Finite Group Databasebeta

← #499590

Subgroups — #499590

Order 512.

507 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#21525C211yesmaximal, characteristic
2#20821C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13529C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic
7#13492C211yesmaximal, characteristic
8#13481C211yesmaximal, characteristic, complemented
9#13383C211yesmaximal, characteristic
10#13383C211yesmaximal, characteristic
11#12674C211yesmaximal, characteristic, complemented
12C4×#366C211yesmaximal, characteristic, complemented
13#12638C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×#405C2211yescharacteristic
19#2323—21no—
20#2314—21no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1969C2211yescharacteristic
34#1942C2211yescharacteristic
35#1935—21no—
36#1894—21no—
37#1846C2211yescharacteristic
38#1845C2211yescharacteristic
39#1836—21no—
40#1809—21no—
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44#1776—21no—
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48#1763—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×#117D411yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#467—22no—
70#467—22no—
71C4×Q16—22no—
72C4×Q16—22no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#103—21no—
85#405—22no—
86#405—22no—
87C2×#97D411yescharacteristic
88C2×#97D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#389—21no—
93#389—41no—
94#388—41no—
95#388—41no—
96#388—41no—
97#388—41no—
98#387—41no—
99#387—41no—
100#387—41no—
101#384—21no—
102#382—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#367—22no—
110#366—22no—
111#366—22no—
112#366—22no—
113#358—22no—
114#358—22no—
115#358—22no—
116#358—22no—
117C22⋊C16—22no—
118C22⋊C16—22no—
119#346—22no—
120C4×C16—22no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×Q8—21no—
129C23×C4C2×D411yescharacteristic
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16#4112yes—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—41no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—42no—
151C4×Q8—42no—
152C4×Q8—42no—
153#117—22no—
154#117—22no—
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—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4D811yescharacteristic
168C2×C4⋊C4D811yescharacteristic
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4D811yescharacteristic
174C2×C22⋊C4D811yescharacteristic
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
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—21no—
190C2×C16—21no—
191C2×C16#4112yes—
192C2×C16#4112yes—
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—21no—
205#107—21no—
206#107—42no—
207#107#4112yes—
208#106—42no—
209#103—24no—
210#103—42no—
211#103—42no—
212#97—22no—
213#97—22no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8#4112yes—
217C4×C8#4112yes—
218#95—41no—
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—
229C24C2×D811yescharacteristic
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C22×D411yescharacteristic
256Q16—22no—
257Q16—22no—
258Q16—41no—
259Q16—41no—
260M16—44no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8#12312yes—
273C2×C8#13512yes—
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—44no—
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—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—82no—
298C4⋊C4—161no—
299C4⋊C4—161no—
300C4⋊C4—161no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—44no—
304C22⋊C4—161no—
305C22⋊C4—161no—
306C22⋊C4—161no—
307C42—21no—
308C42—21no—
309C42—82no—
310C42—161no—
311C42—161no—
312C42—161no—
313C42#12312yes—
314C42#13712yes—
315C16—22no—
316C16—22no—
317C16—41no—
318C16—41no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23C22×D811yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—81no—
330Q8—81no—
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—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
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—81no—
371C2×C4—81no—
372C2×C4#48012yes—
373C2×C4#48012yes—
374C2×C4C2×#12311yescharacteristic
375C2×C4#52511yescharacteristic
376C2×C4#53211yescharacteristic
377C2×C4C2×#13511yescharacteristic
378C2×C4C2×#13711yescharacteristic
379C2×C4#57511yescharacteristic
380C8—22no—
381C8—22no—
382C8—41no—
383C8#52911yescharacteristic
384C8#52911yescharacteristic
385C8#52911yescharacteristic
386C8#52911yescharacteristic
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22C2×#48011yescharacteristic
395C22#310011yescharacteristic
396C22#316911yescharacteristic
397C22#341511yescharacteristic
398C22#341611yescharacteristic
399C22#341711yescharacteristic
400C22#341711yescharacteristic
401C4—22no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—82no—
407C4—82no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4#239511yescharacteristic
415C4#309111yescharacteristic
416C4#310711yescharacteristic
417C4#317211yescharacteristic
418C2—81no—
419C2#2375411yescharacteristic
420C2#3365111yescharacteristic
421C2#3365211yescharacteristic
422C2#3367911yescharacteristic
423C2#3368711yescharacteristic
424C2#3373911yescharacteristic
425C2#3374511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13539C2semidirect
4. #13529C2semidirect
5. #13505C2semidirect
8. #13481C2semidirect
11. #12674C2semidirect
12. C4×#366C2semidirect
13. #12638C2semidirect
14. C4×#358C2semidirect

Maximal subgroups

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

subgroupindex
#215252
#208212
#135392
#135292
#135052
#134942
#134922
#134812
#133832
#126742
C4×#3662
#126382
C4×#3582
#126002