Finite Group Databasebeta

← #491150

Subgroups — #491150

Order 512.

413 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33751C211yesmaximal, characteristic
2Q8×M32C211yesmaximal, characteristic, complemented
3#19264C211yesmaximal, characteristic
4#19262C211yesmaximal, characteristic, complemented
5#12481C211yesmaximal, characteristic, complemented
6#12480C211yesmaximal, characteristic, complemented
7#12477C211yesmaximal, characteristic, complemented
8#12476C211yesmaximal, characteristic, complemented
9#12461C211yesmaximal, characteristic, complemented
10#12458C211yesmaximal, characteristic, complemented
11#12444C211yesmaximal, characteristic, complemented
12#12441C211yesmaximal, characteristic, complemented
13#12025C211yesmaximal, characteristic, complemented
14#11987C211yesmaximal, characteristic, complemented
15#11986C211yesmaximal, characteristic
16#3170C2211yescharacteristic
17#3169C2211yescharacteristic
18#3112—21no—
19#3078—21no—
20C2×C8×Q8C2211yescharacteristic
21#2252C2211yescharacteristic
22#2248C2211yescharacteristic
23#2241C2211yescharacteristic
24#2240C2211yescharacteristic
25#2220—21no—
26#2220—21no—
27#2183—21no—
28#2183—21no—
29#2183C2211yescharacteristic
30#2183C2211yescharacteristic
31Q8×C16C2211yescharacteristic
32Q8×C16C2211yescharacteristic
33#2175—21no—
34#2175—21no—
35#2150—21no—
36#2150C2211yescharacteristic
37C4×M32—21no—
38C4×M32C2211yescharacteristic
39#1684—21no—
40#1684—21no—
41#1678—21no—
42#1678—21no—
43#1676C2211yescharacteristic
44#1676C2211yescharacteristic
45#1676C2211yescharacteristic
46#1676C2211yescharacteristic
47#1670C2211yescharacteristic
48#1670C2211yescharacteristic
49#1668C2211yescharacteristic
50#1668C2211yescharacteristic
51#1568C2211yescharacteristic
52#1567C2211yescharacteristic
53#1551—21no—
54#1550—21no—
55#1372C2211yescharacteristic
56#1372C2211yescharacteristic
57#1371C2211yescharacteristic
58#1371C2211yescharacteristic
59#1365C2211yescharacteristic
60#1365C2211yescharacteristic
61#1365C2211yescharacteristic
62#1365C2211yescharacteristic
63#1362C2211yescharacteristic
64#1362C2211yescharacteristic
65#1361C2211yescharacteristic
66#1361C2211yescharacteristic
67#533—21no—
68#532—21no—
69C2×C4×Q8D411yescharacteristic
70C2×SD32—21no—
71C2×SD32—21no—
72C2×M32—21no—
73C2×M32D411yescharacteristic
74C2×M32D411yescharacteristic
75#500C2311yescharacteristic
76#500C2311yescharacteristic
77#494C2311yescharacteristic
78#493C2311yescharacteristic
79#487—21no—
80#486—21no—
81#480—41no—
82#479—41no—
83#479—41no—
84#477—21no—
85#477—21no—
86#477—21no—
87#477—21no—
88#475—21no—
89#475—21no—
90#467—21no—
91#467—21no—
92#466—21no—
93#466—21no—
94#460—21no—
95#460—21no—
96#448—41no—
97C8×Q8—21no—
98C8×Q8—21no—
99C8×Q8D411yescharacteristic
100C8×Q8D411yescharacteristic
101C8×Q8C2311yescharacteristic
102C8×Q8C2311yescharacteristic
103#440—41no—
104#440—41no—
105C4×Q16—21no—
106C4×Q16—21no—
107C4×D8—21no—
108C4×D8—21no—
109#432—21no—
110#428—41no—
111#423—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8D411yescharacteristic
114#419—41no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117#366—21no—
118#366—21no—
119C4⋊C16—21no—
120C4⋊C16—21no—
121C4⋊C16—21no—
122C4⋊C16—21no—
123C4⋊C16—21no—
124C4⋊C16C2311yescharacteristic
125C4⋊C16C2311yescharacteristic
126#358—21no—
127#358—21no—
128#357—21no—
129#357—21no—
130#346—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#332—21no—
149#332—21no—
150#331—21no—
151#331—21no—
152C2×#41—41no—
153C22×Q8D811yescharacteristic
154C2×Q16—21no—
155C2×Q16—21no—
156C2×SD16—41no—
157C2×SD16—41no—
158C2×D8—21no—
159C2×D8—21no—
160C2×M16—41no—
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—21no—
175C4×Q8—21no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8D811yescharacteristic
179C4×Q8D811yescharacteristic
180C4×Q8C2×D411yescharacteristic
181C4×Q8C2×D411yescharacteristic
182C4×D4—41no—
183C4×D4—41no—
184#117—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4D811yescharacteristic
187C2×C42—21no—
188C2×C42C2×D411yescharacteristic
189SD32—42no—
190SD32—42no—
191M32—22no—
192M32—41no—
193M32—42no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16C2×D411yescharacteristic
197C2×C16C2×D411yescharacteristic
198C2×C16C2×D411yescharacteristic
199C2×C16C2×D411yescharacteristic
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#106—41no—
209#106—41no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—41no—
220C4⋊C8C2×D411yescharacteristic
221C4⋊C8C2×D411yescharacteristic
222#103—41no—
223#103—41no—
224#103—41no—
225#103—41no—
226#103—41no—
227#103—41no—
228#103—41no—
229#103—41no—
230#102—41no—
231#102—41no—
232#102—41no—
233#102—41no—
234C22⋊C8—41no—
235#97—41no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8#4111yescharacteristic
243C4×C8C2411yescharacteristic
244#41—162no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8C2×D811yescharacteristic
252C2×Q8C2×D811yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—81no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—81no—
260C22×C4#12111yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4#13511yescharacteristic
263Q16—41no—
264Q16—41no—
265Q16—81no—
266SD16—82no—
267SD16—82no—
268D8—41no—
269D8—41no—
270D8—81no—
271M16—82no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8#12111yescharacteristic
283C2×C8#12111yescharacteristic
284C2×C8C22×D411yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4C2×D811yescharacteristic
300C4⋊C4C2×D811yescharacteristic
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—81no—
309C42#12111yescharacteristic
310C42C2×D811yescharacteristic
311C42#13511yescharacteristic
312C42C22×D411yescharacteristic
313C16—21no—
314C16—21no—
315C16—21no—
316C16—41no—
317C16—41no—
318C16#13712yes—
319C23—81no—
320C23#46611yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—81no—
326Q8—81no—
327Q8—161no—
328Q8—161no—
329Q8#51012yes—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334D4—161no—
335D4—162no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
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—
366C8C2×#13711yescharacteristic
367C8#57511yescharacteristic
368C22—21no—
369C22—21no—
370C22—21no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22C2×#46611yescharacteristic
375C4—21no—
376C4—21no—
377C4—22no—
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—161no—
384C4—161no—
385C4—161no—
386C4#303711yescharacteristic
387C4#306711yescharacteristic
388C4C2×#51011yescharacteristic
389C4#341411yescharacteristic
390C2—41no—
391C2—321no—
392C2#1926111yescharacteristic
393C2#3360611yescharacteristic
394C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×M32C2semidirect
4. #19262C2semidirect
5. #12481C2semidirect
6. #12480C2semidirect
7. #12477C2semidirect
8. #12476C2semidirect
9. #12461C2semidirect
10. #12458C2semidirect
11. #12444C2semidirect
12. #12441C2semidirect
13. #12025C2semidirect
14. #11987C2semidirect

Maximal subgroups

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

subgroupindex
#337512
Q8×M322
#192642
#192622
#124812
#124802
#124772
#124762
#124612
#124582
#124442
#124412
#120252
#119872
#119862