Finite Group Databasebeta

← #501421

Subgroups — #501421

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2232C211yesmaximal, characteristic
2C2×#2159C211yesmaximal, characteristic, complemented
3#27928C211yesmaximal, characteristic
4#22612C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13178C211yesmaximal, characteristic, complemented
10#13176C211yesmaximal, characteristic, complemented
11#13131C211yesmaximal, characteristic, complemented
12#13130C211yesmaximal, characteristic, complemented
13#13061C211yesmaximal, characteristic
14#12913C211yesmaximal, characteristic, complemented
15#12868C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#482—21no—
18C2×#466C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2500—21no—
22#2345—21no—
23#2232—24no—
24#2159—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38#2094—21no—
39#2092—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1921—21no—
45#1921—21no—
46#1846C2211yescharacteristic
47#1845C2211yescharacteristic
48#1824—21no—
49#1808C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#121—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C2×M32—22no—
67C2×M32—22no—
68C2×M32—22no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#482—44no—
74#466—22no—
75#466—22no—
76#427—22no—
77#427—22no—
78C2×#107—21no—
79C2×#107—22no—
80C2×#107—21no—
81C2×#107D414yes—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#394—41no—
98#394—41no—
99#390—21no—
100#390—21no—
101#389—21no—
102#389—21no—
103#388—41no—
104#388—41no—
105#387—41no—
106#387—41no—
107#384—21no—
108#382—21no—
109#380—21no—
110#380—41no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C22⋊C4—41no—
114C4×C22⋊C4D411yescharacteristic
115#367—22no—
116#367—22no—
117#366—22no—
118#366—22no—
119#366—22no—
120#366—22no—
121C4⋊C16—22no—
122C4⋊C16—22no—
123#357—22no—
124#357—22no—
125#357—22no—
126#357—22no—
127C22⋊C16—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×D4—21no—
136C22×D4—41no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D412yes—
153C22×C8#4112yes—
154C22×C8C2411yescharacteristic
155#121—42no—
156#121—42no—
157#117—42no—
158#117—42no—
159C2×C4⋊C4—24no—
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—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊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×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187M32—28no—
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#4112yes—
199C2×C16#4112yes—
200#107—24no—
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—21no—
213#107—42no—
214#107C2×D412yes—
215#107C2×D412yes—
216#106—42no—
217#106—42no—
218#102—42no—
219#102—42no—
220#102—42no—
221C22⋊C8—22no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95D811yescharacteristic
234#95D811yescharacteristic
235C24—41no—
236C24C2×D811yescharacteristic
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—82no—
244C2×D4—161no—
245C2×D4—161no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—44no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—161no—
259C22×C4—161no—
260C22×C4#12112yes—
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4AGL(1,ℤ/8)12yes—
267C22×C4C22×D411yescharacteristic
268D8—41no—
269D8—41no—
270D8—41no—
271D8—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8#12112yes—
285C2×C8C22×D411yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C82−1+411yescharacteristic
290C2×C82−1+411yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—44no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—161no—
307C4⋊C4—161no—
308C4⋊C4—161no—
309C4⋊C4—161no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—82no—
324C42—161no—
325C42—161no—
326C16—22no—
327C16—22no—
328C16—41no—
329C16—41no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23#45912yes—
344C23C22×D811yescharacteristic
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—164no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
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—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—164no—
389C2×C4#45912yes—
390C2×C4#51012yes—
391C2×C4C2×#12111yescharacteristic
392C2×C4#54711yescharacteristic
393C2×C4#54711yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4#57811yescharacteristic
396C2×C4#57811yescharacteristic
397C8—22no—
398C8—41no—
399C8—41no—
400C8#52511yescharacteristic
401C8#53611yescharacteristic
402C8#55211yescharacteristic
403C8#55211yescharacteristic
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22C2×#45911yescharacteristic
419C22#333411yescharacteristic
420C22#333411yescharacteristic
421C22C2×#51011yescharacteristic
422C22#341411yescharacteristic
423C22#341511yescharacteristic
424C22#341511yescharacteristic
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—82no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4#241311yescharacteristic
438C4#304411yescharacteristic
439C4#334311yescharacteristic
440C4#334311yescharacteristic
441C2—42no—
442C2—321no—
443C2#2457611yescharacteristic
444C2#3358111yescharacteristic
445C2#3358511yescharacteristic
446C2#3391311yescharacteristic
447C2#3391411yescharacteristic
448C2#3392211yescharacteristic
449C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
2. C2×#2159C2semidirect
4. #22612C2semidirect
5. #13507C2semidirect
6. #13505C2semidirect
7. #13476C2semidirect
8. #13474C2semidirect
9. #13178C2semidirect
10. #13176C2semidirect
11. #13131C2semidirect
12. #13130C2semidirect
14. #12913C2semidirect
15. #12868C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22322
C2×#21592
#279282
#226122
#135072
#135052
#134762
#134742
#131782
#131762
#131312
#131302
#130612
#129132
#128682