Finite Group Databasebeta

← #496826

Subgroups — #496826

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2186C211yesmaximal, characteristic
2#21524C211yesmaximal, characteristic, complemented
3#21230C211yesmaximal, characteristic, complemented
4#13238C211yesmaximal, characteristic, complemented
5#13235C211yesmaximal, characteristic, complemented
6#13149C211yesmaximal, characteristic, complemented
7#13148C211yesmaximal, characteristic, complemented
8#13133C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13014C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12991C211yesmaximal, characteristic, complemented
13#12919C211yesmaximal, characteristic, complemented
14#12733C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×D16C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C2×#417—21no—
23C2×#405C2211yescharacteristic
24#2324—21no—
25#2311—21no—
26#2186C418yescomplemented
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
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#1970C2211yescharacteristic
46#1969C2211yescharacteristic
47#1969C2211yescharacteristic
48#1942C2211yescharacteristic
49#1936—21no—
50#1895—21no—
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1762—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×C414yescomplemented
67C22×Q16C2311yescharacteristic
68C22×SD16—21no—
69C22×D8C2311yescharacteristic
70C22×D8C2311yescharacteristic
71C22×M16D411yescharacteristic
72C23×C8C2311yescharacteristic
73C2×#121—21no—
74C2×C4×D4—21no—
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C2×SD32—41no—
78C2×SD32—41no—
79C2×SD32C2×C414yescomplemented
80C2×D16—41no—
81C2×D16—41no—
82C2×D16C2×C414yescomplemented
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#466—22no—
88#466C2×C414yes—
89C4×D8—41no—
90C4×D8—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×#102—21no—
103C2×#102—21no—
104C2×#102—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#390—21no—
116#390—21no—
117#390—21no—
118#390—41no—
119#389—21no—
120#388—41no—
121#386—41no—
122#386—41no—
123#386—41no—
124#385—41no—
125#380—21no—
126#380—41no—
127#380—41no—
128C4×C4⋊C4—21no—
129#366—41no—
130#366—41no—
131#358—41no—
132#358—41no—
133#358C2×C414yescomplemented
134#357—41no—
135#357—41no—
136#357—41no—
137#357—41no—
138#357C2×C414yescomplemented
139C22⋊C16—22no—
140C22⋊C16C2×C414yescomplemented
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×D4—21no—
154C22×D4—21no—
155C22×D4—41no—
156C22×D4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161#135—44no—
162#135—44no—
163#135—44no—
164#135—44no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16C22×C412yes—
171C2×SD16—24no—
172C2×SD16—42no—
173C2×SD16—42no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8C22×C412yes—
185C2×D8C22×C412yes—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—41no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8C22×C412yes—
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8#4112yes—
198C22×C8C2411yescharacteristic
199#121—24no—
200#121—42no—
201C4×D4—42no—
202C4×D4—42no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
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⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4D811yescharacteristic
226C2×C22⋊C4D811yescharacteristic
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230SD32—44no—
231D16—44no—
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×C16—21no—
241C2×C16—21no—
242C2×C16C22×C412yes—
243C2×C16C22×C412yes—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—42no—
254#107C22×C412yes—
255#103—42no—
256#103—42no—
257#102—24no—
258#102—42no—
259#102—42no—
260#102—42no—
261#102—42no—
262#97—41no—
263#97—41no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—41no—
276C24—41no—
277C24C2×D811yescharacteristic
278#41—48no—
279#41—48no—
280#41—84no—
281#41—84no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—161no—
305C2×D4—161no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—44no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—161no—
322C22×C4—161no—
323C22×C4—161no—
324C22×C4—161no—
325C22×C4C4×D412yes—
326C22×C4C2×D811yescharacteristic
327C22×C4C2×D811yescharacteristic
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4#13512yes—
332C22×C4C22×D411yescharacteristic
333Q16—22no—
334Q16—22no—
335Q16—41no—
336Q16—41no—
337SD16—48no—
338D8—22no—
339D8—22no—
340D8—22no—
341D8—22no—
342D8—41no—
343D8—41no—
344D8—41no—
345D8—41no—
346M16—44no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C4×D412yes—
359C2×C8AGL(1,ℤ/8)12yes—
360C2×C8C23×C411yescharacteristic
361C2×C8C2×#4111yescharacteristic
362C2×C8H(5,𝔽2)11yescharacteristic
363C2×C8H(5,𝔽2)11yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C2×C8H(5,𝔽2)11yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—82no—
379C4⋊C4—161no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—44no—
383C22⋊C4—82no—
384C22⋊C4—161no—
385C22⋊C4—161no—
386C22⋊C4—161no—
387C22⋊C4—161no—
388C22⋊C4—161no—
389C22⋊C4—161no—
390C22⋊C4—161no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—161no—
396C16—22no—
397C16—22no—
398C16—41no—
399C16—41no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—42no—
407C23—42no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23C4×D812yes—
421C23C22×D811yescharacteristic
422Q8—42no—
423Q8—42no—
424Q8—81no—
425Q8—81no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—88no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—164no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—88no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—164no—
474C2×C4—164no—
475C2×C4C4×D812yes—
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#53411yescharacteristic
478C2×C4#53511yescharacteristic
479C2×C4C2×#13511yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
481C2×C4#57511yescharacteristic
482C8—22no—
483C8—22no—
484C8—41no—
485C8#51811yescharacteristic
486C8#51811yescharacteristic
487C8#53811yescharacteristic
488C8#53811yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22—161no—
510C22C2×C4×D811yescharacteristic
511C22#318611yescharacteristic
512C22#320511yescharacteristic
513C22#341511yescharacteristic
514C22#341511yescharacteristic
515C22#341611yescharacteristic
516C22#341711yescharacteristic
517C4—22no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—82no—
523C4—82no—
524C4—161no—
525C4—161no—
526C4—161no—
527C4—161no—
528C4#232111yescharacteristic
529C4C4×AGL(1,ℤ/8)11yescharacteristic
530C4#318811yescharacteristic
531C4#321111yescharacteristic
532C2—42no—
533C2—162no—
534C2—162no—
535C2#2204011yescharacteristic
536C2#3345411yescharacteristic
537C2#3345911yescharacteristic
538C2#3376411yescharacteristic
539C2#3376911yescharacteristic
540C2#3378611yescharacteristic
541C2#3379111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2186C2
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. #21524C2semidirect
3. #21230C2semidirect
4. #13238C2semidirect
5. #13235C2semidirect
6. #13149C2semidirect
7. #13148C2semidirect
8. #13133C2semidirect
9. #13130C2semidirect
10. #13014C2semidirect
11. #13012C2semidirect
12. #12991C2semidirect
13. #12919C2semidirect
14. #12733C2semidirect
15. #12600C2semidirect
26. #2186C4semidirect
66. C2×#135C2×C4semidirect
79. C2×SD32C2×C4semidirect
82. C2×D16C2×C4semidirect
133. #358C2×C4semidirect
138. #357C2×C4semidirect
140. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21862
#215242
#212302
#132382
#132352
#131492
#131482
#131332
#131302
#130142
#130122
#129912
#129192
#127332
#126002