Finite Group Databasebeta

← #484038

Subgroups — #484038

Order 512.

425 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33662C211yesmaximal, characteristic
2#18087C211yesmaximal, characteristic, complemented
3#18059C211yesmaximal, characteristic, complemented
4#12084C211yesmaximal, characteristic, complemented
5#12081C211yesmaximal, characteristic, complemented
6#12075C211yesmaximal, characteristic, complemented
7#12075C211yesmaximal, characteristic, complemented
8#12072C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12043C211yesmaximal, characteristic, complemented
11C8×Q32C211yesmaximal, characteristic, complemented
12#12032C211yesmaximal, characteristic, complemented
13#11674C211yesmaximal, characteristic, complemented
14#11592C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2994—21no—
18#2991—21no—
19C4×#135C2211yescharacteristic
20#2237C411yescharacteristic
21#2234C411yescharacteristic
22#2229C411yescharacteristic
23#2228C411yescharacteristic
24#2211C411yescharacteristic
25#2210C411yescharacteristic
26#2189C411yescharacteristic
27#2186C411yescharacteristic
28C4×Q32C2211yescharacteristic
29C4×SD32C2211yescharacteristic
30C4×SD32C2211yescharacteristic
31C4×D16C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic
38#1588—21no—
39#1588—21no—
40#1588—21no—
41#1585—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×Q16C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1567C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1489—21no—
49#1483—21no—
50C8×M16C2211yescharacteristic
51#1371C2211yescharacteristic
52#1371C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1332C2211yescharacteristic
58#1332C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68#529—21no—
69C4×#41—21no—
70C2×Q32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C2×M32—21no—
75#499C2×C411yescharacteristic
76#495C2×C411yescharacteristic
77#488—21no—
78#467C2×C411yescharacteristic
79#466C2×C411yescharacteristic
80#465—21no—
81#446—21no—
82C8×Q8—21no—
83C8×Q8—21no—
84C8×Q8—21no—
85C4×Q16C2×C411yescharacteristic
86C4×Q16C2311yescharacteristic
87C4×Q16C2311yescharacteristic
88C4×SD16—21no—
89C4×SD16—21no—
90C4×SD16—21no—
91C4×D8C2×C411yescharacteristic
92C4×D8C2311yescharacteristic
93C4×D8C2311yescharacteristic
94#436—21no—
95#436—21no—
96#436—21no—
97#436—41no—
98#436—41no—
99#435—41no—
100C8×D4—21no—
101C8×D4—41no—
102#433—41no—
103#431—41no—
104#431D411yescharacteristic
105#427C2×C411yescharacteristic
106#424—41no—
107#423—21no—
108#423—41no—
109C2×C4⋊C8—21no—
110#416—21no—
111C4×M16D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#367C2×C411yescharacteristic
114#367C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116#366C2×C411yescharacteristic
117C4⋊C16C2×C411yescharacteristic
118C4⋊C16C2×C411yescharacteristic
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129#346—21no—
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335—21no—
135#335—21no—
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#335C2311yescharacteristic
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×#41—21no—
153#135—22no—
154#135—41no—
155C2×Q16C22×C411yescharacteristic
156C2×Q16C22×C411yescharacteristic
157C2×SD16—21no—
158C2×SD16—21no—
159C2×D8C22×C411yescharacteristic
160C2×D8C22×C411yescharacteristic
161C2×M16—21no—
162C2×M16—41no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8C22×C411yescharacteristic
169C22×C8#4111yescharacteristic
170#126—41no—
171#125—21no—
172#124—21no—
173#123—41no—
174C22⋊Q8—21no—
175#121—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185#117—21no—
186#117—41no—
187#117—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190Q32—22no—
191SD32—22no—
192SD32—41no—
193D16—41no—
194M32—42no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200C2×C16C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205#106—21no—
206#106—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8D811yescharacteristic
218C4⋊C8D811yescharacteristic
219#103—21no—
220#103—21no—
221#103—21no—
222#103—21no—
223#102—21no—
224#102—21no—
225#102—21no—
226#102—21no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8D811yescharacteristic
232C22⋊C8D811yescharacteristic
233#97—21no—
234#97—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8—41no—
238C4×C8C22×C411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246#41—42no—
247#41—81no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C4×D411yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4#13511yescharacteristic
261Q16—21no—
262Q16—21no—
263Q16—21no—
264Q16#13112yes—
265SD16—42no—
266SD16—81no—
267D8—21no—
268D8—21no—
269D8—21no—
270D8—21no—
271M16—42no—
272M16—161no—
273M16—161no—
274M16—161no—
275M16—161no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—161no—
286C2×C8—161no—
287C2×C8C4×D411yescharacteristic
288C2×C8C4×D411yescharacteristic
289C2×C8C2×D811yescharacteristic
290C2×C8C2×D811yescharacteristic
291C2×C8C2×D811yescharacteristic
292C2×C8C2×D811yescharacteristic
293C2×C8C23×C411yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42C4×D411yescharacteristic
318C42C2×D811yescharacteristic
319C42#13511yescharacteristic
320C42C22×D411yescharacteristic
321C16—22no—
322C16—21no—
323C16—21no—
324C16—21no—
325C16—21no—
326C16—41no—
327C23—41no—
328C23—41no—
329C23C4×D811yescharacteristic
330Q8—22no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—82no—
339D4—82no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4C4×D811yescharacteristic
358C2×C4C4×D811yescharacteristic
359C2×C4C4×D811yescharacteristic
360C2×C4C2×C4×D411yescharacteristic
361C2×C4C22×D811yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C8—22no—
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—41no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8—81no—
376C8C2×#13111yescharacteristic
377C8#56811yescharacteristic
378C22—21no—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22C2×C4×D811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—42no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4#293311yescharacteristic
399C4#295611yescharacteristic
400C4#341511yescharacteristic
401C4#341711yescharacteristic
402C2—41no—
403C2—161no—
404C2—161no—
405C2#1799411yescharacteristic
406C2#3345911yescharacteristic
407C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#2186C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18087C2semidirect
3. #18059C2semidirect
4. #12084C2semidirect
5. #12081C2semidirect
6. #12075C2semidirect
7. #12075C2semidirect
8. #12072C2semidirect
9. #12069C2semidirect
10. #12043C2semidirect
11. C8×Q32C2semidirect
12. #12032C2semidirect
13. #11674C2semidirect
14. #11592C2semidirect
15. #11409C2semidirect

Maximal subgroups

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

subgroupindex
#336622
#180872
#180592
#120842
#120812
#120752
#120722
#120692
#120432
C8×Q322
#120322
#116742
#115922
#114092