Finite Group Databasebeta

← #498333

Subgroups — #498333

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33584C211yesmaximal, characteristic
2C2×AGL(1,ℤ/16)C211yesmaximal, characteristic
3C2×#2180C211yesmaximal, characteristic, complemented
4#23189C211yesmaximal, characteristic, complemented
5#22686C211yesmaximal, characteristic, complemented
6#13214C211yesmaximal, characteristic, complemented
7#13185C211yesmaximal, characteristic, complemented
8#13155C211yesmaximal, characteristic, complemented
9#13154C211yesmaximal, characteristic, complemented
10#13040C211yesmaximal, characteristic
11#13039C211yesmaximal, characteristic, complemented
12#12957C211yesmaximal, characteristic, complemented
13#12956C211yesmaximal, characteristic, complemented
14#12666C211yesmaximal, characteristic, complemented
15C4×#360C211yesmaximal, characteristic, complemented
16C2×#510C2211yescharacteristic
17C2×#509C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×#444C2211yescharacteristic
20C2×#444C2211yescharacteristic
21C2×#442C2211yescharacteristic
22C2×#441C2211yescharacteristic
23C4×#135C2211yescharacteristic
24#2906—21no—
25#2367—21no—
26#2211C412yescomplemented
27#2211C412yescomplemented
28#2210C412yescomplemented
29#2210C412yescomplemented
30AGL(1,ℤ/16)—24no—
31#2180—24no—
32C2×#365C2211yescharacteristic
33C2×#365C2211yescharacteristic
34#2150C2211yescharacteristic
35C2×#362C2211yescharacteristic
36C2×#362C2211yescharacteristic
37C2×#361C2211yescharacteristic
38C2×#361C2211yescharacteristic
39C2×#360C2211yescharacteristic
40C2×#360C2211yescharacteristic
41#2141C2211yescharacteristic
42#2140C2211yescharacteristic
43C2×#347C2211yescharacteristic
44C2×#347C2211yescharacteristic
45C4×M32C2211yescharacteristic
46#1983—21no—
47#1972C2211yescharacteristic
48#1971C2211yescharacteristic
49#1946C2211yescharacteristic
50#1946C2211yescharacteristic
51#1880—21no—
52#1877—21no—
53#1866—21no—
54#1866—21no—
55#1848C2211yescharacteristic
56#1806—21no—
57C4×#106C2211yescharacteristic
58#1770C2211yescharacteristic
59#1386C2211yescharacteristic
60#1386C2211yescharacteristic
61#1383C2211yescharacteristic
62#1383C2211yescharacteristic
63#1375C2211yescharacteristic
64#1375C2211yescharacteristic
65C2×#135C2311yescharacteristic
66C22×Q16C2311yescharacteristic
67C22×SD16—21no—
68C22×D8C2311yescharacteristic
69C2×#131—21no—
70C2×#131—21no—
71C2×#124—21no—
72C4×#41—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75#510—22no—
76#510—22no—
77#509—22no—
78#509—22no—
79C2×Q32C2×C412yescomplemented
80C2×SD32C2×C412yes—
81C2×SD32C2×C412yescomplemented
82C2×D16C2×C412yes—
83C2×M32C2311yescharacteristic
84C2×M32C2311yescharacteristic
85C2×M32C2311yescharacteristic
86C2×M32C2311yescharacteristic
87#495—21no—
88#495—21no—
89#495C2×C412yes—
90#495C2×C412yes—
91#444—22no—
92#444—22no—
93#444—41no—
94#444—41no—
95#442—22no—
96#442—41no—
97#441—22no—
98#441—41no—
99C4×Q16C2×C411yescharacteristic, complemented
100C4×Q16C2×C411yescharacteristic, complemented
101C4×SD16—21no—
102C4×SD16—21no—
103C4×D8C2×C411yescharacteristic, complemented
104C4×D8C2×C411yescharacteristic, complemented
105C2×#108—21no—
106C2×#108C2311yescharacteristic
107C2×#108C2311yescharacteristic
108#427C2311yescharacteristic
109C2×#106C2311yescharacteristic
110C2×#106C2311yescharacteristic
111#424—21no—
112#424—21no—
113#424—41no—
114#424—41no—
115#423—21no—
116#423—41no—
117C2×C4⋊C8—41no—
118C2×C4⋊C8D411yescharacteristic
119C2×#104—21no—
120C2×#104—21no—
121#416—21no—
122C2×#103—21no—
123C2×#103—21no—
124C2×#102—21no—
125C2×#102—21no—
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391—21no—
130#390—21no—
131#388—41no—
132#388—41no—
133#385—41no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D411yescharacteristic
136C4×C22⋊C4—41no—
137#365—22no—
138#365—22no—
139C4⋊C16C2×C412yescomplemented
140C4⋊C16C2×C412yescomplemented
141#362—22no—
142#362—41no—
143#361—22no—
144#361—41no—
145#360—22no—
146#360D414yes—
147#358C2×C412yescomplemented
148#358C2×C412yescomplemented
149#357C2×C412yes—
150#357C2×C412yes—
151#347—22no—
152#347—22no—
153#346—21no—
154C4×C16—21no—
155#340—21no—
156#339—21no—
157#339—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
160C2×#41—21no—
161C22×Q8—21no—
162C22×D4—21no—
163C23×C4—41no—
164#135—21no—
165#135—21no—
166#135C2×D412yes—
167#135#4112yes—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16C22×C412yes—
174C2×Q16C22×C411yescharacteristic
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—21no—
178C2×SD16—81no—
179C2×SD16—81no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8C22×C412yes—
186C2×D8C22×C411yescharacteristic
187#131—42no—
188#131—42no—
189#131—81no—
190#131—81no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—41no—
194C2×M16—41no—
195C22×C8—41no—
196C22×C8—41no—
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2411yescharacteristic
201#124—22no—
202#124—22no—
203#124—41no—
204#124—41no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—42no—
208C4×Q8—81no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—42no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—81no—
215#117—21no—
216#117—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232Q32—24no—
233SD32—24no—
234SD32—24no—
235D16—24no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—21no—
241M32—21no—
242M32—21no—
243M32C2×D412yes—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16C22×C412yes—
247C2×C16C22×C412yes—
248#108—21no—
249#108—21no—
250#108—21no—
251#108—21no—
252#108—81no—
253#107C22×C411yescharacteristic
254#107C22×C411yescharacteristic
255#106—21no—
256#106—21no—
257#106—21no—
258#106—21no—
259#106C2×D412yes—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8—41no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268#104—42no—
269#104—42no—
270#103—21no—
271#103—21no—
272#103—42no—
273#103—81no—
274#102—21no—
275#102—21no—
276#102—42no—
277#102—81no—
278#97—21no—
279#97—21no—
280#97—21no—
281#97—21no—
282C4×C8C22×C411yescharacteristic
283C4×C8C22×C411yescharacteristic
284C4×C8#4111yescharacteristic
285C4×C8#4111yescharacteristic
286#95—21no—
287#95—41no—
288#95—41no—
289#95—41no—
290C24—41no—
291#41—22no—
292#41—22no—
293#41—41no—
294#41—41no—
295C2×Q8—21no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—161no—
321C22×C4AGL(1,ℤ/8)11yescharacteristic
322C22×C4AGL(1,ℤ/8)11yescharacteristic
323C22×C4C22×D411yescharacteristic
324Q16—22no—
325Q16—21no—
326Q16—22no—
327Q16—41no—
328Q16—41no—
329Q16C4×D412yes—
330SD16—22no—
331SD16—22no—
332SD16—82no—
333SD16—82no—
334D8—22no—
335D8—21no—
336D8—22no—
337D8—41no—
338D8—41no—
339D8C4×D412yes—
340M16—41no—
341M16—41no—
342M16—41no—
343M16—41no—
344M16—82no—
345M16—82no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—82no—
357C2×C8—82no—
358C2×C8C4×D411yescharacteristic
359C2×C8C4×D411yescharacteristic
360C2×C8C23×C411yescharacteristic
361C2×C8C22×D411yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C2×C82−1+411yescharacteristic
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—81no—
375C4⋊C4—82no—
376C22⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—161no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—82no—
394C42—161no—
395C42C4×D411yescharacteristic
396C42C4×D411yescharacteristic
397C42AGL(1,ℤ/8)11yescharacteristic
398C42AGL(1,ℤ/8)11yescharacteristic
399C16—22no—
400C16—22no—
401C16—22no—
402C16—41no—
403C23—41no—
404C23—42no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23C2×AGL(1,ℤ/8)11yescharacteristic
412Q8—22no—
413Q8—41no—
414Q8—42no—
415Q8—42no—
416Q8—81no—
417Q8—81no—
418D4—22no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—41no—
423D4—42no—
424D4—81no—
425D4—81no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—161no—
450C2×C4—164no—
451C2×C4#44211yescharacteristic
452C2×C4#44211yescharacteristic
453C2×C4#44211yescharacteristic
454C2×C4#44211yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4D4211yescharacteristic
457C2×C4#54811yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4#57611yescharacteristic
460C2×C4#57711yescharacteristic
461C8—21no—
462C8—21no—
463C8—41no—
464C8—41no—
465C8—81no—
466C8—81no—
467C8C2×C4×D411yescharacteristic
468C8#52011yescharacteristic
469C8#54711yescharacteristic
470C8#55311yescharacteristic
471C22—21no—
472C22—21no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—161no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22C2×#44211yescharacteristic
484C22#327211yescharacteristic
485C22#336611yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—82no—
494C4—81no—
495C4—81no—
496C4—82no—
497C4—161no—
498C4#236111yescharacteristic
499C4#294611yescharacteristic
500C4#327411yescharacteristic
501C4#337311yescharacteristic
502C2—21no—
503C2—21no—
504C2—82no—
505C2—162no—
506C2#2300111yescharacteristic
507C2#3346811yescharacteristic
508C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#509C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2180C2semidirect
4. #23189C2semidirect
5. #22686C2semidirect
6. #13214C2semidirect
7. #13185C2semidirect
8. #13155C2semidirect
9. #13154C2semidirect
11. #13039C2semidirect
12. #12957C2semidirect
13. #12956C2semidirect
14. #12666C2semidirect
15. C4×#360C2semidirect
26. #2211C4semidirect
27. #2211C4semidirect
28. #2210C4semidirect
29. #2210C4semidirect
79. C2×Q32C2×C4semidirect
81. C2×SD32C2×C4semidirect
99. C4×Q16C2×C4semidirect
100. C4×Q16C2×C4semidirect
103. C4×D8C2×C4semidirect
104. C4×D8C2×C4semidirect
139. C4⋊C16C2×C4semidirect
140. C4⋊C16C2×C4semidirect
147. #358C2×C4semidirect
148. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
#335842
C2×AGL(1,ℤ/16)2
C2×#21802
#231892
#226862
#132142
#131852
#131552
#131542
#130402
#130392
#129572
#129562
#126662
C4×#3602