Finite Group Databasebeta

← #496830

Subgroups — #496830

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2189C211yesmaximal, characteristic
2#21525C211yesmaximal, characteristic, complemented
3#21230C211yesmaximal, characteristic
4#13240C211yesmaximal, characteristic, complemented
5#13237C211yesmaximal, characteristic, complemented
6#13149C211yesmaximal, characteristic, complemented
7#13148C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13015C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#12993C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14#12733C211yesmaximal, characteristic
15#12600C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×Q32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C2×#417—21no—
23C2×#405C2211yescharacteristic
24#2323—21no—
25#2311—21no—
26#2189C418yescomplemented
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#1973—21no—
42#1973—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic
46#1969C2211yescharacteristic
47#1969C2211yescharacteristic
48#1942C2211yescharacteristic
49#1935—21no—
50#1894—21no—
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1763—21no—
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×#41—21no—
61C2×#135—22no—
62C2×#135—41no—
63C2×#135—41no—
64C2×#135—41no—
65C2×#135—41no—
66C2×#135C2×C414yes—
67C22×Q16C2311yescharacteristic
68C22×Q16C2311yescharacteristic
69C22×SD16—21no—
70C22×D8C2311yescharacteristic
71C22×M16D411yescharacteristic
72C23×C8C2311yescharacteristic
73C2×C22⋊Q8—21no—
74C2×C4×Q8—21no—
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C2×Q32—41no—
78C2×Q32—41no—
79C2×Q32C2×C414yescomplemented
80C2×SD32—41no—
81C2×SD32—41no—
82C2×SD32C2×C414yescomplemented
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#467—22no—
88#467C2×C414yes—
89C4×Q16—41no—
90C4×Q16—41no—
91C2×#107—21no—
92C2×#107—41no—
93C2×#107—41no—
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99#417—44no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107#405—41no—
108#405—41no—
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#391—21no—
115#391—21no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—41no—
120#388—41no—
121#388—41no—
122#387—41no—
123#386—41no—
124#386—41no—
125#380—21no—
126#380—41no—
127#380—41no—
128C4×C4⋊C4—21no—
129#367—41no—
130#367—41no—
131#358—41no—
132#358—41no—
133#358—41no—
134#358—41no—
135#358C2×C414yescomplemented
136#357—41no—
137#357—41no—
138#357C2×C414yescomplemented
139C22⋊C16—22no—
140C22⋊C16C2×C414yes—
141#346—41no—
142C4×C16—41no—
143#340—21no—
144#340—21no—
145#340—21no—
146C2×#41—24no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—42no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159#135—44no—
160#135—44no—
161#135—44no—
162#135—44no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16C22×C412yes—
174C2×Q16C22×C412yes—
175C2×SD16—24no—
176C2×SD16—42no—
177C2×SD16—42no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8C22×C412yes—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—41no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8C22×C412yes—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8#4112yes—
196C22×C8C2411yescharacteristic
197C22⋊Q8—24no—
198C22⋊Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201#117—41no—
202#117—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4D811yescharacteristic
215C2×C4⋊C4D811yescharacteristic
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4D811yescharacteristic
223C2×C22⋊C4D811yescharacteristic
224C2×C42—41no—
225C2×C42—41no—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228Q32—44no—
229SD32—44no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16C22×C412yes—
241C2×C16C22×C412yes—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—42no—
252#107C22×C412yes—
253#103—24no—
254#103—42no—
255#103—42no—
256#103—42no—
257#103—42no—
258#102—42no—
259#102—42no—
260#97—41no—
261#97—41no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—41no—
276C24C2×D811yescharacteristic
277#41—48no—
278#41—48no—
279#41—84no—
280#41—84no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×D4—22no—
294C2×D4—44no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—82no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—161no—
318C22×C4—161no—
319C22×C4C4×D412yes—
320C22×C4C2×D811yescharacteristic
321C22×C4C2×D811yescharacteristic
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4#13512yes—
326C22×C4C22×D411yescharacteristic
327Q16—22no—
328Q16—22no—
329Q16—22no—
330Q16—22no—
331Q16—41no—
332Q16—41no—
333Q16—41no—
334Q16—41no—
335SD16—48no—
336D8—22no—
337D8—22no—
338D8—41no—
339D8—41no—
340M16—44no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8C4×D412yes—
353C2×C8AGL(1,ℤ/8)12yes—
354C2×C8C23×C411yescharacteristic
355C2×C8C2×#4111yescharacteristic
356C2×C8H(5,𝔽2)11yescharacteristic
357C2×C8H(5,𝔽2)11yescharacteristic
358C2×C8H(5,𝔽2)11yescharacteristic
359C2×C8H(5,𝔽2)11yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—161no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—44no—
379C22⋊C4—161no—
380C22⋊C4—161no—
381C22⋊C4—161no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—82no—
389C42—161no—
390C16—22no—
391C16—22no—
392C16—41no—
393C16—41no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23C4×D812yes—
408C23C22×D811yescharacteristic
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—81no—
414Q8—81no—
415Q8—81no—
416Q8—81no—
417D4—42no—
418D4—42no—
419D4—88no—
420D4—81no—
421D4—81no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—88no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—164no—
463C2×C4C4×D812yes—
464C2×C4C2×C4×D411yescharacteristic
465C2×C4#53411yescharacteristic
466C2×C4#53511yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
469C2×C4#57511yescharacteristic
470C8—22no—
471C8—22no—
472C8—41no—
473C8#51811yescharacteristic
474C8#51811yescharacteristic
475C8#53811yescharacteristic
476C8#53811yescharacteristic
477C22—22no—
478C22—22no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—161no—
488C22—161no—
489C22—161no—
490C22—161no—
491C22C2×C4×D811yescharacteristic
492C22#318611yescharacteristic
493C22#320511yescharacteristic
494C22#341611yescharacteristic
495C22#341611yescharacteristic
496C22#341611yescharacteristic
497C22#341711yescharacteristic
498C4—22no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—161no—
507C4—161no—
508C4—161no—
509C4—161no—
510C4#232111yescharacteristic
511C4C4×AGL(1,ℤ/8)11yescharacteristic
512C4#318811yescharacteristic
513C4#321111yescharacteristic
514C2—42no—
515C2—162no—
516C2#2204011yescharacteristic
517C2#3345511yescharacteristic
518C2#3346011yescharacteristic
519C2#3376711yescharacteristic
520C2#3377011yescharacteristic
521C2#3378811yescharacteristic
522C2#3379111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21525C2semidirect
4. #13240C2semidirect
5. #13237C2semidirect
6. #13149C2semidirect
7. #13148C2semidirect
8. #13132C2semidirect
9. #13131C2semidirect
10. #13015C2semidirect
11. #13013C2semidirect
12. #12993C2semidirect
13. #12920C2semidirect
15. #12600C2semidirect
26. #2189C4semidirect
79. C2×Q32C2×C4semidirect
82. C2×SD32C2×C4semidirect
135. #358C2×C4semidirect
138. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21892
#215252
#212302
#132402
#132372
#131492
#131482
#131322
#131312
#130152
#130132
#129932
#129202
#127332
#126002