Finite Group Databasebeta

← #510430

Subgroups — #510430

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27840C211yesmaximal, characteristic
2#25035C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic
7#13475C211yesmaximal, characteristic, complemented
8#13383C211yesmaximal, characteristic
9#13176C211yesmaximal, characteristic, complemented
10#13133C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic
12#12936C211yesmaximal, characteristic, complemented
13#12912C211yesmaximal, characteristic
14#12834C211yesmaximal, characteristic
15#12812C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19#2497—21no—
20#2423—21no—
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32#2100—21no—
33#2076—21no—
34#2072—21no—
35#2036—21no—
36#1969C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1921—21no—
41#1863—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
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×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C22×C16—22no—
64C22×C16—41no—
65C22×C16—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#467—22no—
71#467—41no—
72C4×Q16—22no—
73C4×Q16—41no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#103—21no—
84C2×#103—21no—
85C2×#103—21no—
86C2×C22⋊C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#400—41no—
90#399—41no—
91#398—41no—
92#396—41no—
93#391—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#388—41no—
98#388—41no—
99#387—41no—
100#387—41no—
101#384—21no—
102#384—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4D411yescharacteristic
108#366—41no—
109#366—41no—
110#366—41no—
111#366—41no—
112C4⋊C16—22no—
113C4⋊C16—41no—
114#358—22no—
115#358—22no—
116#358—41no—
117#358—41no—
118C22⋊C16—41no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C23×C4C2×D411yescharacteristic
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16#4112yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2411yescharacteristic
146C22⋊Q8—42no—
147C22⋊Q8—81no—
148C4×Q8—42no—
149C4×Q8—81no—
150C2×C4⋊C4—21no—
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—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
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×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4D811yescharacteristic
171C2×C22⋊C4D811yescharacteristic
172C2×C42—41no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C16—21no—
177C2×C16—21no—
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—44no—
188C2×C16#4112yes—
189#107—21no—
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—42no—
204#107—81no—
205#103—42no—
206#103—42no—
207#103—42no—
208C22⋊C8—41no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95D811yescharacteristic
223#95D811yescharacteristic
224C24C2×D811yescharacteristic
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C22×D411yescharacteristic
251Q16—22no—
252Q16—22no—
253Q16—41no—
254Q16—41no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C2×#4111yescharacteristic
269C2×C8C2×#4111yescharacteristic
270C2×C8H(5,𝔽2)11yescharacteristic
271C2×C8H(5,𝔽2)11yescharacteristic
272C2×C82−1+411yescharacteristic
273C2×C82−1+411yescharacteristic
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
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—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—161no—
293C4⋊C4—161no—
294C4⋊C4—161no—
295C4⋊C4—161no—
296C4⋊C4—161no—
297C22⋊C4—41no—
298C22⋊C4—82no—
299C22⋊C4—161no—
300C22⋊C4—161no—
301C22⋊C4—161no—
302C22⋊C4Q3214yes—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—82no—
308C42—161no—
309C16—22no—
310C16—41no—
311C16—41no—
312C16—41no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C2×Q3212yes—
320C23C22×D811yescharacteristic
321Q8—42no—
322Q8—42no—
323Q8—81no—
324Q8—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
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—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
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×C4C2×Q3212yes—
367C2×C4C2×Q3212yes—
368C2×C4#53711yescharacteristic
369C2×C4#54611yescharacteristic
370C2×C4#54711yescharacteristic
371C2×C4#57511yescharacteristic
372C2×C4#57711yescharacteristic
373C2×C4#57811yescharacteristic
374C8—41no—
375C8—41no—
376C8—41no—
377C8#53811yescharacteristic
378C8#53911yescharacteristic
379C8#55211yescharacteristic
380C8#56311yescharacteristic
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#324511yescharacteristic
389C22#329411yescharacteristic
390C22#333411yescharacteristic
391C22C22×Q3211yescharacteristic
392C22#341411yescharacteristic
393C22#341511yescharacteristic
394C22#341611yescharacteristic
395C4—22no—
396C4—22no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—82no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4#269511yescharacteristic
409C4#325811yescharacteristic
410C4#331511yescharacteristic
411C4#335011yescharacteristic
412C2—42no—
413C2#3160211yescharacteristic
414C2#3381511yescharacteristic
415C2#3382311yescharacteristic
416C2#3386611yescharacteristic
417C2#3387611yescharacteristic
418C2#3390511yescharacteristic
419C2#3391511yescharacteristic
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
7. #13475C2semidirect
9. #13176C2semidirect
10. #13133C2semidirect
12. #12936C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
#278402
#250352
#135382
#135152
#135052
#134932
#134752
#133832
#131762
#131332
#130602
#129362
#129122
#128342
#128122