Finite Group Databasebeta

← #491192

Subgroups — #491192

Order 512.

397 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33731C211yesmaximal, characteristic, complemented
2#33483C211yesmaximal, characteristic
3#19361C211yesmaximal, characteristic
4#19359C211yesmaximal, characteristic
5#12462C211yesmaximal, characteristic, complemented
6#12461C211yesmaximal, characteristic, complemented
7#12459C211yesmaximal, characteristic, complemented
8#12458C211yesmaximal, characteristic, complemented
9#12456C211yesmaximal, characteristic, complemented
10#12455C211yesmaximal, characteristic, complemented
11#12453C211yesmaximal, characteristic, complemented
12#12452C211yesmaximal, characteristic, complemented
13#12025C211yesmaximal, characteristic, complemented
14#11973C211yesmaximal, characteristic, complemented
15#11972C211yesmaximal, characteristic, complemented
16#3252—21no—
17#3238—21no—
18#3160C2211yescharacteristic
19C2×#500C2211yescharacteristic
20#2966C2211yescharacteristic
21#2255—21no—
22#2255—21no—
23#2254C2211yescharacteristic
24#2254C2211yescharacteristic
25#2252C2211yescharacteristic
26#2252C2211yescharacteristic
27#2183C2211yescharacteristic
28Q8×C16C2211yescharacteristic
29#2169—21no—
30#2169C2211yescharacteristic
31#2168—21no—
32D4×C16C2211yescharacteristic
33#2164—21no—
34#2162C2211yescharacteristic
35#2159—21no—
36#2159—21no—
37#2108—21no—
38C4×M32C2211yescharacteristic
39#1696—21no—
40#1695—21no—
41#1692—21no—
42#1691—21no—
43#1676C2211yescharacteristic
44#1676C2211yescharacteristic
45#1676C2211yescharacteristic
46#1675C2211yescharacteristic
47#1675C2211yescharacteristic
48#1675C2211yescharacteristic
49#1674C2211yescharacteristic
50#1673C2211yescharacteristic
51#1568C2211yescharacteristic
52#1567C2211yescharacteristic
53#1559—21no—
54#1558—21no—
55#1372C2211yescharacteristic
56#1372C2211yescharacteristic
57#1371C2211yescharacteristic
58#1371C2211yescharacteristic
59#1365C2211yescharacteristic
60#1365C2211yescharacteristic
61#1365C2211yescharacteristic
62#1365C2211yescharacteristic
63#1365C2211yescharacteristic
64#1365C2211yescharacteristic
65#1364C2211yescharacteristic
66#1364C2211yescharacteristic
67#533—21no—
68C2×C4⋊Q8—21no—
69#529D411yescharacteristic
70C2×M32—21no—
71C2×M32D411yescharacteristic
72C2×M32D411yescharacteristic
73C22×C16—21no—
74#501—41no—
75#500—21no—
76#500—21no—
77#500—21no—
78#500C2311yescharacteristic
79#500C2311yescharacteristic
80#500C2311yescharacteristic
81#500C2311yescharacteristic
82#499—21no—
83#499—21no—
84#498—21no—
85#489—41no—
86#483—41no—
87#480—41no—
88#479—41no—
89#477—21no—
90#477—21no—
91#477—21no—
92#476—21no—
93#476—21no—
94#476—21no—
95#470—41no—
96#468—41no—
97#464—41no—
98#462—21no—
99#461—21no—
100#446D411yescharacteristic
101C8×Q8C2311yescharacteristic
102#436D411yescharacteristic
103#435—21no—
104C8×D4—21no—
105C8×D4C2311yescharacteristic
106#433—21no—
107#432—21no—
108#432D411yescharacteristic
109#427—21no—
110#427—21no—
111C2×#107—21no—
112C2×#107—21no—
113#423—21no—
114#405—21no—
115C2×C4×C8C2311yescharacteristic
116#367—21no—
117#367—21no—
118#367—21no—
119#367—21no—
120#366—21no—
121#366—21no—
122#366—21no—
123#366—21no—
124C4⋊C16—21no—
125C4⋊C16—21no—
126C4⋊C16C2311yescharacteristic
127C4⋊C16C2311yescharacteristic
128C22⋊C16—21no—
129C22⋊C16—21no—
130C22⋊C16—21no—
131#346—21no—
132#346D411yescharacteristic
133#346D411yescharacteristic
134C4×C16—21no—
135C4×C16—21no—
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335—21no—
139#335C2311yescharacteristic
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#335C2311yescharacteristic
143#334—21no—
144#333—21no—
145#333—21no—
146#333—21no—
147#333—21no—
148#333—21no—
149#333—21no—
150#332—21no—
151#332—21no—
152C22×Q8—41no—
153C2×Q16—41no—
154C2×SD16—41no—
155C2×M16—21no—
156C2×M16—41no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8#4111yescharacteristic
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167#126—21no—
168#125—41no—
169#125D811yescharacteristic
170#124D811yescharacteristic
171#123—21no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C22⋊Q8D811yescharacteristic
175#121D811yescharacteristic
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8C2×D411yescharacteristic
179C4×D4—21no—
180C4×D4C2×D411yescharacteristic
181#117—21no—
182#117—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C42C2×D411yescharacteristic
186M32—22no—
187M32—22no—
188M32—42no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—42no—
192C2×C16C2×D411yescharacteristic
193C2×C16C2×D411yescharacteristic
194C2×C16C2×D411yescharacteristic
195C2×C16C2×D411yescharacteristic
196#107—21no—
197#107—21no—
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#107—41no—
207#107—41no—
208#107—41no—
209#106—41no—
210#106—41no—
211#106—41no—
212#106—41no—
213#106—41no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—41no—
221C4⋊C8C2×D411yescharacteristic
222C4⋊C8C2×D411yescharacteristic
223#103—41no—
224#103—41no—
225#103—41no—
226#103—41no—
227#103—41no—
228#102—41no—
229#102—41no—
230#102—41no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—41no—
235#97—21no—
236#97—21no—
237#97—41no—
238C4×C8—21no—
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8#4111yescharacteristic
242C4×C8C2411yescharacteristic
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—81no—
248C2×Q8—162no—
249C2×Q8C2×D811yescharacteristic
250C2×D4—21no—
251C2×D4C2×D811yescharacteristic
252C22×C4—21no—
253C22×C4—81no—
254C22×C4#12111yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4#13511yescharacteristic
257Q16—161no—
258SD16—161no—
259M16—42no—
260M16—161no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8#12111yescharacteristic
273C2×C8#12111yescharacteristic
274C2×C8C22×D411yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4C2×D811yescharacteristic
293C4⋊C4C2×D811yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—81no—
300C42—21no—
301C42—81no—
302C42#12111yescharacteristic
303C42C2×D811yescharacteristic
304C42#13511yescharacteristic
305C42C22×D411yescharacteristic
306C16—21no—
307C16—21no—
308C16—41no—
309C16—41no—
310C16#13712yes—
311C16#13712yes—
312C23—21no—
313C23#46611yescharacteristic
314Q8—21no—
315Q8—21no—
316Q8—81no—
317Q8—81no—
318Q8—161no—
319Q8—161no—
320Q8—161no—
321Q8—162no—
322D4—21no—
323D4—21no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—162no—
340C2×C4#46611yescharacteristic
341C2×C4#46611yescharacteristic
342C2×C4#46611yescharacteristic
343C2×C4C2×#12111yescharacteristic
344C2×C4C22×D811yescharacteristic
345C2×C4C2×#13511yescharacteristic
346C8—21no—
347C8—21no—
348C8—41no—
349C8—41no—
350C8—81no—
351C8—81no—
352C8—81no—
353C8—81no—
354C8#47312yes—
355C8C2×#13711yescharacteristic
356C8C2×#13711yescharacteristic
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22C2×#46611yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—161no—
372C4—161no—
373C4—161no—
374C4—161no—
375C4#303911yescharacteristic
376C4C2×#47311yescharacteristic
377C4#341611yescharacteristic
378C4#341711yescharacteristic
379C2—41no—
380C2—81no—
381C2#1927111yescharacteristic
382C2#3361311yescharacteristic
383C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#121D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33731C2semidirect
5. #12462C2semidirect
6. #12461C2semidirect
7. #12459C2semidirect
8. #12458C2semidirect
9. #12456C2semidirect
10. #12455C2semidirect
11. #12453C2semidirect
12. #12452C2semidirect
13. #12025C2semidirect
14. #11973C2semidirect
15. #11972C2semidirect

Maximal subgroups

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

subgroupindex
#337312
#334832
#193612
#193592
#124622
#124612
#124592
#124582
#124562
#124552
#124532
#124522
#120252
#119732
#119722