Finite Group Databasebeta

← #491394

Subgroups — #491394

Order 512.

419 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33794C211yesmaximal, characteristic, complemented
2#33502C211yesmaximal, characteristic, complemented
3#19273C211yesmaximal, characteristic, complemented
4#19128C211yesmaximal, characteristic
5#12476C211yesmaximal, characteristic, complemented
6#12475C211yesmaximal, characteristic, complemented
7#12468C211yesmaximal, characteristic, complemented
8#12467C211yesmaximal, characteristic, complemented
9#12459C211yesmaximal, characteristic, complemented
10#12458C211yesmaximal, characteristic, complemented
11#12453C211yesmaximal, characteristic, complemented
12#12452C211yesmaximal, characteristic, complemented
13#12034C211yesmaximal, characteristic, complemented
14#11982C211yesmaximal, characteristic, complemented
15#11939C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3160C2211yescharacteristic
18#3116—21no—
19#3106—21no—
20C8×#41C2211yescharacteristic
21#2254C2211yescharacteristic
22#2252C2211yescharacteristic
23#2240C2211yescharacteristic
24#2240C2211yescharacteristic
25#2234—21no—
26#2234—21no—
27#2228—21no—
28#2228—21no—
29Q8×C16C2211yescharacteristic
30Q8×C16C2211yescharacteristic
31D4×C16—21no—
32D4×C16—21no—
33D4×C16C2211yescharacteristic
34D4×C16C2211yescharacteristic
35#2162—21no—
36#2162C2211yescharacteristic
37C2×C4⋊C16—21no—
38C2×C4⋊C16C2211yescharacteristic
39#1678—21no—
40#1677—21no—
41#1676C2211yescharacteristic
42#1676C2211yescharacteristic
43#1675C2211yescharacteristic
44#1675C2211yescharacteristic
45#1674C2211yescharacteristic
46#1673C2211yescharacteristic
47#1672—21no—
48#1671—21no—
49#1668C2211yescharacteristic
50#1667C2211yescharacteristic
51#1568C2211yescharacteristic
52#1564C2211yescharacteristic
53#1554—21no—
54#1543—21no—
55#1372C2211yescharacteristic
56#1372C2211yescharacteristic
57#1371C2211yescharacteristic
58#1371C2211yescharacteristic
59#1365C2211yescharacteristic
60#1365C2211yescharacteristic
61#1365C2211yescharacteristic
62#1365C2211yescharacteristic
63#1364C2211yescharacteristic
64#1364C2211yescharacteristic
65#1361C2211yescharacteristic
66#1361C2211yescharacteristic
67#533—21no—
68#532—21no—
69C4×#41D411yescharacteristic
70C22×C16—21no—
71C22×C16—21no—
72C22×C16—21no—
73#500C2311yescharacteristic
74#500C2311yescharacteristic
75#500C2311yescharacteristic
76#499—21no—
77#499—21no—
78#498—21no—
79#493C2311yescharacteristic
80#486—21no—
81#485—41no—
82#482—41no—
83#477—21no—
84#477—21no—
85#476—21no—
86#476—21no—
87#466—21no—
88#466—21no—
89#462—21no—
90#461—21no—
91#460—21no—
92#459—21no—
93#452—41no—
94#449—41no—
95C8×Q8D411yescharacteristic
96C8×Q8C2311yescharacteristic
97#441—41no—
98#440—41no—
99C4×SD16—41no—
100C4×D8—21no—
101C4×D8—21no—
102C4×D8—41no—
103C8×D4—21no—
104C8×D4—21no—
105C8×D4D411yescharacteristic
106C8×D4C2311yescharacteristic
107#432—21no—
108#431—21no—
109#431—21no—
110#431D411yescharacteristic
111#427—21no—
112#427—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115#367—21no—
116#367—21no—
117#366—21no—
118#366—21no—
119C4⋊C16—21no—
120C4⋊C16—21no—
121C4⋊C16—21no—
122C4⋊C16D411yescharacteristic
123C4⋊C16D411yescharacteristic
124C4⋊C16C2311yescharacteristic
125C4⋊C16C2311yescharacteristic
126#357—21no—
127#357—21no—
128#357—21no—
129#357—21no—
130C22⋊C16—21no—
131C22⋊C16—21no—
132C22⋊C16D411yescharacteristic
133C22⋊C16D411yescharacteristic
134C4×C16—21no—
135C4×C16—21no—
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335C2311yescharacteristic
139#335C2311yescharacteristic
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#334—21no—
143#334—21no—
144#333—21no—
145#333—21no—
146#333—21no—
147#333—21no—
148#332—21no—
149#332—21no—
150#331—21no—
151#331—21no—
152C2×#41—41no—
153C2×#41D811yescharacteristic
154C2×Q16—41no—
155C2×SD16—41no—
156C2×SD16—41no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—41no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C2×D411yescharacteristic
163C22×C8#4111yescharacteristic
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167#127—21no—
168#125—41no—
169#124—41no—
170C22⋊Q8—41no—
171C22⋊Q8—41no—
172#121—41no—
173#121—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8D811yescharacteristic
177C4×Q8C2×D411yescharacteristic
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4D811yescharacteristic
183C4×D4C2×D411yescharacteristic
184#117—21no—
185#117—41no—
186#117D811yescharacteristic
187C2×C42—21no—
188C2×C42C2×D411yescharacteristic
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—81no—
192C2×C16—81no—
193C2×C16—81no—
194C2×C16C2×D411yescharacteristic
195C2×C16C2×D411yescharacteristic
196C2×C16C2×D411yescharacteristic
197C2×C16C2×D411yescharacteristic
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—41no—
205#107—41no—
206#106—41no—
207#106—41no—
208#106—41no—
209#106—41no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8C2×D411yescharacteristic
217C4⋊C8C2×D411yescharacteristic
218#103—41no—
219#103—41no—
220#103—41no—
221#103—41no—
222#102—41no—
223#102—41no—
224#102—41no—
225#102—41no—
226#102—41no—
227#102—41no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233#97—41no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—41no—
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8#4111yescharacteristic
241C4×C8C2411yescharacteristic
242#41—22no—
243#41—22no—
244#41—162no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8C2×D811yescharacteristic
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—81no—
254C2×D4C2×D811yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—81no—
259C22×C4#12111yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4#13511yescharacteristic
262Q16—82no—
263SD16—82no—
264SD16—82no—
265D8—41no—
266D8—41no—
267D8—81no—
268D8—82no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—81no—
281C2×C8#12111yescharacteristic
282C2×C8#12111yescharacteristic
283C2×C8C22×D411yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4C2×D811yescharacteristic
296C4⋊C4C2×D811yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—81no—
307C42#12111yescharacteristic
308C42C2×D811yescharacteristic
309C42#13511yescharacteristic
310C42C22×D411yescharacteristic
311C16—21no—
312C16—21no—
313C16—21no—
314C16—21no—
315C16—41no—
316C16—41no—
317C23—21no—
318C23—81no—
319C23#46611yescharacteristic
320Q8—21no—
321Q8—81no—
322Q8—81no—
323Q8—161no—
324Q8—161no—
325Q8#50812yes—
326D4—24no—
327D4—21no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4—161no—
333D4—162no—
334D4#50812yes—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—162no—
351C2×C4#46611yescharacteristic
352C2×C4#46611yescharacteristic
353C2×C4#46611yescharacteristic
354C2×C4C2×#12111yescharacteristic
355C2×C4C22×D811yescharacteristic
356C2×C4C2×#13511yescharacteristic
357C8—21no—
358C8—21no—
359C8—21no—
360C8—41no—
361C8—41no—
362C8—81no—
363C8—81no—
364C8—81no—
365C8—81no—
366C8#57711yescharacteristic
367C8#57811yescharacteristic
368C22—21no—
369C22—22no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22C2×#46611yescharacteristic
378C4—21no—
379C4—22no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—161no—
386C4—161no—
387C4—161no—
388C4#304611yescharacteristic
389C4#306911yescharacteristic
390C4C2×#50811yescharacteristic
391C4C2×#50811yescharacteristic
392C2—41no—
393C2—42no—
394C2—321no—
395C2#1941611yescharacteristic
396C2#3360011yescharacteristic
397C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#19128C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33794C2semidirect
2. #33502C2semidirect
3. #19273C2semidirect
5. #12476C2semidirect
6. #12475C2semidirect
7. #12468C2semidirect
8. #12467C2semidirect
9. #12459C2semidirect
10. #12458C2semidirect
11. #12453C2semidirect
12. #12452C2semidirect
13. #12034C2semidirect
14. #11982C2semidirect
15. #11939C2semidirect

Maximal subgroups

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

subgroupindex
#337942
#335022
#192732
#191282
#124762
#124752
#124682
#124672
#124592
#124582
#124532
#124522
#120342
#119822
#119392