Finite Group Databasebeta

← #504186

Subgroups — #504186

Order 512.

528 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#26066C211yesmaximal, characteristic
2#23026C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13514C211yesmaximal, characteristic, complemented
5#13475C211yesmaximal, characteristic, complemented
6#13366C211yesmaximal, characteristic, complemented
7#13277C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic
10#12994C211yesmaximal, characteristic
11#12938C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic
13#12912C211yesmaximal, characteristic
14#12812C211yesmaximal, characteristic, complemented
15#12556C211yesmaximal, characteristic
16C23×C16C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2441—21no—
23#2362—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33#2084—21no—
34#2059—21no—
35#1969C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1927—21no—
40#1873—21no—
41#1865—21no—
42#1854—21no—
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
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×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×#117—21no—
65C2×#117—21no—
66C22×C16—22no—
67C22×C16—41no—
68C22×C16—41no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#499—22no—
74#499—41no—
75#467—22no—
76#467Q814yes—
77C4×Q16—22no—
78C4×Q16—41no—
79#427—22no—
80#427—22no—
81#427—41no—
82#427—41no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×C22⋊C8D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#399—41no—
101#396—41no—
102#391—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—41no—
107#388—41no—
108#388—41no—
109#387—41no—
110#387—41no—
111#384—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—41no—
115C4×C22⋊C4D411yescharacteristic
116#367—22no—
117#366—41no—
118C4⋊C16—41no—
119C4⋊C16—41no—
120#358—22no—
121#358—41no—
122#358—41no—
123#358—41no—
124C22⋊C16—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
132C22×Q8—21no—
133C23×C4C2×D411yescharacteristic
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16C2×Q812yes—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×Q812yes—
148C22×C8#4112yes—
149C22×C8C2411yescharacteristic
150#125—42no—
151#125—81no—
152C22⋊Q8—24no—
153C22⋊Q8—42no—
154C4×Q8—42no—
155C4×Q8—81no—
156#117—42no—
157#117—42no—
158#117—81no—
159#117—81no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
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—41no—
175C2×C4⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4D811yescharacteristic
181C2×C22⋊C4D811yescharacteristic
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—44no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—81no—
213#107C2×Q812yes—
214#107#4112yes—
215#107#4112yes—
216#106—42no—
217#106—42no—
218#106—42no—
219#103—24no—
220#103—42no—
221#103—81no—
222C22⋊C8—22no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95D811yescharacteristic
235#95D811yescharacteristic
236C24C2×D811yescharacteristic
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C22⋊Q812yes—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4#13712yes—
262C22×C4C22×D411yescharacteristic
263Q16—22no—
264Q16—22no—
265Q16—41no—
266Q16—41no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8C22⋊Q812yes—
280C2×C8C22×Q811yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C8H(5,𝔽2)11yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—44no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—161no—
308C4⋊C4—161no—
309C22⋊C4—22no—
310C22⋊C4—41no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—161no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C42—82no—
321C42—82no—
322C42—82no—
323C16—41no—
324C16—41no—
325C16—41no—
326C16—41no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23#47412yes—
334C23C22×D811yescharacteristic
335Q8—42no—
336Q8—42no—
337Q8—81no—
338Q8—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
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—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
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—81no—
379C2×C4—81no—
380C2×C4#47412yes—
381C2×C4#50812yes—
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4#54611yescharacteristic
384C2×C4#54611yescharacteristic
385C2×C4C2×#13711yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57711yescharacteristic
388C8—22no—
389C8—41no—
390C8—41no—
391C8#52911yescharacteristic
392C8#53811yescharacteristic
393C8#55411yescharacteristic
394C8#55411yescharacteristic
395C22—22no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22C2×#47411yescharacteristic
403C22#329411yescharacteristic
404C22#329411yescharacteristic
405C22C2×#50811yescharacteristic
406C22#341411yescharacteristic
407C22#341611yescharacteristic
408C22#341711yescharacteristic
409C4—22no—
410C4—22no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#245011yescharacteristic
423C4#308411yescharacteristic
424C4#331611yescharacteristic
425C4#332011yescharacteristic
426C2—42no—
427C2#2640811yescharacteristic
428C2#3362611yescharacteristic
429C2#3363211yescharacteristic
430C2#3386811yescharacteristic
431C2#3386911yescharacteristic
432C2#3388211yescharacteristic
433C2#3388411yescharacteristic
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. #13514C2semidirect
5. #13475C2semidirect
6. #13366C2semidirect
7. #13277C2semidirect
8. #13110C2semidirect
11. #12938C2semidirect
14. #12812C2semidirect

Maximal subgroups

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

subgroupindex
#260662
#230262
#135382
#135142
#134752
#133662
#132772
#131102
#130602
#129942
#129382
#129212
#129122
#128122
#125562