#1970C2212yes—
35#1969C2211yescharacteristic
36#1957—21no—
37#1942C2211yescharacteristic
38#1933—21no—
39#1892—21no—
40#1845C2211yescharacteristic
41#1808C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#103—21no—
44#1380C2212yes—
45#1380C2212yes—
46C22×Q16C2311yescharacteristic
47C22×D8—21no—
48C22×D8C2312yes—
49C23×C8D411yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#127—21no—
52C2×C22⋊Q8—21no—
53C2×#121—22no—
54C2×#121—41no—
55C2×#121—41no—
56C2×C4×Q8—21no—
57C22×C42D411yescharacteristic
58C2×SD32—24no—
59C2×SD32—24no—
60C2×SD32—42no—
61C22×C16—24no—
62C22×C16—24no—
63C22×C16—42no—
64C22×C16C2312yes—
65C22×C16C2312yes—
66#493—22no—
67#493—22no—
68#467—22no—
69#467D414yescomplemented
70#466—24no—
71#466—24no—
72#466—44no—
73C4×Q16—22no—
74C4×Q16—22no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2312yes—
79C2×#107C2311yescharacteristic
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#102—22no—
84C2×#102—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#390—22no—
93#390—41no—
94#390—41no—
95#389—21no—
96#387—41no—
97#385—41no—
98#384—21no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—41no—
102C4×C22⋊C4—41no—
103#367—24no—
104C4⋊C16—24no—
105C4⋊C16—22no—
106C4⋊C16D414yescomplemented
107#358—24no—
108#357—24no—
109#357—24no—
110#357—42no—
111#340—21no—
112#340—21no—
113#340—21no—
114C22×Q8—21no—
115C22×D4—22no—
116C22×D4—42no—
117C22×D4—41no—
118C22×D4—41no—
119C23×C4D811yescharacteristic
120C23×C4D811yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×Q16—21no—
123C2×Q16—21no—
124C2×Q16—21no—
125C2×Q16—21no—
126C2×Q16—21no—
127C2×Q16C2×D412yes—
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—42no—
135C2×D8—42no—
136C2×D8—42no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D412yes—
148C22×C8#4112yes—
149C22×C8C2411yescharacteristic
150#127—42no—
151#127—42no—
152C22⋊Q8—24no—
153C22⋊Q8—42no—
154#121—44no—
155#121—44no—
156#121—84no—
157#121—84no—
158C4×Q8—42no—
159C4×Q8—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42D811yescharacteristic
179C2×C42D811yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182SD32—48no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187C2×C16—22no—
188C2×C16—48no—
189C2×C16C2×D414yes—
190#107—21no—
191#107—21no—
192#107—22no—
193#107—21no—
194#107—22no—
195#107—21no—
196#107—21no—
197#107—42no—
198#107C2×D412yes—
199#103—24no—
200#103—42no—
201#103—42no—
202#102—44no—
203#102—44no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—22no—
208C4×C8C2×D412yes—
209C4×C8#4112yes—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215C24—42no—
216C24C2×D811yescharacteristic
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—82no—
230C2×D4—82no—
231C2×D4—84no—
232C2×D4—84no—
233C2×D4—84no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—82no—
237C2×D4—82no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4SD3214yescomplemented
253C22×C4#12112yes—
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4AGL(1,ℤ/8)12yes—
260C22×C4C22×D411yescharacteristic
261Q16—22no—
262Q16—22no—
263Q16—41no—
264Q16—41no—
265D8—42no—
266D8—42no—
267D8—42no—
268D8—42no—
269D8—84no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—44no—
282C2×C8#12112yes—
283C2×C8#12112yes—
284C2×C8C22×D411yescharacteristic
285C2×C8C22×D411yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8H(5,𝔽2)11yescharacteristic
289C2×C82−1+411yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C22⋊C4—44no—
304C22⋊C4—84no—
305C22⋊C4—82no—
306C22⋊C4—84no—
307C22⋊C4—84no—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—22no—
312C42—82no—
313C42—82no—
314C42#12112yes—
315C42#13512yes—
316C16—24no—
317C16—42no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23#45912yes—
330C23C2×SD3212yes—
331C23#50912yes—
332C23C22×D811yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337D4—82no—
338D4—82no—
339D4—82no—
340D4—82no—
341D4—164no—
342D4—168no—
343D4—164no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—28no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4#45912yes—
375C2×C4#46612yes—
376C2×C4#46612yes—
377C2×C4C2×SD3212yes—
378C2×C4C2×SD3212yes—
379C2×C4#50812yes—
380C2×C4C2×#12111yescharacteristic
381C2×C4C2×#12111yescharacteristic
382C2×C4#53711yescharacteristic
383C2×C4C2×#13511yescharacteristic
384C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
385C2×C4#57511yescharacteristic
386C8—22no—
387C8—22no—
388C8—41no—
389C8D4211yescharacteristic
390C8#54711yescharacteristic
391C8#54811yescharacteristic
392C8#55311yescharacteristic
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—162no—
400C22—162no—
401C22—162no—
402C22—162no—
403C22—162no—
404C22—162no—
405C22—162no—
406C22#220714yescomplemented
407C22C2×#45911yescharacteristic
408C22C2×#46611yescharacteristic
409C22#324511yescharacteristic
410C22C22×SD3211yescharacteristic
411C22C2×#50811yescharacteristic
412C22C2×#50911yescharacteristic
413C22#341411yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—82no—
419C4—82no—
420C4—161no—
421C4—162no—
422C4—161no—
423C4#221714yescomplemented
424C4#241611yescharacteristic
425C4#303511yescharacteristic
426C4#305511yescharacteristic
427C4#324711yescharacteristic
428C2—24no—
429C2—322no—
430C2#2475011yescharacteristic
431C2C2×#220711yescharacteristic
432C2#3357911yescharacteristic
433C2C2×#221711yescharacteristic
434C2#3360511yescharacteristic
435C2#3381211yescharacteristic
436C2#3381311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#33171C2
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×#2217C2semidirect
2. C2×#2207C2semidirect
3. C22×C4⋊C16C2semidirect
5. C4×#467C2semidirect
6. #13476C2semidirect
7. #13449C2semidirect
8. #13252C2semidirect
9. #13177C2semidirect
10. #13059C2semidirect
11. #12812C2semidirect
69. #467D4semidirect
106. C4⋊C16D4semidirect
252. C22×C4SD32semidirect
406. C22#2207semidirect
423. C4#2217semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
C2×#22072
C22×C4⋊C162
#331712
C4×#4672
#134762
#134492
#132522
#131772
#130592
#128122