Finite Group Databasebeta

← #484139

Subgroups — #484139

Order 512.

399 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33747C211yesmaximal, characteristic
2#18093C211yesmaximal, characteristic
3#18072C211yesmaximal, characteristic, complemented
4#12088C211yesmaximal, characteristic, complemented
5#12087C211yesmaximal, characteristic, complemented
6#12080C211yesmaximal, characteristic, complemented
7#12079C211yesmaximal, characteristic, complemented
8#12079C211yesmaximal, characteristic, complemented
9#12077C211yesmaximal, characteristic, complemented
10#12066C211yesmaximal, characteristic, complemented
11#12065C211yesmaximal, characteristic, complemented
12C2×#1368C211yesmaximal, characteristic, complemented
13#11655C211yesmaximal, characteristic, complemented
14#11651C211yesmaximal, characteristic
15#11413C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3168C2211yescharacteristic
18#2994—21no—
19#2962—21no—
20#2255C411yescharacteristic
21#2251C411yescharacteristic
22#2245C411yescharacteristic
23#2243C411yescharacteristic
24#2217C411yescharacteristic
25#2217C411yescharacteristic
26#2216C411yescharacteristic
27#2216C411yescharacteristic
28#2176C2211yescharacteristic
29#2176C2211yescharacteristic
30#2175C2211yescharacteristic
31#2175C2211yescharacteristic
32#2163C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#1594C2211yescharacteristic
35#1593C2211yescharacteristic
36#1593C2211yescharacteristic
37#1593C2211yescharacteristic
38#1587—21no—
39#1587—21no—
40#1583—21no—
41#1582—21no—
42#1580C2211yescharacteristic
43#1579C2211yescharacteristic
44#1579C2211yescharacteristic
45#1579C2211yescharacteristic
46C2×#334C2211yescharacteristic
47C2×#334C2211yescharacteristic
48#1483—21no—
49#1481—21no—
50#1451C2211yescharacteristic
51#1368C2211yescharacteristic
52#1368C2211yescharacteristic
53#1368C2211yescharacteristic
54#1368C2211yescharacteristic
55#1334C2211yescharacteristic
56#1334C2211yescharacteristic
57#1334C2211yescharacteristic
58#1334C2211yescharacteristic
59#1334C2211yescharacteristic
60#1334C2211yescharacteristic
61#1333C2211yescharacteristic
62#1333C2211yescharacteristic
63#1313C2211yescharacteristic
64#1313C2211yescharacteristic
65#1313C2211yescharacteristic
66#1313C2211yescharacteristic
67#533—21no—
68#529—21no—
69C2×Q32C2×C411yescharacteristic
70C2×Q32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C22×C16C2×C411yescharacteristic
74C22×C16C2×C411yescharacteristic
75#500C2×C411yescharacteristic
76#499C2×C411yescharacteristic
77#495C2×C411yescharacteristic
78#494C2×C411yescharacteristic
79#488—21no—
80#487—21no—
81#467—21no—
82#467C2×C411yescharacteristic
83#467C2×C411yescharacteristic
84#467C2×C411yescharacteristic
85#466C2×C411yescharacteristic
86#465—21no—
87#446—21no—
88#446—21no—
89#446—21no—
90#446—41no—
91#446—41no—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8—21no—
95C4×Q16—21no—
96C4×Q16C2311yescharacteristic
97C4×Q16C2311yescharacteristic
98C4×Q16C2311yescharacteristic
99C4×SD16—21no—
100C4×D8C2311yescharacteristic
101#436—21no—
102#435—41no—
103C8×D4—21no—
104C8×D4—41no—
105#433—41no—
106#431D411yescharacteristic
107#424—41no—
108#424—41no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111#405—41no—
112C4×M16D411yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#367C2×C411yescharacteristic
115#367C2×C411yescharacteristic
116#366C2×C411yescharacteristic
117#366C2×C411yescharacteristic
118C4⋊C16—21no—
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#358C2×C411yescharacteristic
124#358C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16—21no—
128#346C2×C411yescharacteristic
129#346C2×C411yescharacteristic
130#346C2311yescharacteristic
131#346C2311yescharacteristic
132#346C2311yescharacteristic
133#346C2311yescharacteristic
134#334—21no—
135#334—21no—
136#334C2311yescharacteristic
137#334C2311yescharacteristic
138#334C2311yescharacteristic
139#334C2311yescharacteristic
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144Q8⋊C8—21no—
145Q8⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150#322C2311yescharacteristic
151#322C2311yescharacteristic
152C2×Q16—21no—
153C2×Q16C22×C411yescharacteristic
154C2×Q16C22×C411yescharacteristic
155C2×Q16C22×C411yescharacteristic
156C2×SD16—21no—
157C2×D8C22×C411yescharacteristic
158C2×M16—21no—
159C2×M16—41no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8C22×C411yescharacteristic
164C22×C8#4111yescharacteristic
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167#126—41no—
168#125—21no—
169#125—41no—
170#124—21no—
171#123—41no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—41no—
176#121—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—41no—
181C4×D4—21no—
182C4×D4—41no—
183#117—41no—
184#117—41no—
185C2×C42C2×D411yescharacteristic
186Q32—22no—
187Q32—41no—
188SD32—22no—
189SD32—41no—
190C2×C16—22no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—41no—
194C2×C16C22×C411yescharacteristic
195C2×C16C22×C411yescharacteristic
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198#107—21no—
199#107C22×C411yescharacteristic
200#107C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#106—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217#103—21no—
218#103—21no—
219#103—21no—
220#103—21no—
221#103—21no—
222#103—21no—
223#102—21no—
224#102—21no—
225C22⋊C8—21no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228#97—21no—
229#97—41no—
230#97—41no—
231#97—41no—
232C4×C8—41no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8C22×C411yescharacteristic
236C4×C8C2×D411yescharacteristic
237C4×C8C2×D411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8#4111yescharacteristic
242C4×C8C2411yescharacteristic
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4C4×D411yescharacteristic
251C22×C4AGL(1,ℤ/8)11yescharacteristic
252C22×C4AGL(1,ℤ/8)11yescharacteristic
253Q16—21no—
254Q16—21no—
255Q16—21no—
256Q16—21no—
257Q16—21no—
258Q16—21no—
259Q16—81no—
260SD16—81no—
261D8—21no—
262D8—21no—
263M16—42no—
264M16—161no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—82no—
277C2×C8—161no—
278C2×C8—161no—
279C2×C8C4×D411yescharacteristic
280C2×C8C4×D411yescharacteristic
281C2×C8C23×C411yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C4⋊C4—21no—
284C4⋊C4—21no—
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—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42C4×D411yescharacteristic
309C42AGL(1,ℤ/8)11yescharacteristic
310C42AGL(1,ℤ/8)11yescharacteristic
311C42C22×D411yescharacteristic
312C16—21no—
313C16—21no—
314C16—21no—
315C16—41no—
316C16—41no—
317C16#13112yes—
318C23—41no—
319C23#44211yescharacteristic
320Q8—41no—
321Q8—41no—
322Q8—41no—
323Q8—41no—
324Q8—41no—
325Q8—41no—
326Q8—161no—
327D4—41no—
328D4—41no—
329D4—82no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4#44211yescharacteristic
344C2×C4#44211yescharacteristic
345C2×C4#44211yescharacteristic
346C2×C4C2×C4×D411yescharacteristic
347C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—41no—
353C8—41no—
354C8—41no—
355C8—41no—
356C8—81no—
357C8—81no—
358C8—81no—
359C8—81no—
360C8C2×#13111yescharacteristic
361C8#56811yescharacteristic
362C22—21no—
363C22—21no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#218012yes—
368C22C2×#44211yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4#293511yescharacteristic
382C4#295711yescharacteristic
383C2—22no—
384C2—161no—
385C2#1807311yescharacteristic
386C2C2×#218011yescharacteristic
387C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#466C2×C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #18072C2semidirect
4. #12088C2semidirect
5. #12087C2semidirect
6. #12080C2semidirect
7. #12079C2semidirect
8. #12079C2semidirect
9. #12077C2semidirect
10. #12066C2semidirect
11. #12065C2semidirect
12. C2×#1368C2semidirect
13. #11655C2semidirect
15. #11413C2semidirect

Maximal subgroups

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

subgroupindex
#337472
#180932
#180722
#120882
#120872
#120802
#120792
#120772
#120662
#120652
C2×#13682
#116552
#116512
#114132