Finite Group Databasebeta

← #511310

Subgroups — #511310

Order 512.

488 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28532C211yesmaximal, characteristic, complemented
2#28014C211yesmaximal, characteristic
3#13528C211yesmaximal, characteristic
4#13482C211yesmaximal, characteristic
5#13481C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13449C211yesmaximal, characteristic
8#13363C211yesmaximal, characteristic
9#13238C211yesmaximal, characteristic
10#13178C211yesmaximal, characteristic, complemented
11#13133C211yesmaximal, characteristic, complemented
12#13105C211yesmaximal, characteristic, complemented
13#13013C211yesmaximal, characteristic
14#12938C211yesmaximal, characteristic, complemented
15#12890C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#475—21no—
19C2×C4×Q16C2211yescharacteristic
20#2568—21no—
21#2512—21no—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2097—21no—
34#2097—21no—
35#2071—21no—
36#2067—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1925—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#126—41no—
63C2×C4×Q8—21no—
64C2×SD32—22no—
65C2×SD32—22no—
66C2×SD32—41no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—41no—
72#500—41no—
73#475—44no—
74C4×Q16—41no—
75C4×Q16—41no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#106—21no—
85C2×#106—21no—
86C2×C4⋊C8D411yescharacteristic
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#400—41no—
97#400—41no—
98#397—41no—
99#396—41no—
100#391—21no—
101#391—21no—
102#391—41no—
103#391—41no—
104#390—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#388—41no—
109#384—21no—
110#382—21no—
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#367—41no—
115#366—41no—
116#366—41no—
117C4⋊C16—22no—
118#358—22no—
119#358—22no—
120#358—41no—
121#357—41no—
122#357—41no—
123#357—41no—
124#346—22no—
125#340—21no—
126#340—21no—
127#336D411yescharacteristic
128#336D411yescharacteristic
129C22×Q8—21no—
130C22×Q8—41no—
131C22×D4—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16#4112yes—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150#126—84no—
151C4×Q8—42no—
152C4×Q8—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4D811yescharacteristic
165C2×C4⋊C4D811yescharacteristic
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177SD32—44no—
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—21no—
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—42no—
203#107—81no—
204#106—42no—
205#106—42no—
206C4⋊C8—22no—
207#103—42no—
208#103—42no—
209#103—42no—
210#102—42no—
211#102—81no—
212#102—81no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95D811yescharacteristic
230#95D811yescharacteristic
231C24—41no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C22×D411yescharacteristic
261Q16—22no—
262Q16—22no—
263Q16—41no—
264Q16—41no—
265D8—41no—
266D8—41no—
267D8—41no—
268D8—41no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8AGL(1,ℤ/8)12yes—
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C2×C82−1+411yescharacteristic
283C2×C82−1+411yescharacteristic
284C4⋊C4—22no—
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—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—161no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—161no—
308C22⋊C4—161no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—82no—
316C42—82no—
317C16—22no—
318C16—41no—
319C16—41no—
320C16—41no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23C22×D811yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—81no—
332Q8—81no—
333Q8—164no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
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—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4#50912yes—
377C2×C4#53511yescharacteristic
378C2×C4#54711yescharacteristic
379C2×C4#54811yescharacteristic
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4#57711yescharacteristic
382C2×C4#57811yescharacteristic
383C8—22no—
384C8—41no—
385C8—41no—
386C8#53911yescharacteristic
387C8#54211yescharacteristic
388C8#55111yescharacteristic
389C8#56311yescharacteristic
390C22—161no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22#320511yescharacteristic
398C22#333411yescharacteristic
399C22#335811yescharacteristic
400C22C2×#50911yescharacteristic
401C22#341411yescharacteristic
402C22#341611yescharacteristic
403C22#341711yescharacteristic
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—82no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4#272311yescharacteristic
418C4#322511yescharacteristic
419C4#334511yescharacteristic
420C4#336711yescharacteristic
421C2—321no—
422C2#3242611yescharacteristic
423C2#3377911yescharacteristic
424C2#3379111yescharacteristic
425C2#3391711yescharacteristic
426C2#3392011yescharacteristic
427C2#3393511yescharacteristic
428C2#3394411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28532C2semidirect
5. #13481C2semidirect
6. #13451C2semidirect
10. #13178C2semidirect
11. #13133C2semidirect
12. #13105C2semidirect
14. #12938C2semidirect
15. #12890C2semidirect

Maximal subgroups

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

subgroupindex
#285322
#280142
#135282
#134822
#134812
#134512
#134492
#133632
#132382
#131782
#131332
#131052
#130132
#129382
#128902