Finite Group Databasebeta

← #501635

Subgroups — #501635

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C212yesmaximal
2C2×#2208C211yesmaximal, characteristic, complemented
3C22×C4⋊C16C211yesmaximal, characteristic, complemented
4#33172C211yesmaximal, characteristic, complemented
5C4×#466C211yesmaximal, characteristic
6#13477C212yesmaximal, complemented
7#13449C212yesmaximal, complemented
8#13253C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13060C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×SD32C2212yes—
13C23×C16C2212yes—
14C2×#494C2211yescharacteristic
15C2×#467—21no—
16C2×#467C2212yes—
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2857—21no—
21C4×#121—21no—
22#2217—28no—
23#2208—24no—
24C2×#367C2212yes—
25C2×C4⋊C16—24no—
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2212yes—
28C2×C4⋊C16C2211yescharacteristic
29C2×#358C2212yes—
30C2×#358C2212yes—
31C2×#358C2212yes—
32C2×#357C2212yes—
33#1970C2211yescharacteristic
34#1969—21no—
35#1969C2212yes—
36#1956—21no—
37#1942C2211yescharacteristic
38#1934—21no—
39#1893—21no—
40#1845C2211yescharacteristic
41#1808C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#102—21no—
44#1380C2212yes—
45#1380C2212yes—
46C22×Q16—21no—
47C22×Q16C2312yes—
48C22×D8C2311yescharacteristic
49C23×C8D411yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—21no—
52C2×C22⋊Q8—22no—
53C2×#121—21no—
54C2×C4×D4—21no—
55C2×C4×D4—41no—
56C22×C42D411yescharacteristic
57C2×SD32—24no—
58C2×SD32—24no—
59C2×SD32—42no—
60C22×C16—24no—
61C22×C16—24no—
62C22×C16—42no—
63C22×C16C2312yes—
64C22×C16C2312yes—
65#494—22no—
66#494—22no—
67#467—24no—
68#467—24no—
69#467—44no—
70#466—22no—
71#466D414yes—
72C4×D8—22no—
73C4×D8—22no—
74C2×#107—21no—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2312yes—
78C2×#107C2311yescharacteristic
79C2×#103—22no—
80C2×#103—22no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—22no—
91#390—21no—
92#389—21no—
93#389—41no—
94#389—41no—
95#388—41no—
96#386—41no—
97#384—21no—
98#380—21no—
99#380—41no—
100#380—41no—
101C4×C4⋊C4—21no—
102C4×C22⋊C4—41no—
103#367—24no—
104C4⋊C16—24no—
105C4⋊C16—22no—
106C4⋊C16D414yescomplemented
107#358—24no—
108#358—24no—
109#358—42no—
110#357—24no—
111#340—21no—
112#340—21no—
113#340—21no—
114C22×Q8—22no—
115C22×D4—21no—
116C22×D4—41no—
117C23×C4—41no—
118C23×C4D811yescharacteristic
119C23×C4D811yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×Q16—22no—
122C2×Q16—22no—
123C2×Q16—22no—
124C2×Q16—22no—
125C2×Q16—22no—
126C2×Q16—22no—
127C2×Q16—42no—
128C2×Q16—42no—
129C2×Q16—42no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8C2×D412yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—41no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D412yes—
147C22×C8#4112yes—
148C22×C8C2411yescharacteristic
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153#121—24no—
154#121—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—84no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—41no—
177C2×C42D811yescharacteristic
178C2×C42D811yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181SD32—48no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187C2×C16—48no—
188C2×C16C2×D414yes—
189#107—21no—
190#107—22no—
191#107—21no—
192#107—21no—
193#107—22no—
194#107—21no—
195#107—21no—
196#107—42no—
197#107C2×D412yes—
198#103—44no—
199#103—44no—
200#102—24no—
201#102—42no—
202#102—42no—
203C4×C8—22no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—22no—
207C4×C8C2×D412yes—
208C4×C8#4112yes—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—42no—
213#95—42no—
214C24—41no—
215C24C2×D811yescharacteristic
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×D4—22no—
223C2×D4—44no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—82no—
230C2×D4—82no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4SD3214yescomplemented
249C22×C4#12112yes—
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4C22×D411yescharacteristic
257Q16—42no—
258Q16—42no—
259Q16—42no—
260Q16—42no—
261Q16—84no—
262D8—22no—
263D8—22no—
264D8—41no—
265D8—41no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—44no—
278C2×C8#12112yes—
279C2×C8#12112yes—
280C2×C8C22×D411yescharacteristic
281C2×C8C22×D411yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C82−1+411yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—84no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—84no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—21no—
307C42—21no—
308C42—22no—
309C42—22no—
310C42—82no—
311C42#12112yes—
312C42#13512yes—
313C16—24no—
314C16—42no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—42no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23#45912yes—
327C23C2×SD3212yes—
328C23#51012yes—
329C23C22×D811yescharacteristic
330Q8—82no—
331Q8—82no—
332Q8—82no—
333Q8—82no—
334D4—42no—
335D4—42no—
336D4—88no—
337D4—81no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—28no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—164no—
370C2×C4#45912yes—
371C2×C4#46612yes—
372C2×C4#46612yes—
373C2×C4C2×SD3212yes—
374C2×C4C2×SD3212yes—
375C2×C4#50812yes—
376C2×C4C2×#12111yescharacteristic
377C2×C4C2×#12111yescharacteristic
378C2×C4#53711yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4#57511yescharacteristic
382C8—22no—
383C8—22no—
384C8—41no—
385C8D4211yescharacteristic
386C8#54711yescharacteristic
387C8#54811yescharacteristic
388C8#55311yescharacteristic
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22#220814yescomplemented
403C22C2×#45911yescharacteristic
404C22C2×#46611yescharacteristic
405C22#324511yescharacteristic
406C22C22×SD3211yescharacteristic
407C22C2×#50811yescharacteristic
408C22C2×#51011yescharacteristic
409C22#341411yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—41no—
414C4—82no—
415C4—161no—
416C4—162no—
417C4—161no—
418C4—162no—
419C4#221714yescomplemented
420C4#241611yescharacteristic
421C4#303511yescharacteristic
422C4#305511yescharacteristic
423C4#324711yescharacteristic
424C2—24no—
425C2—162no—
426C2#2475011yescharacteristic
427C2C2×#220811yescharacteristic
428C2#3357911yescharacteristic
429C2C2×#221711yescharacteristic
430C2#3360611yescharacteristic
431C2#3381211yescharacteristic
432C2#3381411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
2. C2×#2208C2semidirect
3. C22×C4⋊C16C2semidirect
4. #33172C2semidirect
6. #13477C2semidirect
7. #13449C2semidirect
8. #13253C2semidirect
9. #13177C2semidirect
10. #13060C2semidirect
11. #12811C2semidirect
106. C4⋊C16D4semidirect
248. C22×C4SD32semidirect
402. C22#2208semidirect
419. 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×#22082
C22×C4⋊C162
#331722
C4×#4662
#134772
#134492
#132532
#131772
#130602
#128112