Finite Group Databasebeta

← #498011

Subgroups — #498011

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2177C211yesmaximal, characteristic
2#22807C211yesmaximal, characteristic, complemented
3#22682C211yesmaximal, characteristic, complemented
4#13149C211yesmaximal, characteristic, complemented
5#13147C211yesmaximal, characteristic, complemented
6#13111C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#13012C211yesmaximal, characteristic
9#13012C211yesmaximal, characteristic
10#12890C211yesmaximal, characteristic, complemented
11#12813C211yesmaximal, characteristic, complemented
12#12811C211yesmaximal, characteristic, complemented
13#12663C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2354—21no—
22#2349—21no—
23#2177C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1925—21no—
40#1925—21no—
41#1864—21no—
42#1864—21no—
43#1846C2211yescharacteristic
44#1845C2211yescharacteristic
45#1804—21no—
46#1802—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×#117—41no—
67C2×D16—22no—
68C2×D16—22no—
69C2×D16C2×C414yes—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C4×D8—22no—
75C4×D8—22no—
76C4×D8—22no—
77C4×D8—22no—
78C4×D8—41no—
79C4×D8—41no—
80C4×D8C2×C414yes—
81C4×D8C2×C414yescomplemented
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#388—41no—
105#388—41no—
106#385—41no—
107#385—41no—
108#384—21no—
109#382—21no—
110#380—41no—
111#380—41no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D411yescharacteristic
115C2×#95—41no—
116#367C2×C414yescomplemented
117C4⋊C16—22no—
118#357—22no—
119#357—22no—
120#357—41no—
121#357—41no—
122#357C2×C414yes—
123#357C2×C414yescomplemented
124#346—41no—
125#346C2×C414yescomplemented
126C4×C16—41no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×D4—21no—
134C22×D4—21no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C22×C412yes—
148C2×D8C22×C412yes—
149C2×D8#4112yes—
150C2×D8#4112yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2411yescharacteristic
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—81no—
162C4×D4—81no—
163#117—84no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C4⋊C4D811yescharacteristic
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189D16—28no—
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×C16C22×C412yes—
201C2×C16C22×C412yes—
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#107C22×C412yes—
214#107C22×C412yes—
215#106—81no—
216C4⋊C8—22no—
217#102—24no—
218#102—24no—
219#102—42no—
220#102—42no—
221#102—81no—
222#102—81no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8C22×C412yes—
228C4×C8#4112yes—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95D811yescharacteristic
236#95D811yescharacteristic
237C24—41no—
238C24—41no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—161no—
266C22×C4—161no—
267C22×C4—161no—
268C22×C4—161no—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C22×D411yescharacteristic
276D8—22no—
277D8—22no—
278D8—22no—
279D8—22no—
280D8—22no—
281D8—41no—
282D8—41no—
283D8#11714yescomplemented
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8C4×D412yes—
293C2×C8C23×C411yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C2×C82−1+411yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—82no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—82no—
330C42—82no—
331C42C4×D412yes—
332C42#13512yes—
333C16—22no—
334C16—22no—
335C16—41no—
336C16—41no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23C22×D811yescharacteristic
352D4—24no—
353D4—42no—
354D4—42no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—81no—
359D4—81no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—88no—
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—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—164no—
395C2×C4—164no—
396C2×C4—164no—
397C2×C4C4×D812yes—
398C2×C4C4×D812yes—
399C2×C4#50912yes—
400C2×C4C2×C4×D411yescharacteristic
401C2×C4#54611yescharacteristic
402C2×C4#54811yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4#57711yescharacteristic
405C2×C4#57711yescharacteristic
406C8—22no—
407C8—41no—
408C8—41no—
409C8C2×#11711yescharacteristic
410C8#52011yescharacteristic
411C8#55211yescharacteristic
412C8#55311yescharacteristic
413C22—44no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22C2×C4×D811yescharacteristic
428C22#329411yescharacteristic
429C22#335811yescharacteristic
430C22C2×#50911yescharacteristic
431C22#341411yescharacteristic
432C22#341511yescharacteristic
433C22#341611yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—82no—
441C4—82no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4#235211yescharacteristic
447C4#295211yescharacteristic
448C4#330311yescharacteristic
449C4#336011yescharacteristic
450C2—84no—
451C2—162no—
452C2#2276611yescharacteristic
453C2C4×#50911yescharacteristic
454C2#3345411yescharacteristic
455C2#3387311yescharacteristic
456C2#3388611yescharacteristic
457C2#3393311yescharacteristic
458C2#3394311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22807C2semidirect
3. #22682C2semidirect
4. #13149C2semidirect
5. #13147C2semidirect
6. #13111C2semidirect
7. #13110C2semidirect
10. #12890C2semidirect
11. #12813C2semidirect
12. #12811C2semidirect
13. #12663C2semidirect
14. #12637C2semidirect
15. C4×#357C2semidirect
23. #2177C4semidirect
81. C4×D8C2×C4semidirect
116. #367C2×C4semidirect
123. #357C2×C4semidirect
125. #346C2×C4semidirect
283. D8#117semidirect

Maximal subgroups

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

subgroupindex
C2×#21772
#228072
#226822
#131492
#131472
#131112
#131102
#130122
#128902
#128132
#128112
#126632
#126372
C4×#3572