Finite Group Databasebeta

← #511137

Subgroups — #511137

Order 512.

486 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27930C211yesmaximal, characteristic
2#27839C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13531C211yesmaximal, characteristic, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13477C211yesmaximal, characteristic, complemented
7#13382C211yesmaximal, characteristic
8#13382C211yesmaximal, characteristic
9#13177C211yesmaximal, characteristic, complemented
10#13148C211yesmaximal, characteristic, complemented
11#13132C211yesmaximal, characteristic, complemented
12#13120C211yesmaximal, characteristic
13#12920C211yesmaximal, characteristic
14#12920C211yesmaximal, characteristic
15#12906C211yesmaximal, characteristic, complemented
16C2×#484—21no—
17C2×#467C2211yescharacteristic
18C22×#107C2211yescharacteristic
19#2502—21no—
20#2496—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2099—21no—
36#2095—21no—
37#2095—21no—
38#2075—21no—
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1918—21no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#126—41no—
62C2×C22⋊Q8—21no—
63C22×C4⋊C4—21no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#484—44no—
69#467—41no—
70#467—41no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—41no—
76C2×#107—41no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#103—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×C22⋊C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#400—41no—
91#398—41no—
92#398—41no—
93#397—41no—
94#391—21no—
95#391—21no—
96#389—21no—
97#389—21no—
98#387—41no—
99#385—41no—
100#385—41no—
101#384—21no—
102#384—21no—
103#384—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4D411yescharacteristic
108#367—41no—
109#367—41no—
110#367—41no—
111#367—41no—
112#367—41no—
113#367—41no—
114C4⋊C16—22no—
115#358—22no—
116#358—22no—
117#358—41no—
118#358—41no—
119C22⋊C16—22no—
120C22⋊C16—22no—
121C4×C16—22no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#336D411yescharacteristic
128#336D411yescharacteristic
129C22×Q8—21no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148#126—84no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4D811yescharacteristic
174C2×C22⋊C4D811yescharacteristic
175C2×C42—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16#4112yes—
189C2×C16#4112yes—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
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—42no—
205#107—42no—
206#107—42no—
207#107—42no—
208#103—42no—
209#103—42no—
210#103—42no—
211C22⋊C8—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95D811yescharacteristic
226#95D811yescharacteristic
227C24C2×D811yescharacteristic
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—161no—
250C22×C4—161no—
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C22×D411yescharacteristic
257Q16—41no—
258Q16—41no—
259Q16—41no—
260Q16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8AGL(1,ℤ/8)12yes—
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8H(5,𝔽2)11yescharacteristic
277C2×C8H(5,𝔽2)11yescharacteristic
278C2×C82−1+411yescharacteristic
279C2×C82−1+411yescharacteristic
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—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—161no—
301C4⋊C4—161no—
302C4⋊C4—161no—
303C4⋊C4—161no—
304C22⋊C4—22no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4SD3214yes—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—82no—
314C16—22no—
315C16—22no—
316C16—41no—
317C16—41no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23C2×SD3212yes—
325C23C22×D811yescharacteristic
326Q8—81no—
327Q8—81no—
328Q8—81no—
329Q8—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—164no—
371C2×C4C2×SD3212yes—
372C2×C4C2×SD3212yes—
373C2×C4#51012yes—
374C2×C4#53511yescharacteristic
375C2×C4#54711yescharacteristic
376C2×C4#54711yescharacteristic
377C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
378C2×C4#57811yescharacteristic
379C2×C4#57811yescharacteristic
380C8—22no—
381C8—41no—
382C8—41no—
383C8#53911yescharacteristic
384C8#54211yescharacteristic
385C8#55311yescharacteristic
386C8#56311yescharacteristic
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22#320511yescharacteristic
395C22#333411yescharacteristic
396C22#333411yescharacteristic
397C22C22×SD3211yescharacteristic
398C22C2×#51011yescharacteristic
399C22#341611yescharacteristic
400C22#341611yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4#271811yescharacteristic
415C4#322311yescharacteristic
416C4#334311yescharacteristic
417C4#335311yescharacteristic
418C2—42no—
419C2#3227811yescharacteristic
420C2#3377211yescharacteristic
421C2#3378811yescharacteristic
422C2#3390211yescharacteristic
423C2#3390211yescharacteristic
424C2#3391111yescharacteristic
425C2#3392311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13538C2semidirect
4. #13531C2semidirect
5. #13506C2semidirect
6. #13477C2semidirect
9. #13177C2semidirect
10. #13148C2semidirect
11. #13132C2semidirect
15. #12906C2semidirect

Maximal subgroups

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

subgroupindex
#279302
#278392
#135382
#135312
#135062
#134772
#133822
#131772
#131482
#131322
#131202
#129202
#129062