Finite Group Databasebeta

← #510159

Subgroups — #510159

Order 512.

504 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26065C211yesmaximal, characteristic
2#25035C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic
7#13383C211yesmaximal, characteristic
8#13364C211yesmaximal, characteristic, complemented
9#13282C211yesmaximal, characteristic, complemented
10#13148C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic
12#12993C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12867C211yesmaximal, characteristic
15#12812C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C22×#107C2211yescharacteristic
20#2440—21no—
21#2423—21no—
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#2083—21no—
33#2072—21no—
34#2057—21no—
35#2036—21no—
36#1969C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1863—21no—
41#1852—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C8×C22⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—41no—
64C2×C4×Q8—21no—
65C22×C4⋊C4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#500—22no—
71#500—41no—
72#467—22no—
73#467—41no—
74C4×Q16—22no—
75C4×Q16—41no—
76C2×#107—22no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—41no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#399—41no—
96#398—41no—
97#398—41no—
98#395—41no—
99#391—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#388—41no—
104#387—41no—
105#385—41no—
106#384—21no—
107#384—21no—
108#380—21no—
109#380—21no—
110C4×C22⋊C4D411yescharacteristic
111#366—41no—
112#366—41no—
113C4⋊C16—41no—
114#358—22no—
115#358—22no—
116#358—41no—
117#358—41no—
118C22⋊C16—41no—
119C22⋊C16—41no—
120C4×C16—41no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—41no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16#4112yes—
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
148C4⋊Q8—42no—
149C4⋊Q8—81no—
150C22⋊Q8—42no—
151C22⋊Q8—84no—
152C22⋊Q8—81no—
153C4×Q8—42no—
154C4×Q8—81no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—42no—
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×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—81no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4D811yescharacteristic
179C2×C22⋊C4D811yescharacteristic
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—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—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—42no—
209#107—42no—
210#107—81no—
211#107—81no—
212#107#4112yes—
213#103—42no—
214#103—42no—
215#103—42no—
216C22⋊C8—22no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95D811yescharacteristic
229#95D811yescharacteristic
230C24C2×D811yescharacteristic
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—161no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—161no—
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C22×D411yescharacteristic
263Q16—22no—
264Q16—22no—
265Q16—41no—
266Q16—41no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8#13712yes—
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8H(5,𝔽2)11yescharacteristic
283C2×C8H(5,𝔽2)11yescharacteristic
284C2×C82−1+411yescharacteristic
285C2×C82−1+411yescharacteristic
286C4⋊C4—22no—
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—41no—
299C4⋊C4—41no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—161no—
309C4⋊C4—161no—
310C4⋊C4—161no—
311C22⋊C4—22no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—161no—
315C22⋊C4Q3214yes—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—82no—
321C16—41no—
322C16—41no—
323C16—41no—
324C16—41no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23C2×Q3212yes—
332C23C22×D811yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337Q8—164no—
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—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
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—81no—
377C2×C4—81no—
378C2×C4—164no—
379C2×C4C2×Q3212yes—
380C2×C4C2×Q3212yes—
381C2×C4#50912yes—
382C2×C4#53611yescharacteristic
383C2×C4#54611yescharacteristic
384C2×C4#54611yescharacteristic
385C2×C4C2×#13711yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57711yescharacteristic
388C8—22no—
389C8—41no—
390C8—41no—
391C8#53911yescharacteristic
392C8#54211yescharacteristic
393C8#55211yescharacteristic
394C8#56211yescharacteristic
395C22—22no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#322611yescharacteristic
403C22#329411yescharacteristic
404C22#329411yescharacteristic
405C22C22×Q3211yescharacteristic
406C22C2×#50911yescharacteristic
407C22#341611yescharacteristic
408C22#341611yescharacteristic
409C4—22no—
410C4—22no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#268711yescharacteristic
423C4#324411yescharacteristic
424C4#332011yescharacteristic
425C4#333111yescharacteristic
426C2—42no—
427C2#3135011yescharacteristic
428C2#3379611yescharacteristic
429C2#3381011yescharacteristic
430C2#3386611yescharacteristic
431C2#3386611yescharacteristic
432C2#3387711yescharacteristic
433C2#3389911yescharacteristic
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. #13515C2semidirect
5. #13505C2semidirect
8. #13364C2semidirect
9. #13282C2semidirect
10. #13148C2semidirect
13. #12936C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
#260652
#250352
#135382
#135152
#135052
#134932
#133832
#133642
#132822
#131482
#131202
#129932
#129362
#128672
#128122