Finite Group Databasebeta

← #498365

Subgroups — #498365

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2218C211yesmaximal, characteristic
2C2×#2177C211yesmaximal, characteristic, complemented
3C2×#2159C211yesmaximal, characteristic, complemented
4#23027C211yesmaximal, characteristic, complemented
5#23025C211yesmaximal, characteristic, complemented
6#13235C211yesmaximal, characteristic, complemented
7#13167C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#12992C211yesmaximal, characteristic, complemented
11#12991C211yesmaximal, characteristic, complemented
12#12937C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic, complemented
14C4×#367C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24#2363—21no—
25#2361—21no—
26#2218C418yescomplemented
27#2177—24no—
28#2159—24no—
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×#366C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1926—21no—
48#1925—21no—
49#1875—21no—
50#1874—21no—
51#1864—21no—
52#1862—21no—
53C4×#107C2211yescharacteristic
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×D8C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—41no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C2×D16—22no—
73C2×D16—22no—
74C2×D16C2×C414yescomplemented
75C2×M32—22no—
76C2×M32—22no—
77C2×M32C2×C414yescomplemented
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#466—22no—
83#466—22no—
84#466C2×C414yescomplemented
85#466C2×C414yescomplemented
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—41no—
89C4×D8—41no—
90#427—22no—
91#427—41no—
92C2×#107—22no—
93C2×#107—21no—
94C2×#107—41no—
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#106—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#390—21no—
112#390—21no—
113#390—41no—
114#389—21no—
115#389—21no—
116#388—41no—
117#388—41no—
118#386—41no—
119#386—41no—
120#385—41no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C22⋊C4—41no—
124C4×C22⋊C4D411yescharacteristic
125#367—22no—
126#367C2×C414yescomplemented
127#367D414yescomplemented
128#366—22no—
129#366—22no—
130#357—22no—
131#357—22no—
132#357—41no—
133#357—41no—
134#357C2×C414yescomplemented
135#357C2×C414yescomplemented
136C22⋊C16—22no—
137#346—22no—
138C4×C16—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—41no—
148C22×D4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8C22×C412yes—
164C2×D8C22×C412yes—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C412yes—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8#4112yes—
174C22×C8C2411yescharacteristic
175#121—24no—
176#121—24no—
177#121—42no—
178#121—42no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—81no—
182C4×D4—81no—
183C4×D4—84no—
184#117—42no—
185#117—81no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—81no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4D811yescharacteristic
207C2×C22⋊C4D811yescharacteristic
208C2×C42—41no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213D16—28no—
214M32—28no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16C22×C412yes—
224C2×C16C22×C412yes—
225C2×C16C2×D412yes—
226C2×C16#4112yes—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—42no—
236#107C22×C412yes—
237#107C22×C412yes—
238#107C2×D412yes—
239#107C2×D412yes—
240#106—42no—
241#102—24no—
242#102—24no—
243#102—42no—
244#102—42no—
245#102—81no—
246#102—81no—
247C22⋊C8—22no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95D811yescharacteristic
257#95D811yescharacteristic
258C24—41no—
259C24—41no—
260C24C2×D811yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—161no—
278C2×D4—161no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—161no—
296C22×C4—161no—
297C22×C4—161no—
298C22×C4C4×D412yes—
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4#13512yes—
305C22×C4C22×D411yescharacteristic
306D8—22no—
307D8—22no—
308D8—22no—
309D8—22no—
310D8—41no—
311D8—41no—
312D8—41no—
313D8—41no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8C4×D412yes—
327C2×C8C23×C411yescharacteristic
328C2×C8C22×D411yescharacteristic
329C2×C8C2×#4111yescharacteristic
330C2×C8C2×#4111yescharacteristic
331C2×C8H(5,𝔽2)11yescharacteristic
332C2×C82−1+411yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—161no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—161no—
358C22⋊C4—161no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—82no—
364C42—82no—
365C42—161no—
366C16—22no—
367C16—41no—
368C16—41no—
369C16C4×D414yescomplemented
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—42no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23C4×D812yes—
391C23C22×D811yescharacteristic
392D4—42no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—88no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—164no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—164no—
438C2×C4—164no—
439C2×C4—164no—
440C2×C4C4×D812yes—
441C2×C4#50912yes—
442C2×C4C2×C4×D411yescharacteristic
443C2×C4D4211yescharacteristic
444C2×C4#54711yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57811yescharacteristic
448C8—22no—
449C8—41no—
450C8—41no—
451C8C2×C4×D411yescharacteristic
452C8#52011yescharacteristic
453C8#54811yescharacteristic
454C8#55311yescharacteristic
455C22—22no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22C2×C4×D811yescharacteristic
477C22D4×D811yescharacteristic
478C22#333411yescharacteristic
479C22C2×#50911yescharacteristic
480C22#341411yescharacteristic
481C22#341511yescharacteristic
482C22#341611yescharacteristic
483C4—22no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—161no—
493C4—161no—
494C4#236111yescharacteristic
495C4#295111yescharacteristic
496C4#328511yescharacteristic
497C4#333911yescharacteristic
498C2—42no—
499C2—162no—
500C2—162no—
501C2#2302511yescharacteristic
502C2C4×#50911yescharacteristic
503C2#3345411yescharacteristic
504C2#3384011yescharacteristic
505C2#3384111yescharacteristic
506C2#3391211yescharacteristic
507C2#3391411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2218C2
LCSG ▷ C2×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×#2177C2semidirect
3. C2×#2159C2semidirect
4. #23027C2semidirect
5. #23025C2semidirect
6. #13235C2semidirect
7. #13167C2semidirect
8. #13131C2semidirect
9. #13130C2semidirect
10. #12992C2semidirect
11. #12991C2semidirect
12. #12937C2semidirect
13. #12935C2semidirect
14. C4×#367C2semidirect
15. #12612C2semidirect
26. #2218C4semidirect
74. C2×D16C2×C4semidirect
77. C2×M32C2×C4semidirect
84. #466C2×C4semidirect
85. #466C2×C4semidirect
126. #367C2×C4semidirect
127. #367D4semidirect
134. #357C2×C4semidirect
135. #357C2×C4semidirect
369. C16C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22182
C2×#21772
C2×#21592
#230272
#230252
#132352
#131672
#131312
#131302
#129922
#129912
#129372
#129352
C4×#3672
#126122