Finite Group Databasebeta

← #498502

Subgroups — #498502

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2219C211yesmaximal, characteristic
2#23026C211yesmaximal, characteristic
3#23024C211yesmaximal, characteristic
4#13150C211yesmaximal, characteristic, complemented
5#13148C211yesmaximal, characteristic, complemented
6#13111C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13015C211yesmaximal, characteristic, complemented
9#13015C211yesmaximal, characteristic, complemented
10#12994C211yesmaximal, characteristic
11#12993C211yesmaximal, characteristic
12#12638C211yesmaximal, characteristic, complemented
13C4×#358C211yesmaximal, characteristic, complemented
14#12612C211yesmaximal, characteristic
15#12557C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24#2362—21no—
25#2360—21no—
26#2219C418yescomplemented
27C2×#367C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1928—21no—
42#1927—21no—
43#1873—21no—
44#1872—21no—
45#1865—21no—
46#1863—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C22⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C2×Q32—41no—
70C2×Q32—41no—
71C2×Q32C2×C414yescomplemented
72C2×M32—41no—
73C2×M32—41no—
74C2×M32C2×C414yes—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#467—22no—
80#467—22no—
81#467C2×C414yes—
82#467C2×C414yes—
83C4×Q16—22no—
84C4×Q16—22no—
85C4×Q16—41no—
86C4×Q16—41no—
87#427—22no—
88#427—41no—
89C2×#107—22no—
90C2×#107—21no—
91C2×#107—41no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#106—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×C22⋊C8D411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#391—21no—
110#389—21no—
111#389—21no—
112#389—41no—
113#388—41no—
114#388—41no—
115#387—41no—
116#387—41no—
117#385—41no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—41no—
121C4×C22⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367C2×C414yescomplemented
124#358—22no—
125#358—22no—
126#358—41no—
127#358—41no—
128#358C2×C414yescomplemented
129#358C2×C414yescomplemented
130C22⋊C16—22no—
131#346—41no—
132C4×C16—41no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4—41no—
142C23×C4C2×D411yescharacteristic
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16C22×C412yes—
154C2×Q16C22×C412yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C22×C412yes—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8#4112yes—
164C22×C8C2411yescharacteristic
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—81no—
172C4×Q8—81no—
173#117—42no—
174#117—81no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—81no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4D811yescharacteristic
194C2×C22⋊C4D811yescharacteristic
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202Q32—44no—
203M32—44no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16C22×C412yes—
215C2×C16C22×C412yes—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—42no—
225#107C22×C412yes—
226#107C22×C412yes—
227#107#4112yes—
228#107#4112yes—
229#106—42no—
230#103—24no—
231#103—24no—
232#103—42no—
233#103—42no—
234#103—81no—
235#103—81no—
236C22⋊C8—22no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95D811yescharacteristic
250#95D811yescharacteristic
251C24C2×D811yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—161no—
279C22×C4C4×D412yes—
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4#13512yes—
286C22×C4C22×D411yescharacteristic
287Q16—22no—
288Q16—22no—
289Q16—22no—
290Q16—22no—
291Q16—41no—
292Q16—41no—
293Q16—41no—
294Q16—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8C4×D412yes—
308C2×C8C23×C411yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—161no—
333C22⋊C4—22no—
334C22⋊C4—41no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—82no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—82no—
343C42—82no—
344C42—82no—
345C42—82no—
346C42—161no—
347C16—22no—
348C16—22no—
349C16—41no—
350C16—41no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C4×D812yes—
358C23C22×D811yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—164no—
409C2×C4C4×D812yes—
410C2×C4#51012yes—
411C2×C4C2×C4×D411yescharacteristic
412C2×C4#54611yescharacteristic
413C2×C4#54611yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4#57711yescharacteristic
416C2×C4#57711yescharacteristic
417C8—22no—
418C8—41no—
419C8—41no—
420C8C4×#4111yescharacteristic
421C8#51811yescharacteristic
422C8#55311yescharacteristic
423C8#55311yescharacteristic
424C22—22no—
425C22—22no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22C2×C4×D811yescharacteristic
432C22#329411yescharacteristic
433C22#329411yescharacteristic
434C22C2×#51011yescharacteristic
435C22#341411yescharacteristic
436C22#341611yescharacteristic
437C22#341711yescharacteristic
438C4—22no—
439C4—22no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4—161no—
450C4—161no—
451C4#236411yescharacteristic
452C4#295211yescharacteristic
453C4#330111yescharacteristic
454C4#330111yescharacteristic
455C2—42no—
456C2#2312711yescharacteristic
457C2C4×#51011yescharacteristic
458C2#3345511yescharacteristic
459C2#3387811yescharacteristic
460C2#3387911yescharacteristic
461C2#3389411yescharacteristic
462C2#3389811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
4. #13150C2semidirect
5. #13148C2semidirect
6. #13111C2semidirect
7. #13111C2semidirect
8. #13015C2semidirect
9. #13015C2semidirect
12. #12638C2semidirect
13. C4×#358C2semidirect
26. #2219C4semidirect
71. C2×Q32C2×C4semidirect
123. #367C2×C4semidirect
128. #358C2×C4semidirect
129. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22192
#230262
#230242
#131502
#131482
#131112
#130152
#129942
#129932
#126382
C4×#3582
#126122
#125572