Finite Group Databasebeta

← #498351

Subgroups — #498351

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C211yesmaximal, characteristic
2C2×C4×SD32C211yesmaximal, characteristic, complemented
3C22×#367C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic
5#23024C211yesmaximal, characteristic, complemented
6#13237C211yesmaximal, characteristic, complemented
7#13166C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#12993C211yesmaximal, characteristic, complemented
11#12991C211yesmaximal, characteristic
12#12938C211yesmaximal, characteristic, complemented
13#12937C211yesmaximal, characteristic, complemented
14C4×#367C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24#2361—21no—
25#2360—21no—
26#2217C418yescomplemented
27C4×SD32—24no—
28C2×#367—24no—
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×#367C2211yescharacteristic
33C2×#367C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1928—21no—
48#1925—21no—
49#1874—21no—
50#1872—21no—
51#1863—21no—
52#1862—21no—
53C4×#107C2211yescharacteristic
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×C4×D4—41no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32C2×C414yescomplemented
75C22×C16—22no—
76C22×C16—22no—
77C22×C16C2×C414yescomplemented
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#467—22no—
83#467C2×C414yescomplemented
84#466—22no—
85#466C2×C414yes—
86C4×Q16—22no—
87C4×Q16—41no—
88C4×D8—22no—
89C4×D8—41no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—41no—
95C2×#107—41no—
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#389—41no—
116#388—41no—
117#387—41no—
118#386—41no—
119#385—41no—
120#385—41no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C22⋊C4—41no—
124C4×C22⋊C4D411yescharacteristic
125#367—22no—
126#367—22no—
127#367—22no—
128#367C2×C414yescomplemented
129#367D414yescomplemented
130#358—22no—
131#358—41no—
132#358C2×C414yescomplemented
133#357—22no—
134#357—41no—
135#357C2×C414yescomplemented
136C22⋊C16—22no—
137C4×C16—22no—
138C4×C16—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4C2×D411yescharacteristic
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16C22×C412yes—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8C22×C412yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C22×C412yes—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8#4112yes—
173C22×C8C2411yescharacteristic
174C22⋊Q8—24no—
175C22⋊Q8—42no—
176#121—24no—
177#121—42no—
178C4×Q8—42no—
179C4×Q8—81no—
180C4×D4—42no—
181C4×D4—81no—
182C4×D4—84no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4D811yescharacteristic
205C2×C22⋊C4D811yescharacteristic
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211SD32—28no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—28no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16C22×C412yes—
222C2×C16C22×C412yes—
223C2×C16C2×D412yes—
224C2×C16#4112yes—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—42no—
234#107—42no—
235#107C22×C412yes—
236#107C22×C412yes—
237#107C2×D412yes—
238#107C2×D412yes—
239#103—24no—
240#103—42no—
241#103—81no—
242#102—24no—
243#102—42no—
244#102—81no—
245C22⋊C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95D811yescharacteristic
257#95D811yescharacteristic
258C24—41no—
259C24C2×D811yescharacteristic
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—22no—
267C2×D4—44no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—82no—
274C2×D4—82no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—161no—
293C22×C4—161no—
294C22×C4—161no—
295C22×C4C4×D412yes—
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C2×D811yescharacteristic
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4#13512yes—
302C22×C4C22×D411yescharacteristic
303Q16—22no—
304Q16—22no—
305Q16—41no—
306Q16—41no—
307D8—22no—
308D8—22no—
309D8—41no—
310D8—41no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8C4×D412yes—
324C2×C8C23×C411yescharacteristic
325C2×C8C22×D411yescharacteristic
326C2×C8C2×#4111yescharacteristic
327C2×C8C2×#4111yescharacteristic
328C2×C8H(5,𝔽2)11yescharacteristic
329C2×C82−1+411yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—44no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—161no—
348C22⋊C4—22no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—161no—
354C22⋊C4SD3214yescomplemented
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—82no—
360C42—82no—
361C42—161no—
362C16—22no—
363C16—41no—
364C16—41no—
365C16C4×D414yescomplemented
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23C4×D812yes—
378C23C2×SD3212yes—
379C23#50812yes—
380C23C22×D811yescharacteristic
381Q8—42no—
382Q8—42no—
383Q8—81no—
384Q8—81no—
385D4—42no—
386D4—42no—
387D4—88no—
388D4—81no—
389D4—81no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—164no—
428C2×C4—164no—
429C2×C4—164no—
430C2×C4C4×D812yes—
431C2×C4C2×SD3212yes—
432C2×C4C2×SD3212yes—
433C2×C4C2×C4×D411yescharacteristic
434C2×C4D4211yescharacteristic
435C2×C4#54711yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57811yescharacteristic
439C8—22no—
440C8—41no—
441C8—41no—
442C8C2×C4×D411yescharacteristic
443C8#52011yescharacteristic
444C8#54811yescharacteristic
445C8#55311yescharacteristic
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22C4×SD3214yescomplemented
460C22C2×C4×D811yescharacteristic
461C22D4×D811yescharacteristic
462C22#333411yescharacteristic
463C22C22×SD3211yescharacteristic
464C22C2×#50811yescharacteristic
465C22#341611yescharacteristic
466C22#341611yescharacteristic
467C4—22no—
468C4—22no—
469C4—22no—
470C4—41no—
471C4—41no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—161no—
478C4—161no—
479C4#236111yescharacteristic
480C4#295111yescharacteristic
481C4#328511yescharacteristic
482C4#333911yescharacteristic
483C2—24no—
484C2—162no—
485C2#2302511yescharacteristic
486C2C2×C4×SD3211yescharacteristic
487C2#3346011yescharacteristic
488C2D4×SD3211yescharacteristic
489C2#3384811yescharacteristic
490C2#3390211yescharacteristic
491C2#3390411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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×C4×SD32C2semidirect
3. C22×#367C2semidirect
5. #23024C2semidirect
6. #13237C2semidirect
7. #13166C2semidirect
8. #13132C2semidirect
9. #13131C2semidirect
10. #12993C2semidirect
12. #12938C2semidirect
13. #12937C2semidirect
14. C4×#367C2semidirect
15. C16×C22⋊C4C2semidirect
26. #2217C4semidirect
74. C2×SD32C2×C4semidirect
77. C22×C16C2×C4semidirect
83. #467C2×C4semidirect
128. #367C2×C4semidirect
129. #367D4semidirect
132. #358C2×C4semidirect
135. #357C2×C4semidirect
354. C22⋊C4SD32semidirect
365. C16C4×D4semidirect
459. C22C4×SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
C2×C4×SD322
C22×#3672
#230252
#230242
#132372
#131662
#131322
#131312
#129932
#129912
#129382
#129372
C4×#3672
C16×C22⋊C42