Finite Group Databasebeta

← #499588

Subgroups — #499588

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21524C211yesmaximal, characteristic
2#20821C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic, complemented
7#13492C211yesmaximal, characteristic, complemented
8#13480C211yesmaximal, characteristic, complemented
9#13381C211yesmaximal, characteristic
10#13381C211yesmaximal, characteristic
11#12674C211yesmaximal, characteristic, complemented
12C4×#367C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×#405C2211yescharacteristic
19#2324—21no—
20#2314—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1970C2211yescharacteristic
34#1942C2211yescharacteristic
35#1936—21no—
36#1895—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#1762—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×#117D411yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#466—22no—
70#466—22no—
71C4×D8—22no—
72C4×D8—22no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85#405—22no—
86#405—22no—
87C2×#97D411yescharacteristic
88C2×#97D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#390—21no—
92#390—41no—
93#389—21no—
94#388—41no—
95#388—41no—
96#388—41no—
97#387—41no—
98#387—41no—
99#386—41no—
100#385—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#367—22no—
111#367—22no—
112#366—22no—
113#357—22no—
114#357—22no—
115#357—22no—
116#357—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×D4—21no—
129C22×D4—41no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8#4112yes—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—41no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2411yescharacteristic
151#121—24no—
152#121—42no—
153C4×D4—42no—
154C4×D4—42no—
155#117—22no—
156#117—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
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⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4D811yescharacteristic
177C2×C22⋊C4D811yescharacteristic
178C2×C42—41no—
179C2×C42—41no—
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#4112yes—
194C2×C16#4112yes—
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#4112yes—
210#106—42no—
211#102—24no—
212#102—42no—
213#102—42no—
214#97—22no—
215#97—22no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8#4112yes—
219C4×C8#4112yes—
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—
229C24—41no—
230C24C2×D811yescharacteristic
231C2×D4—22no—
232C2×D4—44no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—161no—
239C2×D4—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—82no—
254C22×C4—161no—
255C22×C4—161no—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C22×D411yescharacteristic
262D8—22no—
263D8—22no—
264D8—41no—
265D8—41no—
266M16—44no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8#12312yes—
279C2×C8#13512yes—
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
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—41no—
301C4⋊C4—41no—
302C4⋊C4—161no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—44no—
308C22⋊C4—82no—
309C22⋊C4—161no—
310C22⋊C4—161no—
311C22⋊C4—161no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C42—21no—
315C42—21no—
316C42—161no—
317C42—161no—
318C42—161no—
319C42#12312yes—
320C42#13712yes—
321C16—22no—
322C16—22no—
323C16—41no—
324C16—41no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—42no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23C22×D811yescharacteristic
340D4—42no—
341D4—42no—
342D4—81no—
343D4—81no—
344D4—164no—
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—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
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—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—164no—
384C2×C4#48012yes—
385C2×C4#48012yes—
386C2×C4C2×#12311yescharacteristic
387C2×C4#52511yescharacteristic
388C2×C4#53211yescharacteristic
389C2×C4C2×#13511yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C2×C4#57511yescharacteristic
392C8—22no—
393C8—22no—
394C8—41no—
395C8#52911yescharacteristic
396C8#52911yescharacteristic
397C8#52911yescharacteristic
398C8#52911yescharacteristic
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22C2×#48011yescharacteristic
414C22#310011yescharacteristic
415C22#316911yescharacteristic
416C22#341511yescharacteristic
417C22#341611yescharacteristic
418C22#341611yescharacteristic
419C22#341611yescharacteristic
420C4—22no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—82no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4#239511yescharacteristic
433C4#309111yescharacteristic
434C4#310711yescharacteristic
435C4#317211yescharacteristic
436C2—81no—
437C2—162no—
438C2#2375411yescharacteristic
439C2#3365011yescharacteristic
440C2#3365311yescharacteristic
441C2#3367911yescharacteristic
442C2#3368311yescharacteristic
443C2#3374211yescharacteristic
444C2#3374411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
2. #20821C2semidirect
3. #13539C2semidirect
4. #13528C2semidirect
5. #13506C2semidirect
6. #13494C2semidirect
7. #13492C2semidirect
8. #13480C2semidirect
11. #12674C2semidirect
12. C4×#367C2semidirect
13. #12637C2semidirect
14. C4×#357C2semidirect
15. #12600C2semidirect

Maximal subgroups

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

subgroupindex
#215242
#208212
#135392
#135282
#135062
#134942
#134922
#134802
#133812
#126742
C4×#3672
#126372
C4×#3572
#126002